OTAN, Outreach and Technical Assistance Network. My name is Marjorie Olavidis. I'm a project specialist for the Outreach and Technical Assistance Network, OTAN, and I'd like to welcome you all to this month's OTAN Tech Talk. Our speaker today is Audrey Deirdor from Sweetwater Union High School District, and she's also one of our newest OTAN subject matter experts. Her topic today is appointment scheduling with the Google Calendar. Go ahead and take it away, Audrey. Thank you. Hi, everybody. So as Marjorie said, this is appointment scheduling with Google Calendar. Again, my name is Audrey Deirdorf. I am a resource teacher at the Sweetwater Union High School District Adult Education, and I'm also a subject matter expert with OTAN, brand new. So today, we're going to go through the different steps on how to create this Google Calendar. So just a little bit about me. I've actually been teaching in adult education for about 29 years now. Seems like a lifetime, but I've been privileged to work in many different areas, and technology happens to be one of them and Google. So I'm excited to share this with you. When we found it, I was trying to help our job coach, and the job coach is going to be meeting with different students, and so he needed an appointment schedule. So we found it by just kind of, oh, look what Google has. So I'm really excited to share this with you. So before we begin, some of the must-hats. One, you have to have a Google workspace for education, which means it's either you have a Google through school or through your district. It has to be workspace for education in order to use appointment scheduling. Or if you have a private account, you can always pay for a subscription that allows you to use this appointment slots feature. So hopefully you all have a workspace for education. If you don't and you find that it's very helpful and it's something that you can use, you can ask your district or your company to go ahead and purchase you a subscription that will allow you to use this feature. So here are our objectives for today. One, we want to learn how to create a bookable appointment schedule, and then how to share that appointment schedule. And then how do you edit? Like your administrator or somebody says, hey, we have a meeting. Oh, I forgot. How do you change your appointment schedule or time slots? So those are the three objectives for today. So we will go ahead and start. So first, I have all the steps to follow on the actual slide step. But I want to do a demonstration of it first by going to a Google tab and showing you. And then we'll come back and review the steps that are actually written. So this is why I'm not in the slideshow part of the slides presentation. So I can go back and forth from tabs to the slideshow. So what I will do is open my new tab. And so you can see here is the Google tab brand new and I will now go to my waffle, which is on the right hand side top corner to the left of my picture or icon. Now you do have to be signed in, signed in with your district or your school account or your public account that has this slot feature. So when I click the waffle or the grid, I will now look for the calendar icon and select the calendar icon. When that happens, you'll have your calendar and on the left hand side, you should see kind of a menu. If you don't see it, it's probable that this main menu, which we call the skinny hamburger, if you click it, it will show up. Or if you select it, the menu will show up. When you have the menu open, you will go ahead and select the create button. And at the very end of the list, at the very end of the list, you will see appointment schedule. So you will select appointment schedule. From here, you will see two things. On the left hand side, you see a menu that's basically a settings menu. And then on the right hand side, you will see the calendar. Now we are at Friday, so I will go ahead and move it over to next week. So on the left hand side, we're gonna direct most of our attention on the left hand side in the settings menu. The first thing you need to do is go ahead and name your appointment schedule. So I will just name it whatever you would like. Then underneath that, you have your appointment duration. How long do you want your appointments to go? Do you want them to go for 30 minutes? Do you want them to go for an hour? So you would select the 30 minute window and you can go ahead and select anywhere from 15 minutes to two hours, or you can custom. So if you custom and you click or select custom, you can now put in 20 minute slots, you can do one and a half hours, and then you would select done on the bottom corner. And now it changes to 20 minutes. So all my appointments will now be 20 minutes long. If you look at the calendar on the right hand side, you can see that it is creating it. As I am changing the settings, it's creating the calendar on the right hand side. So each little darker blue appointment slot, that's 20 minute slots right there. Okay, let's go to the next part underneath. And this is the kind of the meat and potatoes of this calendar. This is your availability, when are you available? So first of all, you can choose whether to repeat it weekly, or if you select this, you can say does not repeat. Let's see what happens when we see does not repeat. The first thing that happens is that week went away and now you're gonna be able to add it date by date. On the right hand side, you can see that the calendar went away. So if you would like to do it date by date, this is what you would select. I will now do it as a repeat weekly. So I went back to repeat weekly, and now you can see I have the Sunday through Saturday schedule. Now, the three letter day is on the left, then you're going to have the appointment times, the start time, and then the end time. After that, you're going to have an icon that means unavailable, the circle with the line through it. After that, you're gonna have an add button. So you can add more times to that day. And then the last icon you're going to see at the very end of this row is copying. You can copy this timeframe for all the days that are available. So let's see how we can adjust it. So I will choose Monday and I will select my start time. And actually I'm only available from 10 a.m. on Monday. And I'm only available until, where is my 11? There it is. I'm only available one hour. If you look on the right hand side, you'll notice that it changed. Look how different it is from Monday to Tuesday. So Monday you can only see 10 to 1020, 1020 to 1040 and 1040 to 11. So the rest of them are still at this 9 a.m. to 5 p.m. time slots. Now, if I wanted to copy, because really throughout my week, I'm pretty busy, these are the only times 10 to 11 is the only time that I'm gonna have available for appointments. I will now use this copy icon at the end of the row and select it and notice that on this settings menu, everything changed to 10 to 11, as well as on the calendar on the right hand side, everything changed from 10 to 11. It's pretty cool. I love that feature. So, but I'm also available usually from one to two each day, Monday through Friday. So what I will do is on this Monday row, I will do the middle icon, which is the add another period to this day and select that. And now I can change the time by selecting inside of the time slot, the beginning time slot, and I'm going to change it to one. And then I have to also do the same for the end time and select it to two. So now you can see on the calendar on the right, it has that one to two time slot just for Monday. So how do I copy it over? Again, gonna go to that Monday row, the last icon selected, and now it's copied to all days. So now Monday through Friday, I have 10 to 11 and one to two available for appointments. Again, remember we did 20 minutes appointment. So now what I'm going to do is for us at adult education, we don't have Friday classes. So I'm going to make this unavailable. So that would be the first icon that you see after the time slots and I will select unavailable and notice that Friday goes away. Now it still has it for the one to two. So I have to also do it for the one to two time slot. So there you go. And so now all my Fridays are gone. So that's how you kind of do your schedule. And then I will show you how to edit or how to change later down the road. So after that, you now have your scheduling window. If you select the scheduling window setting, you can schedule it to either be available now or you can determine what your start date and your end dates are. Let's start with start and end date. So I will select start and end dates and a box will pop open. You can say I want to start the schedule as of today or if you want to start at the following week, I'm going to say, and it pops up, I like this feature that it actually pops up a calendar. So I will now go ahead and say, I'm going to start my appointments as of September 12th, I will select that date by clicking on it or touching it depending on what you're using a computer or tablet. And then the end date, you can say never if you want it to go on forever for the rest of your life as long as you're working or using this or you can decide the end date. So I will decide the end date because we're actually going on fall break next week. So I will go ahead and click the 16th or the 15th to be my last day because fall break is coming. So I will click done. And now I've had the start and end dates. Now, if I wanted it to be available right now, let's go ahead and try that feature available right now. I don't have to do anything else, but now do the next part, which is in advance. How many days in advance? So I've started it today and for 60 days I will have these same appointments available to see by the students or the clients that I'm sending it to. Now, if you only want to do it for 30 days you would again click or select inside and use the up and down arrows. Or if you double select, then I would go ahead and with my keyboard type in 30 and now it only goes for 30 days. So looking to the calendar on the right, using the arrow, the right arrow to the left of September, if I click that notice, I will go 30 days and they will have it there already. Okay. Now let's go to the next part is how much time are you gonna give the student or the client the ability to book the appointment? So if you keep it at the default, which is four hours, that means if they want a one o'clock appointment with you, they have to book it by 9 a.m. the same day. So for me, I like to do it more time so maybe I'll do 12 hours or maybe I'll do 24 hours if you want full day in advance. So it's really up to you. Again, the default is going to be four hours. So going on from there, this is where you can adjust your availability. So I realized, oh no, on this, let me go back. On Tuesday, the 13th, I actually need to change my availability. So what I did was I selected the change a dates availability and I went and selected the 13th and I cannot do one to two. I can only do from three to four. So now I will change it from three and you have to select the end time as well, three to four. And notice on my schedule on the right hand side, it adjusted it for me. And so that's how you could do it in this settings menu. And then this last part of this menu is the booked appointment settings. So what this is great for is buffer time. Do you wanna jump from meeting to meeting? So 20 minutes and go straight to the next meeting. Or do you want time between? So I like to have time between if there's paperwork or I need to run to the restroom or something like that. So I will go ahead and select that as well as change it manually to 10 minutes. Again, you can use the up and down arrows. But if I now select off of that, notice how you now have buffer time on the schedule to the right. So you have this little white area in between the blues, which means that it is buffer time. And then how many do you have per day? So I only have four per day right now, but if I only wanted to schedule three appointments per day, and just give them the option of picking. So I give them all the time options that I have, but I'm only gonna take three appointments per day. This is where you would do that maximum bookings per day. So I will select here and I'm actually only gonna take three appointments per day because of my busy schedule. So now, even though there are four appointment slots on this calendar on the right, only three will be booked. And once that third is booked, the other one will go away. So the next part of this is the coloring of it. If you decide to do different booking schedules, you can color. So notice on my calendar on the right, I do have purple. And so this is a different booking calendar I had. And so this one is now blue. If I wanted to change the color over here on the settings menu, this last portion, you could go ahead and select the color you want it to be. So, and you can actually add a label to it if you would like and just say, this is the test, test label, or whatever you would like it to be. So notice how it changed the color in the menu. Okay, there is a second part of the settings menu. So at the bottom right of the settings menu, you'll have a blue with a white next button. So I will go ahead and select that. So now you will book a page, photo, and name. So this is how you can see what the student or the client sees. So if you select that, right now they see my name, they see my picture. So this is how your identity is gonna actually be displayed on the booking page. So you can always manage it by going to this blue, manage Google account, photo, and name if you wanted to change your name and photo. The next part is the location. So are you gonna meet online? Are you gonna meet in person? So you can choose different locations. So if I selected this choose location button, notice you can do a Google Meet video conferencing. You can decide to do it in person, a phone call, or if you're gonna use a different conferencing platform like Zoom, like we're on right now, or if you're going to use Teams, you can go ahead and put none or specify it later. And then you will have to manually update. And that's okay. So I will go ahead and choose the none for right now. Okay, after I choose the location, you will now have a description. So here is what you can show the student or tell the student what you expect them to bring with them to the appointment or what the appointment is for. So for example, here the note, and I love that it has, you can bold it, you can underline it so that you can basically highlight what you want them to bring. So please bring your homework to the appointment, right? And so I can go ahead and go ahead and bold that and make sure that they know a please bring your homework. Then the next part is the booking form. So when you share your booking page, which I will show you in a few minutes, they have to fill out a form. And so you can determine what goes on that form. Reminder, the little information you ask for probably the better for the students, but you can choose different things. For example, default, first name, last name, and email address, those are required, but you can add an item. If you wanted to add an item, you could add a phone number, or you can customize an item. So like an ID, if you're wanting them to bring their student ID number, or if you want them to put in, I don't know, there's so many different things, like some type of ID or some type of, I don't know. So there's other things that you can do. So you can add the item and then notice that when you add an item again, the only thing left is custom. So you can only do up to five things. They won't let you go beyond that. And then if you want them to require email verification, this is if you have a lot of spam coming to your email. And so you just wanna make sure that you're getting the correct emails from students and not from spammers. You can require email verification. And then the last part of this, before we save all of our settings is going to be your reminders and your confirmation. So you and the person who made the appointment will get a confirmation email. And in it, it will have a calendar invitation so that if you don't use Google Calendar and you use your Outlook Calendar, you can actually save it to your Outlook Calendar. And if the student uses Google Calendar, it can save that or it can save it to whatever calendar the student uses. Then an email reminder, you can set that if you want the email reminder five minutes beforehand, a day before, and you can actually add different reminders. So if you're like me, very forgetful and you wanna add different reminders like a day before, two days before, you can always select the add reminder and it will give you as many reminders as you want. One day before, 30 minutes before and 10 minutes before. Okay, now all you have to do at the bottom right is click your save button. And now it's saved. So let me go back to your appointment scheduling slides. So all of these steps are here. It looks like I'm running out of time. So I'm just going to go to the next part since I just demonstrated all that is about reserving appointments. So after completing your bookable appointment schedule, it's gonna send you the booking page link which looks like this number two here. And so you wanna select this white share button, not open booking page. And then you're gonna share a single booking page which is this picture number three. Then you will select the white button on the bottom left corner of the pop-up box that says copy link, okay? So then you can go ahead and copy that link and paste it into an email, a webpage, a Google classroom at any place that you would actually pay something and anyone can reserve a slot. It's not just for students who have a school account, it can be anyone. Now, it might be tricky if you did that, reserve it by email notification. So you probably would want to deselect that so that anyone can. The only requirement is an email address is required. So if you wanna reserve a time slot. And then we can also add guests to appointments and I can show you that right now by... So if you notice in this, when you see your calendar if it has a grid, this four boxes, that is actually your testing schedule or your booking schedule. So the one that doesn't have it is the actual appointment. So the appointment slot with the grid icon is the booking appointment schedule and the one without is the actual booking that the student did with you. When you do this, when you do this, you click on or select the appointment, the student appointment, the editing tool will pop out and you can see on the right hand side, you can add guests and you're gonna add them by their email address. So your name and whoever booked the appointment will already be on there. Your name and the student's name or whoever booked the appointment will already be on there. But now you can add guests. Maybe you wanna bring in a teacher. Maybe you wanna bring in a support person. Maybe you wanna bring in a counselor, something like that. So you can add guests to the appointment to join you. So let me go ahead and show this real quick. This is what it looks like in real time. So you would click this share button, like we said. So it popped up, share this and then you want the single booking page and you copy your link and then you click done. So that's what that looks like. If you wanted to edit this, you would have this pencil on the left of the delete button. You would select this icon and your settings would then come back and you could go ahead and go through it and change the date as you see fit. So that's how you do it. And now when it sends it to the student, what'll happen is, let me click on this. I'm gonna open the booking page so you can see what it looks like. So I will, when you send it and they open the link, this is what they see. They're going to see your name and photo if you have it and then they'll be able to just select any timeframe and fill out that booking form and then click book. And then it's confirmed you can go ahead and you can cancel the appointment if you want or if you close it, notice that that time now went away. So that's how you do it. I hope you'll find it helpful and thanks for coming today. Thank you, Audrey for all of that great information. Otan would also like to thank all of you for coming to this tech talk. If you have an ed tech tool or some tips that you'd like to share with the adult ed field, email your idea to support at otan.us. We also encourage you to subscribe to the OTAN YouTube channel where you can view archive tech talks as well as other OTAN videos. You can also contact OTAN for additional services including professional development at your site. Just visit the OTAN website at www.otan.us or contact us by phone or email. And thank you again for watching today's OTAN Tech Talk.


Next, we will look at the profit volume ratio. So, profit volume ratio is another useful tool for finding the break even point for sales, especially for the multipurpose firm. So, this is the break even point in the short form is known as BEP that is of sales especially for the multipurpose firm. And what is the PV ratio? PV ratio is S minus B upon S multiplied by 100, where S is the selling price B is the average variable cost. So, the profit volume ratio is the difference between the selling price and the average variable cost. And this is PV ratio is generally used as the break even point for the sales particularly in the multipurpose firm. So, if you the selling price is S is equal to 5 unit, the average variable cost B is equal to rupees 4 unit, then the PV ratio is selling price that is 5 minus variable cost 4 upon 5 multiplied by 100 that is 20 percent. So, we can say that 20 percent is the PV volume ratio. Then we will take another example to understand this PV ratio. Suppose we have product, in two products since we are saying multipurpose firm. So, A and B will take selling price here and will take variable cost per unit and will take machine hour per unit. So, in case of selling price in case of A this is 2, in case of B this is 2.5, variable cost per unit is 1 in case of A, 1.5 in case of B, machine hour per unit is 2 in case of A and 1.0 in case of B. Now, to find out the PV ratio for A, PV ratio for A will take selling price minus variable cost upon selling price multiplied by 100. So, this will 2 is the selling price minus 1 is the variable cost per unit divided by 2 multiplied by 100 that gives to 50 percent. So, 50 percent is the break even level price that is through the PV ratio. So, now, to find out the PV ratio for machine hour because if you look at machine hour required is 2 unit. So, in this case to find out the PV ratio for each machine hour, PV ratio for A for each machine hour this 50 percent will be divided by 2 that is 25 percent. So, 25 percent is the PV ratio and that leads to the break even sales price of the break even price for the sales. Similarly, how to find out the PV ratio for B? PV ratio for B is again S minus B by S. So, there it comes to 2.5 minus 1.5 by 2.5 multiplied by 100 that comes to 40 percent so PV ratio for B will be 40 percent. And since this is the machine hour is only 1 this is the total PV ratio in term of the machine hour also. So, in case of A the PV ratio is 25 percent in case of B the PV ratio is 40 percent. So, in this case B will be preferable to A or the in the other word we can say the firm will get more break even or may get may more profit from B as compared to the A because the PV ratio is more in case of B as compared to A and PV ratio is nothing, but the difference between the selling price and the selling price and the variable cost of production. So, the larger is the larger is the variation more profit to the firm because that that finally, lead to the break even price for the sales for the typical product. Then we will discuss about the so break even point in the sales value is calculated by dividing the fixed expenses F by the PV ratio. So, BEP sales value is the fixed expenses by PV ratio which is equal to F by S minus B divided by S. Then we will talk about the margin of a safety and margin of safety generally represents the difference between sales at the break even point and the total actual cost. So, the difference between the sales at the break even point and the total actual cost leads to the margin of safety. There are three measures to this margin of safety. One that is profit multiplied by sales by PV ratio, second profit by PV ratio and third margin of safety is S A minus S B divided by S A multiplied by 100 where S A is the actual sales S B is the sales at BEP. So, we will just take an example to understand this margin of safety. So, here total revenue is 10 Q, total cost is 50 plus 5 Q and S A is 20. So, given the total revenue and total cost function sale at BEP S B can be if you look at so total revenue is 10 S B and total cost is 50 plus 5 S B because this is the sales at the break even level of output which is equal to Q. Then at the break even point total revenue is equal to total cost. So, substituting the value of total revenue and total cost that is 10 S B is equal to 50 plus 5 S B. So, 10 S B minus 5 S B which is equal to 50. So, 5 S B is equal to 50 and S B is equal to 10. Now, in order to find the margin of safety we know that S B is equal to 10 and S A is equal to 20. So, following the third measure that is S A minus S B by S A multiplied by 100 will give us the margin of safety. Here S A is the actual sales, this is the break even sales. So, corresponding to this we will get marginal load of safety is equal to 20 minus 10 by 20 multiplied by 100 which is equal to 50 percent. So, this margin of safety can be increased by increasing the selling price provided that demand of four product is in elastic because if it is elastic then small change in the price is going to get influenced by the consumer and they will change the demand pattern. So, the margin of safety is the difference between the actual sales and the break even sales and break even sales can be find out from total cost and total revenue. So, margin of safety in the last place if you look at the more is the actual sale that is S A that more is the margin of safety. So, it can be increased by increasing the selling price the sales can be increased by increasing the selling price provide the sales are not seriously affected and it can happen only when the demand for product is in elastic because if it is elastic whenever there is change in the change in the price selling price that leads to affect the demand and in one way that will reduce the sales the quantity of the sales. Also this margin of safety can be increased by increasing the production and sales up to the capacity of the plant even by reducing the selling price provided the demand is elastic. So, it can be also increased by increasing the production and sales up to the capacity of the plant even by reducing the selling price and here again the precondition is the demand has to be elastic. So, this for a margin of safety the other method includes reduction in the fixed absence in order to increase the margin of safety. The other methods to increase the margin of safety is to reduction in the fixed expenses, reduction in the variable expenses or having a product mix with a greater share of one that assure a greater contribution per unit which has a higher PV ratio. So, either the margin of safety can be increased by reducing the fixed expenses, reducing the variable expenses or using a product mix which gives more contribution that is more than the fixed cost and also a higher PV ratio the profit volume ratio. So, margin of safety can be increased by reducing the expenses, both fixed expenses and the variable expenses or greater share of contribution or the greatest value of the PV ratio. So, when it comes to this break even analysis whether it is contribution analysis through the PV ratio or through any other method if you look at it can be only applied to the single product system or cannot be applied when the cost and price data cannot be determined beforehand. So, there is a limitation to the break even analysis if it is applicable only to the single product system and it can be applied only if the cost and price data is known beforehand. Then we will talk about the concept of a learning curve and if you look at we are going on discussing one fact across this session that this long run average cost curve is u shape and this long this u shape the decreasing part is because of economies of scale, the reduced cost is because of economies of scale and the increasing part is because of diseconomies of scale. But if you look at there may be other reason also to understand that this reduced cost it not only because of economies of scale or the increase cost is not because of diseconomies of scale there may be few other factors which influence the increasing and decreasing average cost of production when there is the scale of output increases. So, one of the fact here is that the acquire knowledge and experience help firms in reducing the long run average cost curve most continuously. The learn about time to get work done in shorter period of time reduce cost of production and increase factor productivity like if you take the case of labour if you take the case of capital I think when you the opinion of the firm or the opinion of the learning proponent of learning curve is that when the firm does the same type of production over a period of time they get experience in doing that and that leads to less time that leads to less cost of production and increase the factor of productivity. So, the efficiency of the inputs increases efficiency of the firm increases and that is the reason the cost is decreasing. So, the learning cost the theory behind learning cost is that it is not because of economies of scale or the diseconomies of scale there is one more fact that the knowledge and experience in doing the same kind of activity they learn through it and that leads to the average cost decreases. Suppose someone new to operate that machine and if the labourer is operating that machine over a period of time he gain the skill he gain the exposure to operate this operate the machine and he is doing the whatever the time he is requiring time requires to operate the machine that has come down and maybe also the the productivity of the labour has increased because in the same time he can do something else now. Similarly, if you look at the process itself the process itself initially there is a time to fix up or set up the process, but in the long run when you do it over a period of time the process is set up the system is in place and that leads to less time and the inputs become more productive. So, in case of learning of the opinion is that the acquire knowledge and experience help the firms to reduce the average cost and that leads to the shorter period of time reduce cost of production and increase the factor productivity. And this learning curve is one that is applicable to the cumulative output and it is not the output may be average output the major difference between the long run average cost curve and the learning curve is in case of long run average cost curve we take the average cost of production with the increase in the scale of output, but in case of learning curve we take the cumulative output that is the total product from the beginning of the scale. So, if you look at this curve is showing a declining trend in the. So, if this is the learning curve it is decreasing, but in case of long run average cost curve it is always increases after a point, but is in case of learning curve in case of learning curve it the cumulative cost cumulative output in case of cumulative the average cost goes on decreasing there is no shine to increase the cost of production when the output in the scale of output increases. So, this curve shows a declining trend in long term average cost of production and learning curve is different from the conventional long run average cost curve as long run average cost curve gives the cost of plant wise production learning curve gives the average cost of cumulative output the total output right from the beginning of a production of the commodity. Now, we will say we will take a small numerical to understand this learning curve or we will do the we will estimate the unit of the learning. So, if you take a cost function where c is equal to a q to the power b, where c is the average cost and q is the average cost for unit of output q, a is the cost of first unit of output and b is the rate of decrease in rate of decrease in average cost with cumulative increase in output the value of b is negative because decrease in q will always increase the or may be decrease in the cost with cumulative increase in the output the greater the value of b the faster is the decrease in the average cost. So, this learning of this c a q b can be converted into a logarithmic form that is c a q to the power b. So, this can be log c is equal to log a plus b log q. So, b gives the slope of the learning curve b gives the slope of the learning curve a is the intercept. So, if you look at this if you take this that log c is equal to 5 minus 0.4 log 100 suppose if assign the value of q is equal to 100, b is equal to 0.4 and a is equal to 5 then this equation will be log c is equal to 5 minus 0.4 log 100 and since log of 100 is 2 then we can again reframe this equation as log c is equal to 5.0.4 multiplied by 2 and which is equal to 5 minus 0.8 which is equal to 4.2. So, if you take a antilog of 4.2 that comes to 15849 and if you summarize this then unit cost of 100 unit of output is equal to 15849 is the cost of production. So, similarly you can find out for any level of output that is you can find out the cost for any level of output simply changing the value of q and getting the value of log a and log b. So, if you summarize whatever we have discussed today taking this break even analysis learning curve and PV ratio the business manager must plan for the long run administration of the cost revenue and profit and because in the long run firm will be in a position to expand the scale of production by increasing all inputs. So, it is a kind of scenario analysis it is a kind of long term horizon how the firm because the firm's decision firm's business decision depends upon the cost and revenue because that gives us the profitable and non profitable range of output. So, in the long run with the increase in output the total cost of production first increases at the decreasing rate then at an increasing rate and as a result the long run average cost initially decreases until the optimum utilization of new plant capacity and then it began to increase. And the cost and output relation always follow a laws of return to scale and that is the reason the long run average cost curve is show a u shape. So, the decreasing part of the long run average cost curve is because of economies of scale and long run increasing part is because of diseconomies of scale. So, the cost are assumed to plan their production activity much better if the level of production for which total cost and total revenue break even is known and this implies the profitable and non profitable range of output. So, we analyze the break even analysis for both linear and the non-linear total cost and revenue analysis. So, the break even analysis or the profit contribution analysis is the analytical technique used in studying the relationship between the total cost, total revenue, total profit losses over a range of stipulated output and basically there is a technique of previewing the profit prospect and tool of profit planning. So, if you remember in case of linear cost and revenue we get one break even level of output where the profitable range of output is unlimited. But in the real life this is not possible to get a unlimited profitable range of output that brings the non-linearity in the total revenue and total cost function and in case of non-linear total revenue and total cost function we get two break even level of output. So, rather than getting a profitable range of output we get a upper limit and lower limit for the profitable range of output. Then we discuss about the PV ratio and the PV ratio which is specifically deals and the contribution analysis which is specifically deals with the incremental revenue and the incremental cost. Incremental revenue and incremental cost comes to the fact where we cannot do the analysis with the marginal cost and marginal revenue where the per unit change is not possible. In this case the contribution is the guiding factor for the contribution is the guiding factor for the business manager to decide the range of output. So, in case of contribution analysis beyond the before the break even analysis before the break even level of level of output the contribution is less than fixed cost that is that the firm incur loss. But in case of beyond the break even level of output the contribution is more than fixed cost and that leads to the profit. So, here in case of incremental analysis the guiding factor is the contribution and that helps the manager to decide whether to go for that range of output or not. And finally, we discuss about the learning curve which is alternate to the economies of scale as the reason behind the decreasing cost of production. And here it is different from the long run average cost curve because the learning curve is one where the average cost goes on decreasing for the cumulative output and that achieve through the experience that achieve through the productivity of the factor input. So, we will continue our discussion on this cost again typically on economies and diseconomies of scale in the next session. And for preparing this session these are the session references that is being exclusively followed for this.


you wasted no time jumping on a first pitch there by Trevino did you have your mind made up that you were gonna be aggressive early in that camp yeah I didn't want to strike out again so I was looking for a fastball the Yankees this season is there a better feeling than getting it done like that here at Yankee Stadium no no this is the best it's awesome the A's have given you trouble so far this year what is it about the matchup that has been so tough on you guys I don't know I don't know we just glad we finally got to beat them taking it to us a little bit and hopefully we can go out and win a series tomorrow a little bit of a hiccup for the bullpen but Chad Green came in followed by Chapman and Geeran what did you think of the job they did to shut things down yeah everybody did a really good job coming in they had some traffic a few innings and to get out of there and limit them and keep us in the game is a great job more power from Sanchez and judge what does it mean to this offense when they're going like they are right now yeah I just I mean we weren't getting a whole lot done today so a couple big swings there really kept us in the game and just a really good hitters DJ thanks for the time thank you


OK, so now without further ado, it's my pleasure to introduce the last speaker for this geometry festival, Peter Kronheimer from Harvard. He's going to talk about instanton homology for knots and webs. Thank you. This bangle. I'm very pleased to be here. I'm going to talk about some topics which go back quite a long way, but which have met and interacted rather more recently. Would you speak a little louder? No, that's for the taping. That's just taping. That's fine. I'll try to talk a bit louder. I will come over here, but then get back closer as well. So the two topics of the talk, it's really mostly the first of these that I'm going to be talking about. This is instanton homology as an invariant for knots. I'll also be talking, rather less, about Havana homology. And one of the reasons I'm interested in both of these right now is the way they interact. Instanton homology was originally thought of mostly in the context of the structure group SU2. The one can generalize it to SU where, and I'll talk a little bit about that. Havana homology also has a certain n equals 2 origin. And when one generalizes Havana homology, one obtains, amongst other things, Havana-Rozansky homology. So since both of the items on the first two lines have generalizations, it's natural to look for relationships between instanton homology for SUN and Havana-Rozansky homology. I'll be talking a bit about that too. And this is joint work with Tom Wufger at MIT. Before going on to the next slide, perhaps I wanted to say a tiny bit about instanton homology and how it first appeared in the work of Andreas Fleur. So the starting point here is a close-oriented smooth three manifold Y. And over it a vector bundle, which I think I might suppose is the trivial bundle Cn times Y with the standard Hermitian inner product on the fibers. And then one looks at unitary connections in the fixed vector bundle over Y. So all unitary connections, these form an affine space, infundential affine space on which there's the churn-Symons functional, CS for churn-Symons function on the space of connections with real values. My A is gradually turned into a U area. So the key properties of the churn-Symons functional are critical points, not exactly the flat connections. So the flat connections are rated the representations of the fundamental group in the unitary group. If you introduce a Romanian metric on the three manifold Y, then there's a L2 inner product on the tangent space, this affine space of connections. And then you can consider the formal gradient of the churn-Symons functional and the formal gradient flow for a path of connections A of T. The gradient flow equations, the formal gradient of churn-Symons is the matrix-valued one form, which is the hodge star of the two-form curvature of the connection A. So this is the formal gradient flow. And flow's construction of instant on homology rests on a key observation, which is the following. If I've got a one parameter family of connections on the three manifold Y, then you may look at it as defining a four-dimensional connection, a bundle, connection in the bundle over the cylinder R times Y. So if you think of A of T, this part of connections, instead of defining a single connection A on the cylinder, this equation, equation dagger, tells us that this connection A has an anti-self-dual curvature. So the curvature two-form in four dimensions. Being a two-form, you can split it up the way the two-forms on a four-manifold, manual four-manifold split up into the self-dual and anti-self-dual two-forms. And anti-self-dual curvature is the condition that the self-dual part, F plus, is 0. So the construction of this formal gradient flow and its relation to the representations of the fundamental group of Y is the germ of Flur's instanton Flur homology. And through this observation, one sees that it's related to something which is the anti-self-dual Yang-Mills equations, solutions of which, either on a four-manifold or on R4, often called instantons and this tied Flur's work closely with the then quite recent work of Simon Dahlsen in using instantons to study the topology of smooth four-manifolds. So oops, either 0 or 1 instant. Right, that's this time I haven't skipped. So here the situation is a three-manifold Y. I'm going to talk about an invariant of knots and links, which may lie in a three-manifold, but will usually, for me, lie in R3. This is a variant of the Flur homology, which goes back here to 1986, Flur's work. What is this instanton homology going to be? It's going to be a finitely generated convenient group for the unknotted circle in R3. It'll be Z plus Z. For the trefoil, I've written down the answer. It's four copies of Z and a Z mod 2. One key property that this instanton homology for knots is going to have is, here at the bottom, this unoriented skein relation. So you're supposed to imagine three different knots and links, which outside of a single ball are all the same, but inside that ball, one of them looks like this, like here, and the other two are obtained by this modification. So this is the unoriented skein relation between three knots. That's a bit more symmetric in three dimensions than it looks if you, instead of thinking of that two-dimensional picture there, you think of a three-dimensional ball and think of a tetrahedron there with its six edges. So there are three opposite pairs of edges. If you focus on this opposite pair of edges, or this one, or this one, you obtain three different pairs of arcs inside the free ball. That's another way of looking at what's going on here. So as well as defining this flow homology for three manifolds in 1986, Fleur also defined an instanton homology for knots, one of his last papers in about 1989. Our definition is slightly different from the one that Fleur used, but it actually gives the same answer for knots. So this really is an invariant of knots, which can be seen to go back quite a long way. In Fleur's work, he pioneered looking at exact sequences coming from skein relations in this context. They're not exactly this unoriented skein relation here. Although Fleur's work on Fleur homology has been very influential, many big developments, this particular aspect of it, this instanton homology for knots I think was not a lot developed. There's an exposition of some of those ideas from the first last papers by Peter Bram and Simon Donelson, which appeared in the Fleur Memorial volume in 1995. But very little else has been written based on this 1989 paper. So how do we define instanton-fleur homology for knots? So it starts with the knot group. I sort of define it in a rather backwards way compared to the sketch I gave here for the original instanton-fleur homology case. It starts with the knot group pi, the fundamental group with the knot complement, in which there's a distinguished element or distinguished conjugacy class, at least if it's a knot rather than a link. That's the original conjugacy class. You choose a base point for your pi1. Choose a path from the base point to the knot. Just go around that linking circle and go back. Different paths will give you a different element of the conjugacy class, but I'll write it as if it's a distinguished element here. And I want to look at homomorphisms from the knot group into, in this case, SU2. And I'm going to put the following constraint on these, that this written element or every element in that conjugacy class should map under row to this element of SU2 or something conjugate to it. That tilde sign means conjugate 2 for me here. And I want to write curly r of k as the set of all such homomorphisms, row. That's the starting point of what I want to do. So let's see what that actually does for the unknot k. So the knot group pi1 and the complement is a z for an unknoted circle k. The meridial curve is a generator for that z. So a homomorphism row from pi1 to SU2 is just determined by where m goes. And we've insisted that m go into the conjugacy class of i0, 0 minus i. So this is a little sketch of the 3-sphere, which is SU2. i0, 0 minus i lies on the equatorial 2-sphere. And its conjugacy class is that 2-sphere. So to specify a homomorphism is just to specify where m goes. It goes somewhere on this 2-sphere. So r of k is just a copy of that 2-sphere. So that's this representation variety for this unknot. For the trefoil, it has a non-Abelian knot group now. There are some Abelian representations, those that factor through the h1, which is z. That's, again, this 2-sphere. But there are also some irreducible representations, which are parameterized by a copy of SO3. If you want to think concretely about that, you can think of the presentation of the knot group by the wording of presentation, which is a standard way to write it. For the trefoil, the wording of presentation has three generators, x, y, z. You write it down. This and the three cyclic rotations of this relation are these are the relations of the knot group for the trefoil knot. And each of x, y, and z is an element of the original conjugacy class. So each of those has to go to some point in the 2-sphere. So if you're looking for elements of curly r, you're looking for three points in the 2-sphere. This relation says that if I reflect y through the point x, I'll get the point z. And either x, y, and z can all be the same point. There's a 2-sphere's worth of those. Or x, y, and z can lie on an equilateral triangle on a great circle. And there's a SO3's worth of those. So they're actually very easy to sort of see geometrically these representations. Normally when you talk about representation, the linear representation of a group, what you really mean in terms of homomorphisms is that that would be homomorphisms to the general linear group modulo conjugation by the linear group. But that's not what I'm doing right now. I was talking about just the raw set of homomorphisms not dividing by conjugation. Because if I'm looking at flat connections up to isomorphism, I more naturally get homomorphisms modulo conjugation by the linear group. The reason r of k is going to play a role for us rather than r of k divided by conjugation is the following study ad hoc looking, and indeed I think is rather ad hoc device. So given a not or link k, I'll write k-sharp for the following gadget, it's k, but then disjoint union is some remote ball, a hopflink h, the two-component link there. What's the horizontal bar? So the horizontal bar is an extra, an added extra for this hopflink. What I want, I'll talk more generally about trivalent graphs shortly, but this hopflink has actually been decorated as an extra bar, so it's actually becomes a trivalent graph embedded in three-space. And I want to look at representations of the complement of this augmented guy, which as well as having this condition on the meridian of k that they get mapped to something conjugate to this, I'll make the same insistence for the meridional curves on the hopflink. But I'll ask that this meridian, the meridian of the bar, mapped to this element minus one, zero, minus one, this non-trivial central element of SU2. The reason for choosing in that particular gadget is that there's a unique, essentially up to conjugation, a unique solution on that hopflink to that problem, which if you think of the Caternians i, j, k as elements of SU2, up to conjugation, this is m1 has to be i, m2 has to be j, the commutator is minus one, and that's this element around the bar. So because there's a unique irreducible solution here on this hopflink part, and because pi1 here is just the free product of these two guys, if I look at the original representation variety of k, without dividing by conjugation, that's the same as looking at this more complicated representation variety and dividing by conjugation. So I've enlarged my k to some more complicated gadget. The complicated gadget divided by conjugation is just the original guy without conjugation. So that's, I'll really just be talking about this guy, but the reason it's actually relevant is that we're really doing something a little bit more complicated. So I can think now as representation variety as something a little bit more natural is the space of representation's modular conjugation of this other gadget. So I can now think of it as a modular space of flat SU2 connections on the complement in S3 of this link k together with this hopflink gadget. Flat connections up to isomorphism. So I can now think of that as critical points of this Chern-Simons functional on a certain space of connections. If I take the connections modulo-automorphisms modulo-gauge, I'll write curly B for the connections modulo-gauge. The Chern-Simons functional is real valued on the space of connections, but on the connections modulo-gauge, it's not single value anymore, so it takes values should be normalized in R over Z. So now this representation variety curly R of k has begun to come into the realm of the setup used by Fleur. So I want to define now instant homology of the knot k, and I want to define it as the morse homology of this functional on the space of connections following Fleur's constructions. Technically to do this, after all, I'm looking at the complement of a knot in a three-manifold. A convenient way to do this technically is to view the three-sphere with this embedded knot or this trivalent graph, view it as an orbifold with cone angle pi, so that instead of two pi, the cone angle is pi along these edges, the edges big, but the knot k, the knot or link in S3. And I can view the space of connections as a space of orbifold connections in a bundle with structurally SU2 or SU2 over plus or minus one. So this is the picture of the knot k. It's in a three-dimensional space, but in the normal directions, it's a cone with cone angle pi. And the holonomy around this single locus for the SU2 connection I'm considering will be I00 minus I. If I pass then to the, locally to the smooth branch double cover branch along k, the holonomy will be minus one. So as a SU2 over plus or minus one connection as an SO3 connection, it's that there's no apparent singularity there. It's a sort of orbifold SO3 connection. Does that work for the singular point on them? It does. I'll actually talk about the trivalent graph again later. So now we're all set to sort of follow the outline of Fleur's constructions. We can think of the gradient of the Chern-Simons functional using an L2 in a product. Using orbifold, Romanian metric on this three-dimensional space. Well, here it is. So this guy, although it's a trivalent graph, this thing is also naturally an orbifold. If I take the three-dimensional ball with the three coordinate axes passing through it and divide by the client four group. So the group generated by these matrices, then the quotient is, again, homeomorphic to a ball. But it's a homeomorphic to a ball containing three rays. So that's the appropriate orbifold picture for the trivalent graph. So the critical points of the Chern-Simons functional are exactly our representation variety. And the gradient flow lines are sort of orbifold instantons, orbifold anti-self-tual connections on this cylinder. So I didn't talk much about, I began this talk, I talked about the Chern-Simons functional in the context of Fleur's original constructions. But the outline of what one is supposed to do after that, although the details will vary every time, but the plan is to define a homology theory as a Morse homology. So the standard steps are to perturb the Chern-Simons functional. So the Chern-Simons functional has as its critical points this representation of a R of K, which may be positive dimensional. But after perturbation, the critical points will be isolated and satisfy a Morse condition. I missed what you said when you explained what the tilde over R of K be. I didn't say actually, but nor does it say on the transparency. But the tilde R, I think, must be my notation for the perturbed representation variety. So R of K is the critical points of the Chern-Simons functional. The Chern-Simons functional is then perturbed a little bit, add a small perturbing term, which I won't really discuss. And then there's a new set of critical points, which is now a finite set as in Morse theory. And although there isn't a gradient flow defined by this functional, flow's idea is to construct a chain complex as one would in Morse theory, whose generators correspond to the critical points and whose boundary map counts is defined by counting flow lines, solutions of this equation, which are asymptotic one end to one critical point and at the other to a different critical point that defines a boundary map on this chain complex. Now a particular example that the definition of I sharp of K is that it's the homology of such a chain complex. Some very simple examples, it can turn out that there are no relevant parts of differential, no flow lines between different components of the original representation variety. It may be in some situations sort of perfect Morse function situation, that what this homology group ends up computing is exactly the ordinary homology of this representation variety. That's exactly what happens for the unknot, for example, we saw that the representation variety for the unknot was a copy of the two-sphere. Well it turns out that the instant on homology of the unknot can be thought of as just the homology, ordinary homology of that two-sphere. It's a copy of Z plus Z. For the trefoil knot, the representation variety was a two-sphere and then a copy of SO3, these were the reducible, these are the irreducible representations. The ordinary homology of S2, this joint union SO3 is a Z4 and a Z mod 2. And that is indeed the instant on homology of the trefoil knot. I haven't talked about the gradings here, and of course the ordinary homology is a graded group. The instant on homology as we've defined it is not graded. There is a grading, a cyclic grading by Z mod 4, but not an integer grading. And this isomorphism here, you wouldn't expect it to preserve the grading, you'd expect some shifts, like as you would even in a familiar finite dimensional morse theory. So this is a slide you've seen before repeated here. We've talked about instant on homology. I haven't talked about why there is this unoriented skein relation. That's an interesting story. It's closely related to a surgery exact triangle which Fleur proved for the three-manifold instant on homology that he defined in 1986. But I'll explain briefly this invariant of knots and links, the fact that it's more or less by construction of finitely generated Abelian group, and we saw briefly why the unknot and the trefol give you these two. So Tom Moffko and I were playing around and we wrote this definition down quite a few years ago now and we computed these two examples. And then there's something which is quite striking. There's something else which is gonna feature in this talk now, which is Havana homology. It was defined at the turn of the millennium by Mikhail Kovanov. It's an invariant of knots and links in R3. Can't be generalized in any easy way to knots and links in general three-manifolders, R3 or S3. Like instant on homology, it's a finitely generated Abelian group. And it so happens that for the unknot, it's Z plus C and for the trefol, it's Z to the fourth plus Z mod two. And more or less from its definition and construction, I'm not gonna talk much about the definition of Havana homology, but this unoriented skein relation is satisfied by Havana homology. If you have three knots related by this unoriented skein relation, then there's a long exact sequence relating their Havana homologies. So this was sort of defined by Havana off in 1999 or so. And by the time Tom Rufka and I were trying to play around with instant on homology and do these calculations, the Havana homology of the trefoil was I think well known to people who knew anything about Havana homology that didn't include us at the time. The similarity is very striking and it's particularly striking in that these instant on homology and Havana off whoops homology would have such different origins. So Havana homology grew from the Jones polynomial which was defined in 1984. It comes from the world of quantum algebra. The definition of Havana homology is very algebraic. You can present it to smart undergraduates quite easily. Instant homology is defined at a similar time to the Jones polynomial. There's two big developments in the 1980s defined using gauge theory through connections. And although there were some terms in common, for example, the Jones polynomial was considered by Whitton as coming from the Chern-Simons functional. Instant homology and the Jones polynomial were really rather separate things for a long time. So the fact that they turn out to be so closely related is a relatively recent discovery. So the fact that they're the same for the trefoil and the unknot is a consequence of a particular relationship here. This is the existence of a spectral sequence. The spectral sequence which starts with Havana homology is its E2 term and which abuts to the instant homology that we've been discussing. The reason that they're the same for the trefoil is that nothing happens in the spectral sequence after the E2 term. I'm not gonna define Havana homology in detail, but given a planar diagram of a knot or link with n crossings, then you can smooth out these crossings, they're n of them, each one can be smoothed out in one of two ways, like so, like so. And you'll obtain two to the n different circles, smoothings of this knot diagram.  So for those of you that have sort of seen Havana homology either firsthand or presented in a talk before, what one does, a bit of algebra to compute Havana homology is you draw the n dimensional cube with its two to the n vertices here, n is three. At each vertex, you put a vector space determined in a simple way from this bunch of circles.  It turns out that you can compute instant homology from exactly a similar picture. It's just that the instant homology has some additional differentials which will run not just along the edges, but also along the diagonals and great diagonals. So from that point of view, Havana off, excuse me, differential on the same complex as some D one and for the instant on case, we've got a D one. In fact, it's only the odd diagonals and the next one in that picture would be the great diagonal D three and then there'd be a five dimensional great diagonal D five and so on. So there's a larger differential on the same complex which computes instant on homology. So a consequence of the existence of this spectral sequence is as an application that Havana homology detects the unknot. That is to say, you can tell whether or not it is modded by computing the Havana homology. If you get Z plus Z, it's the unknot and if you don't, it's not. This is an attractive application because Havana homology is such a sort of basic simple thing to compute. I mean, it's a very ungeometric environment. It's an open question whether the Jones polynomial detects the unknot. Some sense, this is a related question, but I don't expect that it's a useful step in the direction of that unsolved problem. The existence of this spectral sequence and do that cube picture, it's really, it's an idea related to a similar idea of Osvat and Sabo who considered the Hagar homology of branched double covers and really what's going on there, but the core, mathematical core of it is the fact that instant homology satisfies this skein exact sequence. It's like generalization, that picture's really what makes that work. So why does that imply that the Havana homology detects the unknot? So from the spectral sequence it immediately follows that the rank of the Havana homology is at least as big as the rank of the instant homology. An earlier work shows that the instant homology detects the unknot. The rank is at least two always and strictly greater than two for non-trivial knots. That in turn comes down to really an interesting state about instant homology. It's, you can think of it as, it's a corollary really, that if you have a non-trivial knot then the representation variety is not gonna just be this two sphere. It must contain some other stuff. There must be some irreducible representations. The reason one can prove something like this for instant homology is because it's really a three-dimensional thing. You can put it in an arbitrary three-manifold. You can cut your three-manifold along incompressible surfaces. You can glue your three-manifolds. You have all this flexibility of three-dimensional topology. In particular, a notion of sutured manifold decompositions which are introduced to topology by Gabai and were used in the context of Haigard for homology first by Kajini and then Ni and Yuhaj. So it's this extra flexibility of instant homology that allows you to do this. We defined our representation variety using SU2 connections and we asked that the marital element be mapped to the equatorial two-sphere, the conditional class of I00 minus I in SU2. I could have defined it and it would have been more equivalent to ask that the marital element map to this element of U2. So multiplied by I. Think of the representation variety defined in a slightly different but equivalent way. Having done that, you can see one of many possible ways to generalize this story. Suppose I want to replace N equals two by larger N. I think of this as the essentially only interesting involution in U2. Let's look at the involutions in UN.  So let's write sigma sub K for that involution in UN. It's a reflection in a N minus K plane. So now I need a not K in R3 and a choice of weight lowcase K, somewhere between one and N minus one. And let's look for representations of the not group in UN mapping the marital element to this chosen involution. So we've got a choice of K there. Previously that would have been N equals two and K equals one. So there's a representation variety we can consider here just as we did in the previous case. It turns out it's profitable to generalize this picture a little bit to trivalent graphs rather than embedded curves, which would be the not K. So suppose I have a trivalent graph and a weight on each edge. So here there's a single weight on the not K. We've got now a graph with edges and a different weight K for each edge. So then you're looking at a representation variety which was also recently considered by Andrew Lobb and Raphael Zetner. I've drawn a sort of example here. So this is a trivalent graph. I've associated a weight to every edge. Not arbitrarily, at every trivalent vertex here there are three weights K1, K2, K3. And in some ordering I haven't written down the condition. In some ordering I've always arranged that K1 plus K2 is equal to K3. So, although there may not be a global orientation if you look at it locally as a sort of flow that the flow is preserved. If the K1 and K2 are incoming the total flow K3 is outgoing. So one plus one is two here and two plus three is five at the top vertex there. And let's look at representations of pi one of the complement. I wonder if I didn't miss a, I think I've lost a transparency at some point. I want to look at representations of the complement pi again into UN such that each meridional curve there's a meridian for each edge, m sub e for the edge e should map to an involution whose negative eigenspace has got dimension K sub e. And if you look at these conditions K1 plus K2 is equal to K3. I've got a block of size K1. If I put the block of size K2 with its minus ones here and multiply those then I'll get a block of minus ones of size K3. So that's the model that's going to happen near each trivalent vertex that act on the fact that for some conjugate of sigma K2, sigma K1 times that conjugate of sigma K2 is sigma K3. I'm writing gamma here for such a decorated trivalent graph. K is a trivalent graph, lowercase K underlined as its assignment of weights. Again there's a gadget I can put out to this bit like the Hopfling of before. So I can think of actually r of gamma as representation's modulo conjugation on some slightly larger object. I can think of this again as a modulized base of flat SUN connections in some orbifold context as critical points of an orbifold of a Chern-Simons functional. And there's really very little difficulty just generalizing the previous constructions to this i sharp n of gamma. So more some ology again of the Chern-Simons functional. So trivalent graphs appeared in the generalization of Havanov homology earlier. They appeared in the work of Havanov and Rosansky on their Havanov-Rosansky homology of knots and links. They appeared, for example, in a version of the unoriented skein sequence. If I have three knots or links related in this unoriented skein way, for Havanov-Rosansky's generalization of Havanov homology, it's no longer true that there's a skein relation related, a long exact sequence relating the homologies of these guys. And if you assign weights one, one here and one, one there. There is, however, a long exact sequence relating three guys and knot or link here and knot or link here. But the last guy is a trivalent graph with a edge of multiple, a wedge of weight two there. So trivalent graphs were a natural thing to look for in the SUN situation. So then it's natural to try and compute these things in the very simplest cases for the unknot. The representation variety is going to be the conjugacy class of this involution, which is, for example, if K equals one, it's a CpN minus one. For larger K, it's going to be a grass manion. The instant homology will now turn out to be Z to the n. Previously, it was Z2, Z plus Z. What about the trough oil? The same way we looked at the representation variety of the trough oil before, we can analyze what the representation variety is, in this case, in a very similar way. If I give it weight one, I again get the abelian, the reducible representations, a copy of CpN minus one. The irreducibles, up to conjugation, it's again a unique irreducible representation, just coming from the two-dimensional representation. Moving it round by conjugation, though there's this, that S is the sphere bundle, the sphere bundle inside the tangent bundle to CpN minus one. So the unit tangent vectors to complex projective space. Well, what is the instant homology then for the trough oil? So our first guess and second and third guess are quite what we expect to see is the ordinary homology of this representation variety, as in the case n equals two. But there's a surprise which happens here. The instant homology, generalized away from n equals two, you expect to see the same patterns. But for n bigger than two, it turns out the instant non-homology of the trough oil is just z to the n, exactly the same as for the unnot. And in fact, for any not k, if I take gamma to be k decorated with weight one, the instant non-homology just turns out to be z to the n. So that's a puzzle and a potential disappointment, I'm going to return to shortly. I've mentioned Havana-Frozansky homology. We still expect I sharp of n to be related to Havana-Frozansky homology. SU2 led to the Jones polynomial, led to Havana homology. If SUn, in its n dimensional representation, there's a corresponding quantum invariant which leads to Havana-Frozansky homology. So it's a very natural thing to look for here. There are many similarities between these. There's, again, a skein sequence for both of them, a long exact sequence relating to the homologies of those three knots. Both are functorial for these interesting gadgets called foams. And the Havana-Frozansky homology, these trivalent graphs are called webs. Foams are certain singular co-boardisms between trivalent graphs. You can see, I tried to sketch an example of a foam here. The interesting point is there's one in the middle here where there are six two-dimensional facets meaning at that central vertex. I expect both to give the same answer for planar webs, that's trivalent graphs decorated with arbitrary integers by lying in the plane. But that's a conjectural statement still. Foams, by the way, are overfolds too, just in one dimension higher. If I have a foreball enacted on by the diagonal matrices of determinant plus one with minus ones and ones there, it's a group of order eight. The quotient is, again, topologically homomorphically a foreball, but it's got these strato of non-trivial stabilizers. And what that foreball is, it's a cone. Take the one skeleton of a tetrahedron in the free sphere and then take the cone on that in the foreball. So the tetrahedron has six edges. You have these six two-dimensional facets in the ball. For planar webs, I expect the answer to be related to, well, the answer, Havana-Ferozansky homology, which is a vector space whose dimension is what's the number of so-called moi states. There's a very simple combinatorial thing you can do. Looking at the representations of pi one of the complement is rather tricky in un. But if you just look at representations where everything maps to this group of diagonal elements with plus or minus ones on the diagonal, then it's just a simple combinatorial map problem of the number of minus ones has to equal the weight and at each vertex the minus ones have to join up correctly. So a combinatorial thing, you don't have to talk about un at all, but those combinatorial things are the moi states, certain assignments to the edges. So moi states are the fixed points of a torus action, diagonal matrices. So this representation value, it's Euler characteristic as the number of moi states. This is a result that Lob and Zettner proved independently. I will just spend a few moments at the end just talking about why something changes for n bigger than 2. Perhaps I'll only try and say this briefly. So if I told you that some proper Morse function on some non-compact space had as critical points, a single point and then a circle here and I asked you what you thought the Morse homology was going to be. So you wouldn't be able to tell me because you wouldn't know what the differentials were, what the gradient flow lines looked like. You might guess it's a perfect Morse function and you just get the homology of the circle and the homology of the point. But you might also imagine that there's a function of this shape with a single critical point and then also a circle's worth of critical points. In which case there'd be a flow line here and in fact the homology would just be Z. When you look at what I told you the representations of Vardy was for SUn and even for SU2, I told you the representation of Vardy was a copy of Cpn-1 and then a copy of the unit sphere bundle in the tangent space of Cpn-1. Now that's very much the picture you'd expect if you had a function on the tangent space of Cpn-1 which on the vector spaces, each tangent space at a point had this shape that had a critical point at the origin and then a critical set on the unit sphere in that vector space. So rather than get the homology of this guy plus the homology of that guy, it's quite natural just to get the homology of this guy alone as the Morse homology. So if you think that's what's going on then the question is not why for larger n is the homology so simple. The question more is why is it for n equals 2 that there's more than one? Why aren't we just getting Z plus Z in the n equals 2 case? So let's think about the n equals 2 case. This point here is supposed to think of that as being the 2-sphere. Think of this circle here as the SO3. It's the unit circle bundle in the tangent space of the 2-sphere. The whole space here is the tangent space of the 2-sphere and its homology is Z plus Z. So why is that not the answer for the instant homology for SU2? The answer is that for SU2 and SU2 alone this trajectory has a companion which you can't really see in this finite-dimensional analog but there are actually two cancelling trajectories here one of which is a rather non-trivial path of connections other which is a path of connections which you'd expect to see here in this finite-dimensional picture. What happens for n equals 2 which doesn't happen for n equals 3 is a symmetry. If I have a connection on the complement of the knot whose monodromy around this little circle is I0, 0 minus I I can multiply it by minus 1 here. So if I think of a flat bundle E over the complement of the knot I can change it to C tensor E with the original flat connection on E but on C, C is a line bundle and its homology is minus 1 on the link of the knot. This is a slightly non-trivial operation on flat SU2 connections of the sort we're considering. It doesn't apply for any larger n if I take minus 1, 1, 1, 1 and multiply it by minus 1 I haven't got something which is conjugate to this guy that these guys are playing a different role there's a symmetry here for n equals 2 which isn't present in the other cases. This was a little bit more about that situation. So I want to just end with this last thought. So the instant homology I've been talking about just comes out to be Z to the n is when the weight associated with the knot K is just 1, the simplest case. For other weights I think this guy might be something in some sense more interesting. The particular case which I think is interesting to look at is the case where p equals n over 2 which is somewhat much closer to the SU2 case from the point of this symmetry. Again, multiplying by minus 1 will map this guy to something conjugate to it at the size of the minus 1 and plus 1 blocks are the same. And that's the story of a relationship between instanton and Havana homology. Two things which trace their origins back to the 1980s and which came together about five years ago. What's the level of complexity in computing the Havana homology for a knot with n cross 6? Does it make sense to ask that? You mean in the computational sense. That's an interesting question because there's more than exponential in theory in the number of crossings n. There's the amount of time needed. But in practice, algorithms are actually quite quick and there's no trouble to compute the Havana homology of a 40 crossing knot. The algorithms are implemented by hand or by computer? No, they're computer, but there's a... If the knot is long and thin, then you can compute efficiently. For a knot which isn't long and thin, you can try and break it up into tangles which are long and thin and have sort of divide and conquer strategy. And you can you can put that... You can implement a little heuristic for that on the computer so you can compute quite fast. But still, as far as we know, the theoretical worst case is still just the naive more than exponential time algorithm. Yeah. Other questions? In that case, let's thank the speaker again.


Hey everyone. I am Ghadal Mashaqba from the University of Connecticut. In this talk, I will present our work on unclonable polymers and their cryptographic applications. This is a joint work with Ran Kaneki, Yeniv Erlach, Jenthanger Shani, Tal Malkin, Itzik Pierre, Anna Reutbert-Bermann, and Iran Traumer. Imagine we have memory devices that are unclonable and they self-destruct after retrieving the stored data. Also, if this memory stores several messages, only a few of them can be retrieved after which the device will be fully distracted. So you cannot retrieve all the messages that you see in the photo. Such bounded query memory devices can be used in many applications. Among them, we have bounded execution software known as one or K-time programs. These are programs that can be run over only a few inputs, which is impossible to do using software alone. Even in the quantum model, it was shown that without these special memory devices, it is impossible to build bounded execution quantum programs. This idea was first put forward by Goldwasser, Keli, and Rothblum, who assumed the existence of simple one-time memory devices, which imitate the functionality of non-interactive oblivious transfer tokens, which they used to build one-time programs from garbage circuits. That was an assumption without any real-world realization. The only way we know to construct these devices relies on tamper-proving a whole computation, while assuming that these sophisticated hardware tokens resist side-channel attacks and reverse engineering. So we wondered if we can build these memory devices based on minimal and better understood hardness assumptions. To achieve this goal, we joined forces with top-notch and brave biologists to find an alternative technology to build unclonable and self-destructive memory devices. We do that in a rigorous way, laying down foundational modeling and analysis of the capabilities and security guarantees that we are able to achieve. We also introduced novel amplification techniques, so we can use the weak and simple memory devices we build in provably secure cryptographic applications. Our request was inspired by recent advances in biochemistry and engineering that allowed storing digital data in the form of DNA. So a digital message can be encoded into a set of nucleotides that is synthesized into DNA material. By the way, don't worry about the biology details here, just focus on the big picture that DNA can represent digital data, and given a DNA sample, we can retrieve the original message or the digital data from that sample. However, DNA evolved to be clonable. So a sample of DNA can be replicated as many times as you want, meaning that you can read the stored data as many times as you want. This led us to consider another biological polymer. Proteins also can be used to store digital data. So in this case, the digital message will be encoded into a sequence of amino acids, which are the basic building blocks of proteins. And then this sequence is synthesized into protein material. And here, the magic has started. First of all, proteins are unclonable. The central dogma of molecular biology states that once information has got into a protein, it can't get out again. So given a protein sample, we cannot replicate it or get the genetic information out of it.  To us cryptographers, this is a biochemical one-way function, and we know what to do with hardness assumptions. We turn hard lemons into lemonade. Another amazing feature is that sequencing a protein to retrieve the digital message is destructive. So you cannot get the protein sample back after feeding it to the mass pick instrument or machine that is designed to read proteins. Also, this machinery provides an output only if the protein is pure enough. So if you feed a mix of random proteins, or if you feed this machine with a mix of random proteins, it will output nothing. Based on these observations, we propose another construction for consumable storage tokens. At a high level, as before, we take a digital message and transform it into a protein. Then we connect the target protein with another short protein sequence called header, such that this header can be recognized by matching antibodies. So knowing the header description, which can be digital information, by the way, allows identifying the matching antibodies. This header to us represents the secret key tied to our secret message. After that, we mix the target protein or the protein that represents our secret message with a massive set of random decoy proteins that are attached to different random keys. So the vial containing the mix, where here in the photo, our co-author Anna is holding one, is the consumable token. Now, to retrieve the message, again, remember, if you just give the mix to the mass spec machine, it will not be able to identify the secret protein or the secret message. This sample must be purified first. To do that, we apply the matching antibodies, the ones that match the secret header to the vial, to pull down the target protein with high enough purity. Then we cleave the header and then use the mass spec machinery to read the amino acid sequence of this target protein, which will be decoded into the digital message. After months and months of designing the token, we spend more months distilling the model that best represents biology. Our goal was to require the minimum on the biology side to produce the simplest possible construction. In particular, our consumable token can store only a small number of short messages using short keys. So here we need amplification techniques in order to deal with that. Also, the only meaningful interaction with the consumable token is by applying antibodies. So present a key, either you get the message tied to that key if the key is correct, or you will get nothing. Also, each retrieval attempt will consume part of the vial. Even when you are applying the matching antibodies, these antibodies will pull down the target protein with high amount. Also, it will pull down or consume parts of the other proteins in the mix. So there will be a degradation for the whole mix. In our construction, we designed the token in a way that under that non-technology that is available now, an honest party will be able to perform one data retrieval query. So it can apply one key and get the corresponding message if that key is valid. But to account for more powerful adversaries, maybe I don't know, some adversaries out there may have more advanced machines that allow them to use the sample to perform multiple data retrieval queries. We say that, okay, this consumable token and honest can perform one data retrieval query, while an adversary can perform up to n data retrieval queries, where n is a small integer. Also, our consumable token has non-negligible soundness error gamma. So applying an incorrect key that is close enough to one or to the correct key may retrieve or may return the secret message with probability gamma at maximum. Also, we extended our construction to build what we call partially retrievable memory. So a consumable token can store a vector of the messages using v keys. And even knowing the set of keys, the v keys, an adversary can retrieve only up to n messages out of the v messages stored in the token. After that, we put our cryptography toolbox on the table and asked two questions. How to amplify this weak device that supports constant size storage to obtain powerful functionalities that can deal with arbitrary size data? And how to do that in a rigorous, provably secure way? This took us a long journey. Our first step was building a mathematical model for the biological construction. This produced what we call the vector model in which a vial or a consumable token is represented as a vector of protein amounts. We also modeled each of the biochemical procedures performed in the wet lab as an algorithm operating on this vector. Then we defined an ideal functionality for consumable tokens with clean interfaces and formally showed how it is realized by the vector-based construction. Then we developed several algorithmic and cryptographic techniques to amplify the weak properties of the token and showed how to use these tokens in various cryptographic applications. In this paper, we show two of these applications, namely digital lockers and one-in-time programs. In the interest of time, I will discuss these applications briefly. A digital locker is simply encrypting a secret message using a low entropy key or a human generated password such that we build digital lockers that resist exhaustive search attacks. An adversary can try to decrypt the messages only up to in time, so he can try only up to in password guesses. Using our token, we were able to construct lockers that are resistant to exhaustive search attacks. This application required additional techniques to amplify the soundness error to be negligible. These techniques relied on sharing the message into U-shares and store each share in a separate token. Now instead of sending one consumable token that stores the encrypted or the secret message M, now we are sending U tokens. Also we had or we changed these tokens together to enforce sequential operation. The latter is needed to preserve the number of password guesses to be N despite sending multiple tokens, all of them tied to the same password. The second application, one in time programs, here we have a secret program or one that contains some secret data. Ascender sends that program to the recipient such that an honest recipient will be able to execute the program over only one input while a more powerful adversary can execute it over in inputs. As you might realize, this is a modified version of one time programs introduced by GKR, but it is one that is based on real-world weak memory devices rather than just a strong assumption. The core idea of our program or our construction, sorry, is to obfuscate a program containing the intended functionality F such that evaluating F over an input X requires a corresponding secret message M. So the program first will check, okay, do you have the secret message M corresponding to X? If that's true, it will output F of X, otherwise it will output nothing. And what we do is that we store the messages corresponding to the domain of F in a consumable token. And remember, because now actually this depends on having a consumable token containing or storing several secret messages that again correspond to the domain of the function. And now the adversary can only retrieve up to N secret messages from the consumable token. And so it can run this program only up to or over only up to N different inputs. We faced a problem here, which is related to the weak properties of the consumable token. A consumable token can store only a small number of messages denoted as Q here, which means that we cannot deal with functions that have domains or domain size larger than Q. In order to deal with functions with exponential domain size, we use linear error-correcting codes. So instead of sending one token, we send omega tokens. And now instead of mapping X itself to a secret message, we first encode X and take the code word, which is of length omega. Each symbol in that code word will tell us which secret message to retrieve from each consumable token. So now in order to execute the program over input X, we need omega secret messages from the omega tokens sent to the recipient. And now the obfuscated program will check that is this the set of correct secret messages corresponding to the code word of the input. If that's the case, it will output f of X, otherwise it will output nothing. And based on that, we can cover a domain size of Q to the power d plus one, where d is the dimension of the linear code. We set the code distance in a way that only invalid code words can be retrieved. So despite sending omega tokens to the adversary and now there is more cancer protein material available with that adversary, still by setting the distance correctly, we can guarantee that it's still the same guarantee. Only in inputs or only encode words can be retrieved and the program can be executed over only up to n inputs. To conclude, this work introduced an innovative real-world construction of unclonable and self-destructive memory devices. This was done along with formal treatment and showing proof of obscure cryptographic applications. For our ongoing and future work, the directions are to fall. On the biology side, we are working on a sister paper showing the detailed biological construction along with empirical results. And on the cryptography side, we are refining our model strengthening them and we are developing more cryptographic applications. Thank you so much for listening and happy to take questions. Thank you.