Chad Peleshek the city planning and development director and I want to take everybody for joining us today on this important release of the information on an affordable housing study that the city commissioned back started this process back in October. A few housekeeping things that I just want to share with the group is that if you do have comments through this thing we would be happy to try to answer them so Becky Benz is joining us from MSA and she is the consultant that worked on the study and is going to do the presentation on the results. But if there is questions that arise throughout the study I would encourage you to use the chat function and put your comments in the chat and we will try to answer the questions as we go along. I mean at the end of the presentation I should say and if we don't get to all the questions we will put them into a question and answer format and get the answers to you and make it available on the city's website in the next day or so. So I would encourage you if you have to check out early and you want no more information to visit the city's website at sheboyganwi.gov for the information on it. So just a little bit background of how we got here the city has built a number there has been a number of new apartments that have been built in this in the community particularly in the downtown in the past 5 to 6 years and we wondered for a long time what the affordable housing demands were both from the single family side of things as well as the multi-family so we went out on the request for proposals early last fall interviewed a few firms and ended up choosing MSA to help put this together MSA as a consulting firm out of Madison and Becky men's is joining us today and she will give us the report and like I said if you've got any questions please use the chat function on the go to meeting and with that I will turn it over to Becky. All right thanks Chad and thanks everyone for being here and finding out a little bit more about what's been going on with the sheboygan housing study and hopefully asking some questions too. So today we're going to go through the findings and also the recommendations and strategies that we found as we work through the studies. Generally what we're going to go through is just a little bit of background on the process and the data that we used and we'll talk a little bit about findings of the housing market in general and then we'll further break those findings down between the renter and owner market. After that we'll follow up with some demand projections looking at numbers of units by price point general locations. So those recommendations and then finally we'll finish up with a whole toolbox of strategies for how the city can help get those units implemented. So study process. So this whole thing really began with a big data dump and us sifting through a lot of different data sources. If you look at the report there it's very long there are a lot of numbers in it. So that was really the first step for us. Our next step was to meet with staff a few times throughout the process to look at the data we were finding confirm that these were things the city was also seeing talk to the city a little bit about their concerns and what they were worried about with the housing market and then review findings and recommendations with them as well. We did interviews with those we view as local housing experts to better understand the housing markets and its shortcoming. So these interviews were with realtors lenders builders landlords and nonprofits primarily. We also did a community survey to try and better understand what residents were experiencing and the issues that they were having with housing in the city and we had over a 100 or sorry 850 responses to the community survey. We are doing a public review right now. Council viewed a similar presentation last week. After this presentation hopefully council will be approving the study in the next couple of weeks that they're next meeting. So all the different data that we used a lot of it came primarily from the American community survey data from the U.S. U.S. Census Bureau. We also use quite a bit of data from the hood conference of housing affordability strategy. And that data is primarily derived from a CS data but it just allows us to filter that data even further by affordability measures which is really helpful. We also got access through local realtor to the multiple listing service to get our pulse on what was going on with the real estate market primarily the single family condo and duplex duplex markets in the city. And then also that community insight piece that's a really important part of all the housing studies we do because data tells us a lot. But often we need to truth that data with what people are actually seeing. So again that's those experts we talked to the residents and then staff input as well. So general findings. I think population is a good place to start. So over the past 10 years the city has experienced a population decrease while surrounding communities policy increases in population similar to the county. And if we look out over the next 20 years to see what might happen. So the population projections we're using are all based on Department of Administration population projections and those are based on status quo. So I think these estimates were done in like projections in 2013. So they're a bit dated but they're based on what was happening at the time. So looking out over the next 20 years DOA is projecting the city to have a decrease in population of 3% and the county's population is projected to increase by another 1.4% during this 20-year time frame. So it isn't all doom and gloom for the city. I think this just shows that there is an opportunity here. So with the right types of housing with the right types of amenities all the revitalization you guys are doing downtown and making the city a really attractive place for people. I think that's going to go a long way to capture some of the anticipated regional growth. So hopefully this housing study is part of getting you some of that growth to happen. Age cohort projections are another important piece of the puzzle. So the DOA also projects by age but they only do this at the county level. However, we did apply the general percentage increases to the city. But if we're looking strictly at the county numbers, the line I want to draw your attention to is the blue line which is those aged 85 plus. So DOA is projecting that this age cohort is going to increase by 112% over the next 20 years. So this is about 3000 additional seniors in this age group. So there's going to be a growing senior population in the county and also are anticipating the same thing at the city level. So the city does have a lack of senior living options. So with this increase more units are going to be needed for seniors both assisted and independent living facilities. So this is going to provide an additional type of housing that some seniors are looking for. And it's also going to have the added benefit of putting additional usually single family more affordable older type housing on the market for let's say first-time home buyers or those who are looking to buy at a lower price point. So it's going to have a double double benefit for the city. We also looked at incomes. That's really the most important determining factor in how much a household can pay for housing. So if we look at incomes in the city, the median household income and per capita incomes are relatively low compared to surrounding communities and to surrounding counties. So that's just telling us hey generally price points are probably going to need to be lower in the city. There's less money to spend on housing. So that's something important to note. We also looked at what employment growth is going to be like. We know there's a really strong manufacturing base in the city and in the county and surrounding area. So what we looked at is so the Bay Area Workforce Development Board, they're a nonprofit created to address local economic workforce issues. And what they do is they project employment growth for the 11 county northeastern Wisconsin region. So we looked at their numbers to see what they were projecting for employment growth. And what they're looking at over the next 10-ish years in the northeastern region is about a 4% growth in employment. And this is really consistent with what we were hearing when we talked to the county to SCEDC. And when we talked to the local employers that we interviewed, they all felt yes, we are hiring. We are anticipating hiring in the future. Outlook is really good. So this next slide, this is taking those projections that the Workforce Development Board did. And this is looking at the most in-demand occupations for the region and what those salaries are going to be on average. So I calculated a weighted average salary for these most in-demand occupations for the Bay Area region. And we're looking at a weighted average of about 50,000, which is about the median household income in the city. But generally these wages range between 20,000 and 100,000. So there's a really big difference and those are really different affordability levels that need to be considered. Other things of note, so over half of the homes in Sheboygan are detached single family homes. This includes both owner and renter occupied units. 80% of the homes in Sheboygan were built prior to 1980. So it's generally an older housing stock that we're seeing in the community. And this is typical in a lot of Wisconsin communities are generally older types of housing. And while age isn't a direct measure of condition, we often use it as an indicator. So older homes tend to have poor energy performance, higher maintenance costs, and sometimes maintenance and upkeep might even be impossible for some buyers. So that's an area of concern. And age of housing, quality of housing is something that came up in the interviews we did and also throughout the survey responses. Another point to note is just the development of housing in the city over the past six years. So on the right, we have some permit data on the topics for single family and duplex duplex units on the bottom for multi family units for both apartments and for condos. You'll see the development of single family homes has remained relatively low. And this has really been the case in the city since about 2000. Things have just really slowed down. But however, we saw in the past six years, as Chad mentioned, there's been a big increase in the number of apartments and multifamily buildings being built. So a big factor in this is that the city doesn't have a lot of greenfield space for new subdivision development. So it's really hard to get subdivisions with all those single family homes built. A lot of what the city is doing is infill development, which generally lends itself more towards the multi unit types of uses. So that's that's just one big reason that that has been happening. Diving a little bit deeper into the rental market. So just an under 40% of the housing stock in the city is rental units. Again, generally older age housing stock. So it's not really surprising that over over half of the rental units in the city are considered affordable for a household that's earning the median income. Although affordability generally seems good, there is still about a third of the renter population. Who is cost burdened? And when I say cost burden, this means that these households are paying more than 30% of their gross income towards towards rent. So I want to draw your attention to the table on top. So this rental unit mismatch table. So in this table, we break down by income levels. So I'm just going to walk through the first line of the table just to tell you what's going on. So we're looking at zero to 30% AMI, which is zero to 30% of the area median income. So this is the very lowest income population group in the community. So what we're doing in this table is we're breaking down the number of rental units who are technically affordable to households at those income levels. That's the second column. The third column is the number of renter households who are actually at that income level. And then the fourth column calculates the difference between the two. So we're seeing either an under supplier over supply of units. What this table is showing us that generally there's an over supply of units that are available at lower price points, which again isn't surprising due to the age of housing in the city. And generally an under supply of higher price units, which we know units haven't been been built as often recently. So we're expecting to see that this isn't table is not telling us that we need 3000 higher price units, but generally showing us what exists today and where there might be an opportunity for more units. The figure at the bottom, the rental unit consumption is showing us actually who's consuming the rental units by the by the different price points. So I'm going to draw your attention to the bottom column. This is for the rental units that are affordable at 30% AMI. So who's actually living in those units? So 38% of the households living in those units are at the 30% AMI income level. The 60% of people are at higher income levels and could technically afford to pay more. Now that's not saying that they would choose to pay more, but they could technically afford to pay more. So what this this figure is showing us is that households have generally been renting down in the market. And again, reinforcing that idea that there might be some room for higher priced units, which would free up some of the more affordable units to households who actually need units at those at those price levels. We also looked at rental unit vacancy. So a healthy vacancy rate is between 5% and 7%. This generally will signal a healthy balance between supply and demand. So there are enough units available to renters that they have a choice and provides a downward pressure on rent. And this also serves an incentive for landlords to be competitive with what they're offering and at what price. So the city's vacancy rate is at 3.3%, which is a little bit lower than desirable. So there are new units needed to bring vacancy rates up to a healthy level. Another interesting statistic is the cap rate. So the cap rate is the net operating income divided by the purchase price for property and really gives you a measure of risk or a measure of risk for an investor. And when we looked at the cap rate for multifamily within the city of Sheboygan in 2020, it was 7.4% which is the lowest the rate has been in the past 10 years. So this is really telling us that maybe multifamily housing isn't as attractive to developers and investors right now as it has been and it might be a little more tricky to get this type of housing built. The third point I wanted to make is just the point of construction costs being really high right now material costs are really high right now. So in building new units, especially affordable housing unit subsidies are needed to offset costs and ensure that units are affordable for lower income households. So this this slide is showing just a typical example of what it costs. This is for one bedroom rental construction. This includes the cost of land. So what we're seeing on average is the cost of about $130,000 per unit. And if you do the numbers and figure out what the break even monthly rent is on this type of unit, it's $1300. And next slide I'm going to show you. So this is HUD different income levels again, these are those zero to 30% 30 to 50% AMI those different income levels I was talking about a couple of slides ago. And these these squares are all showing what an affordable rent would be at this income level, based on number of people in the family. So if you look at that, I've basically crossed out all the income levels and family sizes for what this type of unit would not be affordable for. So there are very few low income families who could afford a new one bedroom rental construction, again reinforcing the need for some sort of subsidy to make it make it happen and make it affordable. Moving on to the ownership market. So median home value in Sheboygan is relatively low compared to surrounding communities. Again, this is just a reflection of the older housing stock. But with the lower price point, and it's also not surprising to find that most current owners can comfortably afford their homes. So only 18% of owner households in Sheboygan are cost burdened or paying more than 30% of their income towards their mortgage. And if we look at those households who are cost burdened, it's primarily households at the lowest income level at that 50% of the area median income or below. It isn't a typical to see a smaller percentage of owner households who are cost burdened as compared to renters, because there is that barrier to entry for owning a home. You need to have a savings. You need to have a good paying job, credit score, all those types of things. So that generally is a barrier to some lower income families. So not surprising there. We also did the same kind of ownership unit mismatch. So we broke down by the different income levels, how many homes are available at that price point, and how many owner households are at those income levels. And we're seeing something very similar to what we saw in the renter market again, due to the age of housing. So we generally have an oversupply of lower valued homes and an undersupply of higher valued homes. Again, not saying we need 6000 new higher priced homes, but just that there might be a room room for some newer newer units. And we also did the ownership unit consumption breakdown to see who's actually living in homes by different affordability levels. Again, I'm going to use the affordable at 50% AMI the lowest bar on this graph. So for units that are affordable at that level. 20% of households consuming those units are at the 50% AMI or below level. 80% of households consuming those types of homes are at higher income levels. And again, could technically afford to pay more not that they would, but they could. So again, the same kind of idea. There's there's a market for higher price housing. If we could get some of those households who could technically afford to pay more into more expensive housing that might free up some of the housing that's most affordable for the lowest income families in the city. Here's that MLS data. I talked about the real estate data. So we looked at single family statistics and generally probably not surprising to anyone. Supply of single family homes is really tight. So this is led to an increase in sales prices. So over the past 6 years, sale prices have increased 52% for single family homes. Average days on market decreased from 77 to 29. So very, very tight supply definitely room for more. We looked at the same statistics for condos and duplex homes. And we're seeing similar we increasing demand, though not to the same level as single family homes. So median sale price for duplexes increased 75% number of sales increased 27% over a 6 year period and days on market decreased from 98 to 48. For condos, median sale price increased 31% over 6 years. Number of sales went up 45% average days on market dropped from 90 to 72. Demand projections. So this is where we're going to get into the numbers of units. I'm going to walk through just kind of generally what went into the calculation. But I'm not going to do this for every slide. So first of all, we broke down the number of units by owner occupied units versus renter occupied units. So these first slides are owner occupied units. And we did two different estimates. So we did a conservative estimate. So using that DOA population projection for the city. And then the higher estimate is using the percentage increase projected for the county, but just applying that to the city. So owner occupied conservative estimate what went into it. So we incorporated projected demand from new households within the city. And we're specifically looking at households who are under age 65. We have separate calculations for senior units. So new households, we also took into account demand from existing residents who are likely to move. ACS has a nice number you can use for the annual turnover within the city. So we also looked at that. And then we broke the projections down into single family detached versus single family attached. And the percentage breakdown we used is 75 detached 25 attached. And when I say attached, I really mean like town homes, duplex types units that so keep that kind of thing in mind. So we broke it down by that. And then we also included what it's going to take to get the city to a healthy vacancy level for owner units. So I didn't talk about it in earlier, but the vacancy unit for ownership units is 1.1% in the city. Healthy rate is 2%. So we included some additional units to bring that that vacancy rate up. So we crunched the numbers and the total single family detached need. This is for the next 10 years. We're anticipating 244 units. And for attached units over the next 10 years 81 units. So in total 326 units or 33 owner occupied units per year. This is the high demand projection for owner occupied units. So I'll just do total we're looking at 750 units total or 72 units per year for owner occupied. The next two sides are for renter occupied units. So I guess the only things are different about these calculations. So we did the new households. We looked at demand from existing households, but we also took into account we adjusted the recommendations down by the number of units that the city currently has under construction, which is 399 units. So we did a reduction there. And then we took those units and we broke the recommendations into three different price categories. So we have affordable units, mid-level units and high market units. If you look in the report, those are all tied to different AMI levels slash price levels. So we broke it down by that. And then we also added in the number of units that it was going to take to bring the city from the 3% vacancy up to 5% vacancy. So just overall the number of rental units that we're projecting the conservative estimate is 401 units over the next 10 years or 40 units per year. And then the high demand projection is upwards of 1,000 units or 100 per year. These last two slides are geared specifically for seniors. We broke out the 65 plus population. So this first slide is looking at independent living options. So we projected the number of senior households age 65 plus over the next 10 years. And the slide is specifically looking at the percentage of households who do not need assistance with daily living. So we calculated total market potential. And then we further broke down the projected units into subsidized units and market rate units based on current income levels for those 65 plus in the city. And the thing to note about these projections is there's not an official count available for the number of independent living units in the city right now. So this 1,000 subsidized 284 market rate units. This is total number of units needed in the city by 2030. It's not taking into account any existing units. And then on the flip side, we also looked at seniors who would need assistance with daily living and looked at the number of assisted living units that the city would likely need over the next 10 years. So this number does take into account existing units. The state has a nice count of assisting living nursing home units. So we calculated that there would be a need for an additional 196 assisted living units over the next 10 years. Part of our recommendations was looking at specific types of units and locations in the city. So the first type of unit that we're recommending a focus on is condos. So the benefit of condos that there is that they're attached ownership units, they require less land and therefore less costs. So they're generally a more affordable home ownership option, which is a good fit for those with lower incomes who maybe don't have what they need to afford a detached single family home. So that's a great ownership opportunity there. Also the senior housing piece. So for those seniors who are looking for independent or assisted living units, there aren't a lot in the city right now. So building more of those, it's going to build the need for the seniors. And it's also going to like I talked about before, free up some additional affordable owner occupied units, which will be great for starter homes or those with lower incomes. So that's kind of a double whammy there. The third recommendation we had is for missing middle housing. There's a graphic below on the side to show you kind of what we're looking at. So these are the 2 to 16 unit attached and they can be either condo or rental units. But these types of units have the benefit of addressing affordability. And they're also compatible with a lot of different neighborhoods. You still want to be mindful of where you're placing them and making sure it makes sense. But they're they're generally compatible in a lot of different locations. The fourth type of unit that we're recommending a focus on is 3 plus bedroom rental units. This is something we've heard in interviews in Sheboygan and we've heard this in other communities we've talked to as well. It's it's hard to house lower income families. And those units that are available. There's generally I mean just due to the age of concern about the quality of housing and just they're not being many options. So families don't really have a choice. So that would be the last type of unit that we recommend to focus on. And then finally just a location specific recommendation to focus on infill development. So we know the city has limited greenfield growth opportunities so placing a high priority on infill especially downtown especially along certain corridors and the unit types that that we recommended a high priority focus on really lend themselves well to infill development. So the condos the senior housing really that attached type of housing that works well as infill. We also included a map for target areas for affordable housing. So the lighter areas on this map are the higher priority areas for affordable housing. And this map was based on so there's a state and federal program low income housing tax credit which is the most common type the developers used to help fund affordable housing projects. And it can there's a competitive part of to that program. And so developers can score higher by meeting certain criteria and so we took some of those criteria and put them on a map. So some of those criteria are on a long bus routes near parks schools within tax increment districts. So we highlighted those and put those into a map for the city to use and also developers to use to highlight those areas that might score higher when going for funding for affordable housing. Now the strategy that I really get into the meat of what what it's going to take to to make this happen. So the first strategy I want to recommend and the first thing I would suggest to the city to do is to establish a housing committee. So this committee should be in charge of implementing this plan overseeing any funding programs that results as from this plan. Doing public outreach about what the city is doing what the city's priorities are and especially when it comes to affordable housing. Doing an outreach about about who's living at these types of units. That's a common barrier. Communities are seeing to getting affordable housing built as people just feel a little uncertain about it. So using this housing committee is really a public outreach piece as well. And then this group could also be in in charge of updating this plan as new data becomes available and new priorities arise. The second recommendation under communication would be to just take a have the city take a stance in the messaging stance. But the city believes in access to safe and affordable housing for everyone. Especially those who struggle to find housing more than others. So low income resident residents homeless those who have a criminal record prior evictions and really prioritize programs that help these groups. The third recommendation would be to attend we did events. We did events. So this recommendation actually came from an affordable housing developer who's worked with the city. And their their general thought was the city's been great to work with. So the city's been really flexible in working with developers especially to get affordable units construction constructed which is great. So this developer suggested go to the we did events network with developers let them know what what you're interested in talk about potential project ideas just have a conversation. So relatively easy easy thing to do but could have a potential big impact. The fourth recommendation would be to continue the developer summits that the city is doing to show developers what's going on in the city what kinds of new and exciting things are happening and why it's a great place to live and really market those redevelopment properties. Under initiatives. So the first one would be to focus on neighborhood associations. So this came up especially during interviews. And in the survey as well I would say. So neighborhoods are really really important to people. They feel it creates a sense of identity and really helps people feel like they're they're part of something. And so we would suggest to continue building neighborhood associations and potentially engage them in different planning efforts and new developments. To get them on board and really you know from the beginning of a process to get people's input. The second initiative we are recommending is to purchase and market city owned redevelopment properties. So identify properties that are primed for redevelopment and release RFPs tell developers exactly what you want and where you want it. That's something that they like to see and we've seen other communities do that with success. A third recommendation would be to develop neighborhood master plans. So I mentioned the city doesn't have many opportunities for new greenfield development. But there are opportunities within the city's extraterritorial jurisdiction. So if you look at the city's comprehensive plan and the future land use map there's one area to the north and one area to the south that are identified as planned neighborhoods. So we really see these two areas as opportunities for the city to set a vision for the neighborhood make these places walkable include a mix of uses and a variety of housing types at different price points. So lots of different people can live in the community live in the neighborhood and they can stay in those neighborhoods as they age or as their circumstances change to focus on housing focus on just quality of life in the neighborhoods providing parks providing gathering spaces make these really a model for what you want to see as other neighborhoods might redevelop within the city. So people what what it could be. One thing I didn't know in the plan though is careful site design is going to be a key part to the success of these neighborhoods and making them work and function like they should. Last initiative recommendation would be to create a tenant resource center. So landlord tenant relations with something that that came up a lot in interviews and just talking talking with different people. So creating something like this promote promote positive relationships between landlords and tenants. This could be you know the city already has access to some of this may be expanding some of these services. So the housing counseling eviction clinic clinic exploring landlord and tenant rights and just helping make those relationships work better and giving people a resource to do that do so. As far as regulation. We had a number of recommendations. So probably one of the biggest ones is just allowing multifamily as a permitted use. So right now multifamily is a conditional use in the residential neighborhoods that it is listed under. So just making it easier to develop and showing people yes, this is something that we want. That's going to save developers time and money as well. A second recommendation would be to update the zoning code to reduce front inside yard setbacks and all residential districts to decrease the number of variances that are needed. That's something that came up when we're talking to staff and developers is because of the older generally older lots. They're smaller. So there are generally a lot of variances that are requested and that just adds more process and time slash cost to developers into the city. The third recommendation is kind of going one step further than that. But just creating a new residential district that's going to enable smaller lots perhaps on alleys by right to just try and streamline the process and make it easier for that to happen. The fourth recommendation is about the city's current code enforcement program. So that's something that came up in interviews and it was recognized as a success for improving the appearance and safety of neighborhoods and some main corridors. But it was also brought up that one downside to that program is that for older homeowners or those with lower incomes and little equity in their homes. It can create hardships when they're already struggling. So for this reason it's important to have staff and maybe it's one maybe it's more people who can be empathetic and take the time to work with property owners and try and find solutions. We're trying to avoid putting additional hardship on the property owner. Last recommendation here just would be something the city has done recently is just being flexible on its infrastructure requirements. The city has seen some reduced cost due to curb requirements modified cross sections of streets for some some green field development that happened recently and that cost savings is passed on to homeowners eventually so that's going to bring in homes at lower price points which is going to be a benefit to everyone. Lots of funding recommendations. I guess that's really what a lot of it comes down to. So the first probably one of the biggest recommendation is to expand the use of the city's revitalization fund. So using this as a general purpose vehicle to fund affordable initiatives around the city. These funds could be used as matching funds as land purchase for new construction renovations down payment assistance you name it. And this fund that what we're seeing communities. How they're seeding this type of fund is kids. Bonding sale of land general funds private contributions. This is often a good way to help developers be more competitive when they're applying for funding programs. This is a really nice match to have set up for them that's ready to go. Another recommendation so this is based off of something that Dane County is doing that developers across the work across the state are really excited about. So what they've done is they've gotten together a bunch of major employers in the county and investors. And they started this fund to build housing that's affordable at the 80 percent AMI or lower level. And when investors are getting out of it is a modest return. But employers see the value in having housing that's affordable for people working at their businesses. They want people to like where they live. They want people to be able to afford where they live. And they're hoping that'll get them invested in the community and where they're living. So I think that's a really interesting thing to maybe put some focus on and something that we've heard developers are interested in. And we're increasingly seeing employers realize that housing having the right type of housing at the right affordability levels is important for what they're doing too. The next two recommendations are focused on TIF. So the first would just be to use TIF to construct infrastructure to offset some of that cost to developers, which eventually leads to offset costs for home homeowners. And then the second TIF recommendation is to use the TIF affordable housing extension. So the state allows you to hold TIFs open for an additional year and to use that increment primarily for affordable housing initiatives. So the city has four TIFs closing over the next 10 years. So we really see these as great opportunities to seed the neighborhood revitalization fund and make a really big impact for the city as far as what's available for affordable housing funding. Also recommend promoting the use of the low income housing tax credit again that neighborhood fund could be used as match which would be very attractive to developers. Increase the use of the WEDA 710 Flex financing program. So that program is not as attractive when interest rates are low like they are now. But when they're when they're higher again, that's an attractive program to use and it requires a set aside of affordable units at the 80 percent AMI level. There are several down payment assistance programs. So just promoting the use of those programs, there's one local program and then WEDA has one at the state level and the region has one at the federal home loan bank of Chicago. The last recommendation is maybe something to tackle with other other communities or your representatives on this is something that Illinois was looking at. But off offering a bill that offers rental incentives to landlords who rent or renew leases to someone convicted of a crime. But again, this is something here time and time again in every community we work with. So a recommendation is in there to partner with some of those communities and and see what can happen. And then last but not least recommendations related to partnerships. So the first one just be working with affordable housing providers and developers that are already located in the region. So habitat partners and others by having a supply of available lots or even rehabs of existing buildings. The second piece is again related to the landlord education. We know the city does that already but just continue to advertise those programs and reach out to landlords. The third recommendation again relates to major employers in the region. So this recommendation is getting at just convening maybe a working group maybe once a year to see what's going on. What kind of employment projections on these businesses have what price or sorry salary levels. They're anticipating our hourly wages. They're anticipating hiring at and then talking about initiatives that the city is spearheading and seeing if there might not be some overlap of interest and maybe pulling some of those employers on board and on board and getting them to champion some of the initiatives and work with the city. And then the last recommendation is to explore opportunities to partner with property management companies who offer on-site child care or health care. I'm just really those things that we often see families struggling to find access to offering them on-site or even what we're seeing some communities do is offering them adjacent to a new development. So right now I can think of one in my community where a new apartment building was built right next to a health care clinic. So looking at models like that that other communities are trying out. And then just finally to wrap it all up so that's a lot of different strategies. A lot of those things are within the city's control. But there are a lot of things that are outside of the city's control. So it's going to take a lot more than just those strategies to solve the city's affordable housing issues and other housing issues. And this is the same with every community. So it's going to take talking to neighboring communities to talk about what types of housing they're providing what what need they're seeing and having some partnerships there. And it's also going to be more state and federal policies and programs. Lytec is awesome. But that's probably the main thing that that is out there right now. So hoping hoping we'll see more of that and more more programs and policies and maybe funding in the future. That would certainly help. So with that that is all I had. I guess we'll open it up to questions now. Chad are you you're on mute. Well sorry there was three questions in this in the chat and then we'll open up to other questions. So addressing the three questions in the chat. One of them was is a study available and I said put the link in the chat. It's also available on the city's website. It's a fairly large document. So we can't email it but you can get it out of the city's document software board docs. So that's available. There's another question in terms of rental units. Does it seem like the answer is actually for the city to continue to build the luxury more expensive rental units? Question mark we've heard time and time again that there are too many. So this seems to be count. This seems to counter what we've been hearing the market deserves. How is the city managing that. So I guess I'm looking for I don't know if Tara if you're still on and if you'd be willing to unmute your line and just get I'm a little confused over what you're asking. Yeah hi Chad this is Tara. Can you hear me OK. Yep. Perfect. Well thank you. Yeah I just know like it's one of the conversations that we've been hearing are you know people saying there are too many luxury apartments being built. We want to see some more affordable options. So I just think it's kind of interesting to see the figures that maybe actually the answer is building more of those to kind of free up those affordable units. That's kind of the conclusion that I'm hearing. I just want to make sure is that is that kind of correct or can you expand upon that a little bit. So I would say the answer is both. I wouldn't say that simply more luxury units are going to do it. But we know it's hard to get affordable housing built. So definitely yes I would say priority on on more affordable units. But at the same time we know that it's easier to get market rate units constructed and that's going to have the benefit of attracting some of those households who can pay more. So I would say it's definitely both. I didn't mean to come off as it's it's it's we need more luxury units. That's definitely not what I meant to say. I think both are part of the solution. Okay. Thank you for that. And Emily asked we've been told that the creation of luxury apartments would free up lower cost apartments in the community. But boots on the ground observation is that landlords are asking higher rents on the now quote unquote freed up apartments. Is there a way to balance the market to avoid the inflation of older lower cost apartment rental expectations? So certainly I think increasing the supply. I think supply is still quite tight for rental units. So landlords can do things like that which which is not right. But I think adding more units is going to place more pressure on landlords to to to make them have to do better and provide higher quality units or provide more competitive price points. So those are the questions we got. Is there anybody else on the line that has additional questions? You don't have to type it in the chat. You can just unmute and ask the question. There is a question that just came in. Does the city have in its planning at present more senior options? And the answer is yes. We recently finished a neighborhood development plan for the River Bend neighborhood and looked at what the former may line property along the Sheboygan River could be. And the recommendation that came out of that was senior housing options related to both independent and assisted and then some type of memory care. So that's just one of the locations. There's multiple other locations that this could be at. So yes we are we see the need in the study confirms the need and we're just looking for interested parties that would like to develop that type of housing. Anyone else have any questions? What about another question? What about 55 and older condos? Yes, that plays into the whole thing. It's you know, it's independent and market rate and condos. So it's finding the right developer that can make the numbers work and that you know the right location. But yes, that is part of it. The question about the slideshow posted anywhere. We will get it on the city's website and then we will also get the this is being recorded by W S C S our local cable station and we once it's posted on their YouTube channel, we will post that on the site as well. So you can get it right on our homepage. The next question did the consultant analyze not only quantity but also quality of low cost housing overall in the city. Yeah, so I guess most of our analysis there came from anecdotal data from talking to staff from talking to the housing experts and it also shine through in the survey that we did. So there's definitely a concern about quality. And I think continuing to fund provide funds grant funds for people to rehab their homes. I think that neighborhood revitalization fund expanding the use of that and providing additional seed money for that. I think funding rehab is going to be an important part of that just because of the age of housing stock. I think that's going to be a big, big focus too. There's another question. Does the city plan to create a housing committee? And at this stage, I am unable to answer that. We probably will. This document will be report will be referred to the committee of the whole when the new council is seated and the new mayor is on board. And then that will be one of the discussion items is whether the council would support the creating of another committee to address housing. Are there any other questions? I'll come off mute and ask a question verbally. First of all, I just wanted to say thanks to Becky because I know I talked to her as part of this process and to Chad for all the information and being able to, you know, put it into a 40 minute presentation. But kind of along the lines of what Heather asked, could you talk a little bit more about next steps and how the city will be communicating whether or not, you know, what's going to be followed through on as far as some of the recommendations that were put forth. So like any study that we do, we will work with the council to work through these, you know, to work through the recommendation. Some of these some of these require the council to take action when it comes to changing ordinances and processes. We can this is a document that we have already shared with a number of interested parties that are considering developing affordable housing. So it's going to be a working living document in the planning department and the city, you know, and the city document as well. So I and then we'll be reaching out to our partners. We are working on staff in the city planning with a number of partners that are on the phone are working on a housing solutions, housing collaborative grant application where they would bring in some technical assistance from a housing affordable housing consultant to kind of give us some help on what kind of drilling into more of the landlord kind of tenant problems. So there's a number of initiatives underway and we will continue to use the opportunities that they present themselves and also to look at. You know, look at how we use this to develop more housing. I think the the challenge we had and I we had a meeting earlier this week with our state and federal legislators and I shared the concerns that we have in the fact that, you know, if the if it's costing 1300 plus dollars per unit to build this type of housing. And we want to keep rents in the lower range of the 4, 5, 6, 700 dollars. How do we build the bridge a gap in between that because the city can't the city can't fund that gap that level. So, you know, I know that in talking with Senator Baldwin's office and the contact there that there was some there's some stuff coming through its federal legislation that might help with some of this, but it's going to take as one of the last slides that Becky laid out it's going to take the, you know, entire in the state, the federal and the city to, you know, kind of fill these demands because frankly, we just don't have enough resources on the local level to, you know, completely move forward and get these units built based on current construction costs. So there's another question that that low vacancy rate of 3.3 causes great problems for finding housing for low income people that is very important to bring that up. And then another question, do we know what the selection process will look like for the housing committee? And I can't answer that at this stage. It's really going to be up to the committee, the council and the council members to decide that. But I would believe that it would be open to interested parties and people that want to serve on it. So we will keep the communication open and work through that process with the council. Are there any other questions? Hi, my name is Emily Cundey and I'm actually currently the chair of our Sheboygan Housing Coalition. I just kind of want to bring this up to the community at large here that there is a housing coalition that meets monthly in Sheboygan. We're meeting virtually now. But if anyone's interested in joining that, it's a pretty open forum where we discuss the housing issues in Sheboygan and throughout Sheboygan County. It seems like this is a pretty interested group and not all of you are at the table. I will in the chat put my email address and I can add you to the list if you're interested in learning more about that. But I am pretty sure that members of the housing coalition would love to be part of any sort of committee that could help with this housing moving forward as well. Thank you, Emily. We're well aware of the good work that your collaboration does and we will definitely keep you guys involved. So are there any further questions? If not, hearing that there aren't, if you have anything further, my contact information is on the city's website and be happy to answer any further questions of anybody that might have anything. Otherwise, watch the city's website for some additional information. You can download the study and read it there and I hope you enjoyed today's presentation on the affordable housing market study results. So thank you for your time tonight. Yeah, thank you, everyone.


Today was day two. I just came back from IBC and today I saw Canon's booth and we checked out the virtual production morning details where there is like a pitch 2.1.9 screens to be able to like do virtual production and track using the Cineflex on the Jib. I saw that briefly yesterday but I had more time to actually get into it today. From there we actually checked out the 3D experiential zone that was actually they had like a booth and we checked out the live multicam broadcast solutions that Canon implemented and also had like setting conversations on applications and the workflow or still what may be beneficial and setting plans that we could actually integrate related to specific of my needs and shortly after that we went I went to Grip Factory Munich and checked out a couple of their dollies and the sliders and the payloads and a lot that all related to that and they're dizzy on the morphic that's now out it's quite interesting the clear the neutral flare version not the blue flare actually find out very interesting because of some of the things that you could actually do and that was also what I find interesting was the 28th had like edge very noticeable edge distortions so you have to like bring the characters a lot more to center but the close focus was good it was interesting on the widest focal length I could see how with the doubt and that you could actually make interesting things much more nice I think a lot of entry into the lens space is really really really a great thing that we get to like have options for the affordable and super high end thereafter tried out geckos cams and lens they actually are inspired by the Canon K35 lenses that actually no longer available and if you try to pick up one on the market I think they are like one of the most expensive lens more expensive than Ari because of the limited in circulation and yeah the lens was designed based with that and they were quite interesting they had characters they had texture to it you get and all of them were handmade by this boutique company called Gecko and they have like three other lenses they make one for the alexa 65 wide the gecko 35 so the normal full frame and they have like a new lens that they also like made I forgot the name based like that and after that we went to try out the roll rig from confetti move max and it was quite interesting it has like an auto leveling mode which which may be interesting useful steady cam wave to take out like the wobble and the roll axis and you can also use it like for music videos very they have like put together the rig that will enable it um carry the right camera you want to use for it and after that there are a couple of companies that had battle lights that we checked out had extensive conversation see the off-frames and but a lot of them seem very similar because they had both bicolor and ROGB but the wattage was where the difference is coming so the only thing I do not know if they have serum x I picked a couple of calculus so I'll probably study that in my own alone time but that was like some of the things actually got to like notice we I got some very interesting jib from this company called the sceny tic tac um normally a jib goes like this like up and down axis left and right axis but this jib can go up and down left and right forward and backward but not in like the techno current way but like the mechanical style and it's quite light like I was pretty shocked at the weight when I actually like picked it up you get I was pretty shocked but if you look at it it looks like it's heavy and it's very quite light but very steady and that was like an interesting discovery after that I had like an experience with light bridge and funny enough just three boots away after like checking the sceny reflect the sierra less system from light bridge there were other systems that was also present goddard's had one I saw that yesterday in a company called coalites has another one too again but having seen it I still think um sierra less is a lot more um curated to the experience of diffusing light and keeping it controlled you get it's not just about how frosted the reflection is but um how the light scatters as it's from some some from what I saw the one from coalite then have that I didn't have to put them to test goddard's because it says the prototype better than what they have like a small display that they actually put up to be able to test there so I was able to see a couple of things just quite the experience um day three hopefully I get like I did get to check out black magic and also saw a couple of iterations around the black magic space I was really interested in the ultimate workflow that was used that and it was green screen to be removed so that that that was like something I was interested in and yes I was able to like satisfy my curiosity based on that so yeah that was that that happened and that's all that happened so tomorrow is day three and we continue from where we'll stop see you tomorrow


Hi everyone, this is Nan Wang from the Australian National University. We are going to present our paper Flash Proofs, Efficient Zero Knowledge Arguments of Range and Polynomial Evaluation on the 6th December in Asia Crypt 2022. Flash Proofs are three round sigma protocols and the discrete logarithm assumption with a transparent setup. They do not require any pairing operations. First, we will present a new type of zero-knowledge range argument. We devise a new variant of the bit decomposition approach to prove that a committed value is in the range between zero and two power n minus one. Our range argument is tailored to confidential transactions on blockchain platforms. It achieves n to the power of two-third sublinear efficiency in communication and verification. The major breakthrough is that verifying our range argument consumes a comparable amount of gas costs to that of the most efficient ZK snark without resorting to any trusted setup on Ethereum. Our argument also supports the aggregation of multiple arguments for further improvement in efficiency. The table at the bottom shows the proof size and verification gas costs on Ethereum for a 32-bit and a 64-bit range respectively. Second, we will introduce a new zero-knowledge argument for polynomial evaluation. The argument aims to prove a public polynomial relation between two committed values. We provide two zero-knowledge protocols, which are optimized for lower and higher degree polynomials respectively. Our protocols build on the work of buyer and growth. To the best of our knowledge, our argument can instantiate the most communication-efficient membership argument in the discrete logarithm setting among those not requiring trusted setups. It also supports the aggregation of multiple arguments sharing the same inputs. The table illustrates an efficiency comparison of VG13 and our two arguments for lower and higher degree polynomials. It can be observed that our argument achieves a non-trivial improvement, improving verification and proof size. If you're interested in our work, please come and watch our presentation on 6 December. Thank you.


OK, then we can start. So this is the third lecture on algebraic topology, where we're talking about the homotopy invariance theorem. So this was a theorem. If f and g, x to y, are homotopic maps, then it follows that they induce the same map on homology. So then f star is equal to g star on hn x to hn of y, for all n. So we had first introduced the chain homotopies. Maybe I will not, or maybe I will repeat it. So this was, if you have two chain maps, f star and g star from a certain chain complex, c star d to another chain complex, d star d. Then we said they were chain homotopic if one had some kind of diagonal maps which interpolate between the two. So then f and g are chain homotopic. If there exist maps s i from ci to di plus 1, such that we have that d composed with s i minus s minus 1, s i minus 1 composed with d, is equal to g i minus f i for all i. So somehow the difference between these two can somehow be obtained as having to do with something composed with a differential. And this will serve to prove that this lemma, which we proved, if f and g are chain homotopic, when the induced map f star is equal to g star from hn of c to hn. So then they induce the same map on the homology of the chain complex. Now we wanted to use this to prove the theorem. And I have to repeat the setup I expect. So we go to the proof. So we had first seen that one can reduce to the case enough to show the theorem in case, say, f is equal to zero and g is equal to e1, where e0 and e1 are the inclusions of x. So x times i, x goes to x comma zero and similar for i1. We had seen that this is enough to prove it in this case. And then we were making this setup. So we want to, out of a simplex, we somehow make a prism and divide it into smaller simplices. So this was the following. So in delta n, so the standard simplex times i, i is always interval of 0, 1. And we take the kind of end points. So we put vi equal to ei comma 0. So somehow the one's in the bottom. And wi equal to ei comma 1. And so if this one is delta 1, then we get basically another copy of delta 1 here. And we have here this case is square, but in general prism. And we define some maps from, given a map from delta n. So if we have a homotopy, we get from this a map from n plus 1 simplices to x as follows. So first we define, we look at these maps, w0, v0 until vi, wi until wn. This, by definition, is a map from delta n plus 1 to delta n times i, whose image is just the span of these points. So for instance, it's not v0, v1, w0, w1. And so here we would have this thing would be v0, w0, w1. So the image of this map and the image of the map, the second one, would be v0, v1, w1. And then using these maps and our homotopy, we can, OK. So you don't need the homotopy. Using our map, we can define this prism operator. So let's just write it down. So we want to first define it on simplices. So we want to define an operator p from cn of x to cn plus 1 of x times i, which in the end is supposed to give us a chain homotopy between i0 and i1, OK. But we first have, so at least it goes in the correct way. It goes kind of diagonally. And we do this first for a simplex. We define it by this formula. We just use all of these, compose it with sigma times the identity on i, and take the alternating sum. So this is sum i equals 0 to n minus 1 to the i. Sigma composed with the identity on i v0, i, w i, until wn. Yes, indeed, it is cross. It is misplaced in the notes, which now I copied. Here it is the composition. So we just have, so this is a map from delta n plus 1 to x times i, and then we compose it with the identity on i times the map sigma. So this is a very different thing. And we put these signs as usual so that in the end something will cancel. And then for, this is just for a singular simplex, then in general case, we do it by linearity. So for alpha, so for we make p of sum over simplices a sigma times sigma. So this is an integer. This is a singular simplex. We define this as usual to be the sum over all sigma a sigma times p of sigma. So always by linearity. So then we have defined this map, which is a homomorphism between these two ABing groups. And we want to show p is a chain homotopy between the maps induced by i0 and i1. From i0 lower star to i1 lower star, where i0 lower star is just a map from the chain complex c star of x to c star of x times i, which is obtained in the usual way by composing with i0. So sum a sigma a sigma times sigma is mapped to sum over all sigma a sigma times i0 of sigma. i0 composed with sigma. That's how this induced map on the level of change was done. We just take the map to x and we compose it with the i0 into x times i. OK, and we want to end the same obviously for i1. And so now we want to prove that this is indeed the case. And you know, so in some sense, this is it. So the proof is difficult because you have to come up with the idea of using this. Now the rest is you just compute. And it's either true or false. And if it's true, then you have proven it. So how do we prove it? So what does it mean that it's a chain homotopy? It means that dp minus pd, always in the correct degrees. So I could call this, maybe I could call this pn, should be equal to i0 star minus i1 star or the other around. It doesn't matter. So that's what we have to show. So we just compute what we have here. So it's enough to show this on simplices, obviously. So let sigma from delta n to x, singular simplex, then by linearity, it's enough to show that if we apply this thing to sigma, then we get this. So let's compute. What is dp sigma? Now I forget the n because there's. Well, we know the definition of d and we know the definition of p, so we just write it out. So this is d of sum i equals 0 to n minus 1 to the i, sigma times the identity on i v0, v i, w i, w n. And you recall the definition of d, which is, again, an alternating sum where one removes the i's of it and takes this or the j's of the things which are written here and multiplies it by minus 1 to the j. It takes the alternating sum. So this is sum. And now there are obviously either one is among the first or one is among the last. So we distinguish these cases. So first the sum j is smaller than i. Do I want this one first? Then the sum j is bigger equal to i. So minus 1 to the i, minus 1 to the j, sigma times the identity on i v0, v i, forget vj. And then if we are, so this is for the first here, now if the number is bigger so that we get to the last one, one has to remember that this one is actually not the i plus first. This one is the i plus first is the i plus second. So it has an extra sign if one makes the alternating sum. So this is minus 1 to the i, minus 1 to the j plus 1, sigma times 1i v0, v i, w i, wj, wn. Because the sign take minus 1 to the which point in this list it is, starting with 0. So this is the jth one. But this is the j plus first. Because first we go until i and then we go on with i again. So this is what it is. So this is the notation which I use, but maybe it's too much of a short end. I sum with respect both to i and j. So maybe I should write it better. Although it is also standard. So it's sum i equal 0 to n, sum j smaller equal to i, and sum i equal 0 to n, sum j bigger equal to i. It's a double sum. And before I meant just wrote just this, and I meant I'm summing over both. But you know, that's maybe OK. So we just note the terms with j equal to i more or less the same. So if j is equal to i, then it just means we don't have this twice the same thing. Because we leave out one of them. So if you have j equal to i here, we throw away the first one. If you have j equal to i here, we throw away the second one. But the effect is the same. You know, we just get the elements that we have, and there's no doubling. But it's not so they cancel. And so these essentially all occur in both sums. But we see that the sign is different. So they will cancel. But it's not completely that they cancel because we have, so they cancel except for two terms. Namely, the case when i is equal to j is equal to 0 here. Then here, when i is equal to j equal to 0, then in the first sum, we have just from w0 to wn. In the second sum, we have the sum from v0, then w1 to wn. It's not the same thing. So one of them occurs in the second, the other one doesn't. So only one of them survives. So the thing which does not cancel here is in the case i equal j equal to n or i equal j equal to 0 or equal to n. And I claim that the only thing that survives is the following two terms, sigma times the identity composed with w0 to wn minus sigma times the identity composed with what now I'm quite sure I got it right. So I think I have to see. And then the other one is v0. So because all the other ones that we get here occur twice, once here and once here. But this one here will not occur in the second sum because we always start with v0 in the second sum. And this one here will not be in the first sum because the first sum, every element, starts with wn, stops with wn. So these are the only ones which don't cancel. But all the other ones, it's always it comes once here and once here with opposite signs. And so the sum, the terms here with i equal to j are only those. Why do I care about this? Well, we want to compute the difference. So I claim that if I do it the other way around, I get only the terms where i is different from j. So let's see why this would be true. On the other hand, if I take pd sigma, well then we do the same thing. First we apply d. So this is so here sigma is just so this will be as follows. We have p of sum j from 0 to n of minus 1 to the j. Sigma composed with E0, Ej. So this was how the map was defined. Now we have this simplex sigma, which is a map from delta n to this. And this is the composition with this base map. It gives us this thing. And we take p of that. And now we have to do the p. So the p is linear, so we have to apply it to each of them. And we do the same thing. So how does it go? So there are two possibilities. So I claim I can write it as follows. This is sum i equal 0 to n minus 1 to the i. So now we replace the sigma. So we also have this sum. So first we assume, again, the case that j is now smaller than i. Obviously, so let's see. So we take E0, Ej, Ei, Wi, Wn. So we take whatever we have. And we double put in the ith position. We put one up, and the other one we leave down. This is what it is. And I forgot, obviously, the whole thing that was written here. This was minus 1 to the i, sum j equal 0 to n minus 1 to the j sigma times the identity on i composed with this. And then, so I wanted to have j smaller than i. And now, so j is the one we leave out. So obviously, now, j cannot be equal to i, because i is the one we double. We cannot double the thing which is not there. So this is plus sum i equal 0 to n, sum j is bigger than i. So we have again minus 1 to the i. And we have again to see that, so we have to think about the sign anyway, but to this we can do in moment sigma times the identity V0, Vi, Wi. And we leave out the Wj until Wn. And so what about the sign? So you see that now we have not left out the jth one, but the jth plus first again, because this one is doubled. So this is minus 1, so the j plus 1. And so if you look at this, this is the same sum, this is the same thing here as what is written here, except to that we don't have the term j is equal to i. So we see that the difference between these two is precisely the terms with j equal to i, which is this. So we see this is the same as dp sigma, except for the terms with j equals to i, which are only in the first sum and not in the second. And so this improves, so maybe this is what we want to prove, you can leave it there. Over the opposite sign it does not matter. So this proves that this thing dpn of sigma minus pn minus 1p sigma is equal to this minus sigma times the identity v0n. And now you have to just remember how this goes. So I mean how everything was defined. So we had this thing that we have here. So we have that w, so vi is equal to ei comma 0 and wi is equal to ei comma 1. So thus it follows. So this thing here is just the image, just this thing is just the map. So w0 mu n is just the map e1 e0 to en. So this is in delta n times i. So the 0.1 here, and this is just the identity. And so this is just the identity. So in other words, this is identity of delta n times 1. And the other one, v0, is just the identity of delta n times 0. So every point gets mapped to itself times 1 here times 0. And so in other words, this is equal to, so therefore this map sigma times the identity of i composed with w0 to wn is equal to i1 composed with sigma. That's what it says. And in the same way, well, and so the map, so thus we see that this difference is equal to i1 composed with sigma minus i0 composed with sigma. So the sign is precisely the other round, which is the same by definition as i1 lower star of sigma minus i0 lower star of sigma. So we have shown that on simplices, this formula holds, and everything is linear, so it holds in general. So thus p is a chain homotopy from whatever, either e0 to 1 or e1 to e0. So e1 lower star to e0 lower star. And so therefore by the previous theorem, then e1 lower star and e0 lower star induce the same map on homology. And therefore by the case that one could reduce to this, it follows that homotopic maps induce the same map on homology. OK, so this was a slightly tricky proof. And you should maybe, the combinatorics of this sum is maybe also something one has to study with some, take a little bit of time to digest. But at any rate, it is not particularly difficult. It's just many little steps, and you have to always know where you are. But this kind of trick of making all these n plus 1 simplices out of the n simplex in this way, this is actually quite difficult to think of something like that. So now we have learned that if you have two homotopic maps between topological spaces to give the same map on homology, so then we can also ask ourselves, I mean, in what kind of other relations are there between homology and homotopic? For instance, we have this other invariant, which is the fundamental group. So how is it related to the homology? And this we want to briefly investigate now. I will not completely. I will prove a little bit about it and state the precise result. But I will not prove it completely because it's too long. But at least I will not miss out anything which is difficult. So relation between fundamental group and homology. So as before, I will briefly recall the definition of the fundamental group, although you just had it, but so we fix notations. So let x be a topological space. And we fix the base point x0 in x. So a loop in x at x0 is a map from the interval 0, 1, which into x, continuous map, which at both endpoints maps it to x0. So is a continuous map, say sigma from 0, 1 to x, with sigma of 0 is equal to sigma of 1 is equal to x0. So we have our space, have our point x0, and the loop is somehow a loop. And then the fundamental group is about homotopy classes of loops. So two loops at x0. Anyway, maybe I call them sigma and tau from 0, 1 to x are homotopic. So not when they are homotopy, not when they are homotopic, homotopic just as map to x, but also fixing x0, as you know. So they are called homotopic if there is a continuous map, maybe f, from the product of these intervals. So interval was also called i, after all, from i times i to x, such that if I take f restricted to, say, which we want to have wanted, maybe i times 0 is equal to sigma, f restricted to i times 1 is equal to tau. And in addition, at all times, the point x0 is fixed. So f restricted to 0 times i is equal to x0. I mean by this, it's the constant map which sends everything to x0. And f restricted to 1 times i is also equal to x0. So this is this homotopy. So if you want to make some kind of picture, we have this product of these two intervals, i times i, of the 0, 1, 1 here. And we have that, say, here, the map is sigma. In the bottom, in the top, the map is tau. It goes from here to here. Here, it's all the time x0. And here, it's all the time also x0. And this makes sense because, after all, both end points here are mapped also to x0. OK. And we also had some, there was some composition of loops. So if sigma from 0, 1 to x and tau, 0, 1 to x are loops at x, loops at x0, then we can define the composition. Maybe sigma composed tau. So I'll make this as a star. I don't know what the standard notation for you is. This is, again, a loop where we just run through one after the other. And we re-parameterize so that we get along from 0 to 1. So for instance, you say t is mapped to tau of twice t if t is smaller than 1 half and sigma of twice t minus 1 if it is on the other, in the other half of the interval. And as at the end point here, one gets always to x0, this matches and gives us a continuous map. One just does one after the other and re-parameterizes. And so the equivalence classes of loops at x0 in x form a group under homotopy. So we have this homotopy of loops. Look at the equivalence classes. Form a group, p1 of x, x0, the fundamental group. And if I write the equivalence classes, for instance, so I write something like that, gamma is the equivalence class of gamma. Then the group structure is just that if I take gamma times tau is defined to be gamma composed with tau. So this was how this was defined. And this is a group. In general, it will be a non-commutative group. So now we want to relate this to homology. And the reason why this will be related to homology, in fact, to the first homology group is that after all this i, the interval 0, 1 is just the standard one simplex. So a loop in x is a singular one simplex in x. So therefore, we have somehow an obvious maps which relate them. Then we have to see what these maps do. So note our definition delta 1 is just the interval 0, 1. Thus a loop in x at x0 or whatever is a singular one simplex in x. And there's more. So we have a loop, say, at x0, if you call it gamma. So gamma is a map from 0, 1 to x with gamma of 0 is equal to gamma of 1 equal to x0. And now, what is it if we take the differential of gamma as a one chain? So then d gamma is, if you remember, the end point viewed as a constant map, the beginning point, no, the end point minus the beginning point. So this is gamma 1 minus gamma 0, which means x0 minus x0 to 0. So in this case, I didn't have to remember who was first because they're equal, but still one should. So thus gamma is a cycle, c1 of x, so one cycle. So as it's a one cycle, we can compute, we can define its homology class. It's just the equivalence class, modular boundaries. So we have a map, which I may be called chi, from the set of loops in x at x0 to the first homology group of x, which sends gamma to its class in homology, modular boundaries. And we want to, and so we basically have the same space, we have the loops, certain equivalence classes of loops give us the fundamental group with respect to certain equivalence relation. And with respect to another equivalence relation, it gives us the first homology. I want to see how these two equivalence relations are related to see how the groups are related. And also somehow, the group structure should be related. And this is the subject of the following proposition. So if gamma, so there's two parts, if, say, gamma and tau are homotopic loops in x at x0. So we have two loops at x0, which are homotopic as loops. Then it follows that they define the same homology class. And the second statement is that this is also compatible. So this means, in other words, note this means we have a map from pi1 of x at x0 to h1 of x, a well-defined map, by sending the class of a loop to the class of a loop as a singular one chain. Because we know that if they are in the same class, they are also in the same class here. And the second statement is that it's compatible with the group structure. So if, again, so maybe gamma, tau again loops, then if I take the class of the composition, this is a class gamma class, a class of tau. So in other words, it means if we put these two together, it means we get a homomorphism from pi1 to h1 of x, a homomorphism of groups. So there is a homomorphism from the fundamental group to h1 of x. So I want to prove this proposition. And then I will just state what precisely is true. Because you would want to know you have a homomorphism. Is the homomorphism injective? Is it surjective? What is the kernel? And I will state afterwards. But first, I just have this. So in other words, I say it again, chi from pi1 of x at x0 to h1 of x, gamma maps to gamma is a group homomorphism. So the proof is somewhat similar to this homotopy invariance theorem, a little bit simpler. So we have two parts. So assume we have a homotopy between these two. So now these two change miraculously that names into sigma and tau. And then, well, anyway, it doesn't matter. So let f from i times e, i, to x be a homotopy. So this means that f restricted to i times 0 is, say, I don't know which one I want first. Gamma, how are they called? f restricted to i times 1 is tau. And f restricted to 0 times i is equal to x0. And this is also true for f restricted to 1 times i. So this was this thing. And we do the same thing as we did for the homotopy invariance. We, again, make a map from other simplices in the same way. So we define v0 equal to 0, 1, 1 equal to 0, 0, 0, 1. I hope I get it right. Yeah, anyway. So I'm not quite sure whether I might have exchanged these two factors, but I think you can manage that. So w0 equal to 1, 0, w1 equal to 1, 1. So we have this square i times i with v0, v1, w0, w1. And we have this map, f, which somehow maps from this. So x. And if I restrict it to the bottom, if I'm lucky, and I think this precisely means I always have to exchange these, then I get sigma, actually gamma. If I restrict it to the top, I get tau. And here's the constant map x0. And here's the constant map x0. And then we subdivide this thing in the same way as for the homotopy invariance theorem. So we define a two-chain, say alpha in c2 of x, by alpha is equal to f composed with v0, w0, w1, minus f composed with v0, v1, w1. So this means we have, so the map which maps the standard simplex first on this half is this map. The map which maps the standard simplex on this half is this map. And we take this one with plus. And so we take this one with plus, this one with minus, and we apply f to it, and this gives us a two-chain. So this is our two-chain. And we want to claim, obviously, that we want to claim that two things are homologous. So we want to show that the difference between the two, these two, is precisely whatever, tau minus gamma or the other route. And it's kind of obvious that this would be the case, because the thing in the middle cancels out. It's x0 constant. It will also cancel out. So we write it down. So what is d alpha? Where do I want to do this now? So this, I write it out, because anyway then. So what is it, d alpha? This is, how does one do d? One leaves out the i-th one and takes it with minus 1 to the i, where i starts with 0. So the first one is, so here we have this one.  And then minus, and so we have a, so the whole thing minus, because we have a minus here.  And then we do the same thing here. So f composed with v1, w1. f composed with v0, w1. f composed with v0, v1. So these two cancel. And what else do we have? This one here is this w0, w1. So this is tau. So this composition, f composed with this is tau. This map, we are here on this side, is the constant map x0. Here we have w1, that is x0. And here we have v0, v1. That is gamma. So we see that indeed the difference of tau and gamma is a boundary, namely that of alpha. So we find tau is equal to gamma. So this proves the first part. In the second part, I will do slightly more, in a slightly more hand waving way. I just draw your picture, and you would have to turn this into a formula to show that the picture describes something continuous. So we have to find, you know, we want to prove that these two classes are that the class of this and the classes are equal. So we have to find, again, a singular two chain whose boundary is the difference between these. And we do this actually given by just one simplex. So let's say sigma from delta 2 to x be given as follows. And I will now just write it. So we take our simplex. I write it in the, you know, I draw it in this way, although the angles are maybe not correct. So this one is e0, e1, e2. So this is a two simplex. And now I want, on this part, I want the map to be tau. So the map from here, so I have a map from here to x. And on this part, the map should be tau.  On this, it should be gamma. And then I draw a line here. And I just, you know, whenever I have a line here connected like this, like this, then, oh, it's correct way around. I will find out that it's precisely wrong way around. But anyway, so maybe I call this gamma and this tau. This will not make a difference. So the tau after all comes first. So then on this part, it should be tau, again, reparameterized. So if you take the map from the interval to here, which goes from here, then composed with the map here, it should be tau. And here it should be gamma. And it's always for all these lines, always until the corner. So I hope you understand what I mean. So here, we take, you know, a map from the interval 0, 1 to this line and compose with the map to x. And this composition should be tau. If we take, instead, this line, until this middle line here, then the composition of a kind of linear map from here to here with x, again, should be tau. And on the other hand, for this part, should be gamma.  So you can somehow see that if you just take this composition here, you can really see that this is essentially gamma composed with tau. I just make a picture. So and then, obviously, you see, then if I take, what is d sigma? This is the map which you obtain by leaving out to 0. This is gamma, the one which you get by leaving out d1 minus gamma tau plus, so it follows that, OK? So obviously, so the exercise, so to turn this into a proof, you have to write down a formula for this. So as to show that in this way, I have defined the continuous map. So for sigma, so as to see, it is continuous. And that's kind of trivial. You just have to look at this picture and see what you have to do. And then this proves it. But anyway, here I didn't want to do the details. It's obviously often with homotopies that if you actually want to write down a formula, it gets a bit complicated. But I think, I mean, I hope you can imagine what I mean. No? I don't know, you look, I cannot know. I mean, I just say it and nobody says either yes or no. So what I just mean, I have here, I have the map sigma. If I restrict sigma, so I have a map here from, so I have a map, one map from the standard interval to this. For instance, if I just take the map from a standard interval to this with just maps, this with just maps in the obvious way from this 0, this 1, and then compose it with sigma, then I get tau. If I do it here, again, starting with E1 to E2, then the composition with sigma is gamma. And here, I do it in such a way, the map from E0 to E1. I take a standard interval here, the composition with sigma is this. And then I say, I can extend this to a continuous map on the whole thing by always saying that the map, which is the composition of the embedding of the standard interval into this here, into this part, if the composition of the standard, the embedding here with sigma should be tau, and here it should be gamma. And I claim, and I do this for every such. I have the whole thing is exhausted by these. And this gives us, the claim is this is a continuous map from this whole triangle to x. And it's kind of obvious that it should be so. But obviously, to prove it, you have to write down a formula. If you can write down a formula which is everywhere defined and continuous, then it is a continuous map. This proves this proposition. And then I want to just state what the result is. So without proof, so the precise relation between fundamental group and h1. So you can see, we have this map, chi from p1 of x, x0, to h1 of x is a group homomorphism. So in best case, you could hope that they are isomorphic. But this can in general not be the case because you know that h1 of x is commutative. And p1 of x, x0 does not need to be commutative. So that means that if you have a commutator of elements, it always lies in the kernel of this map. So if you write down something, so if you put k to be the subgroup generated by alpha, beta, alpha to the minus 1, beta to the minus 1. So the alpha and beta are elements in p1 of x, x0. So this is the subgroup generated by all such elements. So this subgroup is the commutator subgroup. Then because this group is abelian, it follows that all elements in k will lie in the kernel of this map. Because they are just mapped to the image of A plus the image of B minus the image of A minus the image of B of beta. So then clearly k is contained in the kernel of this chi. And the theorem is that indeed, k is precisely the kernel and the map is surjective. So chi is surjective and k is equal to the kernel of chi. So thus we have that h1 of x is isomorphic to the fundamental group of x. Maybe I should make some assumptions. Anyway, divided by k. And obviously the fundamental group does not work so wonderfully if the space is not path connected. So we assume x is path connected. So then the statement is this. And so you could also say, so often sometimes you say that if you take a non-commutative group and you divide it by the commutator subgroup, you get the abelianized version of the group. Because you get the biggest quotient, which is abelian. You throw away everything. So the first homology group is obtained from the fundamental group by making it abelian. OK, so this is this result. So the proof is not very difficult. You just have to, it's similar to what we did here. So for every thing in h1, you have to somehow construct a path which maps to it. And that's a loop which maps to it that's not very difficult. And it's a bit more, it's not particularly difficult, but it's a little bit lengthy to see that the kernel of the map precisely consists of those. You can see that this requires making some nice two chains of which you get that the difference has to be the boundary and so on. So that might be a little bit complicated. But it's all quite simple. And you can find it in the books. And at any rate, I just wanted to state what the real result is. So in particular, for instance, we get that the first homology of s1, so s1 is the one sphere. So the first homology of that thing is equal to z, because we know that the fundamental group is z. And if we abelianize it, it doesn't change anything. And we have that h1 of sn, sn is the n sphere, is equal to 0 if n is bigger than 0. And as we already know, to z if n is equal, bigger than 1, is equal to 1. So because again, we know that the fundamental group of this thing is 0, if n is bigger than 0. So this was as much as I wanted to say about this topic. Now I want to, so this was slightly more concrete than what we had before. Now again, we began a little bit more abstract. So we want to introduce relative homology. So if you have a topological space and a subspace, you have the homology of the topological space, the homology of the subspace. And there's actually something which kind of is in between these two, which is relative homology. And we will prove that they are connected by something which is called the long-exact homology sequence. And for that, we will need again some homological algebra. First I introduce the thing, and then we will go to this homological algebra, which is what? OK, so let's see. So we want to talk about relative homology. And the long-exact, I actually didn't give you the notes, no? Anyway, homology sequence. So after these lectures, or maybe tomorrow if I don't manage, I will give you the continuation of the notes. So let x be a topological space. And a in x be a subspace. So it means it's a kerosene-induced topology. We define the relative homology. So that's the definition. So we have to define some chain complex. So if we look at the chain complex, cn of a. So these are maps from the n-simplices into a. This is in a natural way contained in cn of x. This is just the maps from the linear combinations of maps from simplices to x, which happen to map into a. So cn of a is equal to the set of all some sigma a sigma times sigma in cn of x, such that sigma of delta n is contained in a for all sigma that we have here. And we also see that if we look at the map d that we have here, if we have d from cn of x to cn minus 1 of x, if we have a chain which happens to lie in a, so that the map actually goes to a, all these maps go to a instead of to the bigger space x, and obviously the d of it, which is just the restriction to some phases, will also map to a. So maps cn of a to cn of a to cn minus 1 of a. So in some sense, so we have this chain complex and cn of x with the d's and we have another chain complex. And this is really, in some sense, a sub-complex. So we can say it as follows. We say we define cn of xa to be the quotient. We know that cn of a is contained in cn of x. We can take the quotient group. And as the d here maps cn of a to cn minus 1 of a, we get an induced map d on the quotient. So if I write for the moment, say, alpha for the class of this in this quotient, then we can define d from cn of xa to cn minus 1 of xa by sending the class of alpha to, I mean, modulo cn of a, to the class of d alpha modulo cn minus 1 of a. So we get a chain and obviously, d composed with d is 0 because d composed with 0 was d composed with d was 0 to begin with. It certainly doesn't become more by taking the quotient by something. We take after the clause. So the class of d composed with d here is obtained by taking d composed with d applied to some alpha. And that's already 0. And if we take a class of it, it stays 0. So thus, cn of xa, so I say c star of xa with d is a chain complex. And the relative homology is just the homology of this thing. So denote, I don't know whether I need this or whatever, cn of xa to be the kernel of d from cn of xa to cn minus 1 of xa. And so these are the cycles, relative cycles. And bn of xa is defined to be the image of d from cn plus 1 of xa to cn of xa. These are the relative boundaries. And then the homology is, as usual, the cycles divided by the boundaries. So the nth relative homology xa is hn of xa, which is defined to be the quotient. So that was rather simple and relatively formal. So actually, but now we want to see how this is related to the usual homology of x and of a. And this is by means of some exact sequences. I want to show that the hi of x, hi of a, and hi of xa are closely related by a long exact sequence. I have to tell you what that is. Roughly, it says it's not quite true. What this will imply is that if you know two of them, you know the third. If you know two of them, you know the third. But in a somewhat intricate way. And so let's see what this is. So I talk about this long exact sequence. So as I do not assume that you know what exact sequences are, I have to review this first. And then I have to set up the homological algebra to this. I have defined, this was a definition. I just said that zn of xa was defined to be the kernel of d from cn of xa to cn minus 1 of xa. So and I had called in the setting where we don't have these two things, but just one, we had called this zn and I had called them cycles. And now if I have the pair of x and a, I say it's relative homology. I say it's relative cycles and it's relative boundaries. But this is just a definition. This is the name how I want to call it. So yes. OK. So you wanted to know what n cycles and n boundaries are. So I maybe can briefly review this. So you remember we had defined these groups cn of x, which were some sigma, a sigma times sigma, where a sigma is in z and sigma is a map from delta n to x. And the n refers to the dimension of the insimplex from which we map. This we call n chains. Now we have a map d from cn of x to cn minus 1 of x. And so we call zn of x to be the kernel of d. So this is some subgroup of cn of x. So these are also n chains. So therefore I call them n something. And I call them cycles. I mean, this is just the word that people call them. Such things, the kernel of d is called cycles. And as they are n chains, I call them n cycles. The property that it has to satisfy to be called an n cycle and n boundary is that it is a cycle or a boundary. So a cycle means that's in the kernel of d. And that's it. And the boundary means that's the image of d. So bn of x. These are called n cycles. So zn of x is the group of n cycles. And it's just defined like this. It's the kernel of d and nothing more. And bn of x is the image of d. This is also a subgroup of the same n chains. And it is called the group of boundaries. This, again, just a name that people give to it. I mean, it's not a name that I invented. And now we know that d composed with d is equal to 0. So it follows from this that bn of x is contained in zn of x. And so therefore we can take the quotient, which is the homology. So this is, with the cycles and the boundaries, these are just words. You just call it like this.  This I said the first time. But you could have asked then. But now I, so you see, these are formal expressions. So therefore I can take the formal sum. I mean, obviously, this you have to know. Otherwise it cannot do anything at all. So you have an element. So if you have a class alpha in cn of x, then alpha can be written as sum a sigma times sigma. So sigma and where sigma runs through. So each sigma is a map from delta n to x. But it's for different ones. So you just have this that you have a certain finite number. So you have to each map from delta n to x, to each continuous map from delta n to x, you associate an integer, an a sigma is an integer. So in such a way that only finitely many of these integers are non-zero. Because if you write sum, you always mean that the sum is finite. So you have for finitely many of these maps, you associate an integer. But you view this in such a way that you associate an integer to every simplex. And for most of them, the number you associate is 0. That's a simpler way of seeing it. Because then you can sum it very easily. If you have sum sigma alpha sigma times sigma, plus sum sigma beta sigma times sigma, how do you sum them? Well, like you would sum, if these were really sums of things, you just sum the coefficients. This is defined to be the sum over all sigma, a sigma plus beta sigma times sigma. So let's look at the, so we can, and the difference is, you can also take mine. So this is the sum in the group. And the way of doing, so the standard way how you would call this construction is you'd have, this is the free ABN group generated by these singular simplices. Just take the, and the minus the element, the additive inverse, you can easily see. So minus of sum sigma, a sigma times sigma, obviously is sum over all sigma minus a sigma times sigma. So let's look at a very small example, just to make sure. So obviously, if you look at a real topological space, I mean some of real life, then it will have very many points. And there will be very many maps from delta n to x. So the set of the sigma over which you sum is huge. It's some uncountable set. But you only take finite sum. So it's not, there are only finitely many coefficients non-zero. But let's look at a very small case. So assume that x is equal to set of two points. See, it's called, it consists of a point x and a point y. So this is a small case. So in that case, so what are, so if I want to look, what is cn of x? So delta n is connected. So here we have a, these are supposed to be a discrete set of two points. So it means that delta n can only map to x or it can map to y. So that means what? So we have two maps. We have the map from delta n to x, the constant map, which I could call, well, maybe I call it x underlined. So which maps delta n constant to x. Every point of delta n goes to x. And we have y underlined, which is the map delta n to y. So every point t is mapped to x. And here t is mapped to y. So what is then cn of x? Well, this sum, this is so, is a set of all formal sums. We have a times x plus b times y, where a and b are integers. That's what it is. And so it is just, and the addition is the usual one. You are just at the coefficients. So this is isomorphic to z squared. But obviously, if you take something like rn, and you want to say concretely what this is, this is some a being group which has uncountably many generators. So it's not so easy to write concretely what it is then. I mean, the miracle is that if you do the homology, so you take this thing, you take the d, and then you take the kernel of the d divided by the image of d, you usually get some finitely generated a being group. At least if you have a reasonable space. So as we for instance saw that if the space is connected, then h0 is equal to z. This is quite miraculous if you think what a big thing you take of which you take a quotient. So you take something huge, and you divide it by something as huge, and it turns out that the quotient is very small, and also still interesting. Anyway, so that's what we have there. So I hope this is a bit clearer then. Okay, thank you. So next time we talk about relative homology.