Hey, Doug, you're going to party the whole lot of night? You bet. I wouldn't miss that. Sounds great. Let's hurry up and wrap this up so we can get out of here and get some beer. Okay. I like wine. All right. I thought we barricaded that off. I wonder what's up with that. Ah, they probably got some bad ground or something out there. Well, let's not worry about it. You go get the air hose and I'll go bar down. Okay. In the last 10 years, in the United States' loan, nearly 200 underground miners were killed. And over 30,000 were badly injured by falls of rock and ceiling-related accidents. For a workforce our size, that's a lot of people. Why is this? A lot of this has to do with the fact that we have an ever-present hazard that no other industry has. That is the potential for falling rock. Nearly 60% of the fatalities in underground metal and non-metal mines are roof fall-related. The truth is that most, if not all of these fatalities, could have been avoided by following some basic guidelines. In this video, we'll be going through a complete mining cycle and identifying some simple steps which you can take to keep yourself from getting injured and that might just save your life. Mining is the backbone of this industrial and technical world we live in. Without mining, we would not have many of the things we take for granted every day. Everything on this planet is either grown or it originates from a mine. Mining is not for everyone. It is one of the most demanding jobs you can do for a living. It is also one of the most dangerous jobs in America today. Working in an underground mine can be particularly dangerous. Underground workers like ourselves are constantly exposed to the threat of falling rock. More underground miners are injured and killed in rockfalls than in any other type of accident. Let's go back and analyze the rockfall incident that you saw earlier in this video. Take a close look at the work area and see if you notice anything that could have contributed to such an incident. After a quick review, we'll be going through a typical mining cycle focusing on work habits which can prevent rockfall-related injuries. No two mines are alike and there are many different methods for mining. We will therefore go through a fairly generic mining cycle and highlight some basic principles which every underground miner should know and use. We'll start off the cycle just after the blast which is when the ground is the most unstable. We'll cover the importance of ventilation and wash-down followed by a close visual inspection. Identifying loose material using both visual and sounding techniques is probably the most important skill you can master to prevent a rockfall. We'll also look at some simple but crucial tips for barring down which is your first line of protection against rockfalls and we'll discuss the importance of rockfall awareness during the mucking process as well as during installation of roof support. Finally, we'll highlight alertness by all people in the mine regardless of their job description. In all the steps of the mining process, the bottom line is being alert to the potential for falling rock and constantly aware of your position in relation to the roof support which is your last line of defense. Just when everything is supported and relatively stable another round is loaded and the blast goes off, changing everything and the cycle starts again. The most dangerous part of the mining cycle is right after a blast. That's when the rock's going to be very unstable from blasting and several people have gotten hurt and even killed for not taking good care of the loose ground before they walked in there. After the blast the area should be well ventilated. Ventilation removes fine dust and particles from the air and more importantly dilutes noxious gases such as carbon monoxide and oxides of nitrogen which may have resulted from the blast. The initial entry into a blasted heading is one of the most potentially dangerous situations you'll encounter. There will certainly be loose rock near the face and often the shock from the blast will loosen other rock in the heading sometimes even causing ground support to fail. As you enter a freshly blasted heading stay beneath supported ground and carefully inspect the entire area for signs of loose rock or any shifting which may have occurred. During this initial visual inspection look closely at both supported and unsupported ground. The shock from a blast can easily cause shifting of stable or supported rock. You know when an opposite crew comes in and blasts a course is going to loosen up rock. It's just the way it goes with the territory. So every day I come through and I'll walk back while I'm watering and check my stope area to make sure it's free of any fallen rock. If you see loose rock on the floor make sure you stop and check to see if it came out of the back or rib. If it did either bar down the area or barricade it and report the situation to your supervisor. Once the initial visual inspection is done wash down the entire area. This not only improves air quality by reducing dust but also cleans the rock allowing for easier identification of loose material as well as geological features such as fractures, faults slips and bedding planes. This is a good time to look for missed rounds and bootlegs. Be sure and mark the bootlegs and wash out the explosives if there is a missed round. Otherwise the driller could potentially drill into a live charge. Once we go in after a blast first thing we do is check things out, wash things down and start barring down the back and ribs. And most miners agree that the barring down is probably the most important thing we can do to prevent rockfall accidents. Although some mines use large mobile scaling machinery and others allow scaling of the lower ribs with a mucker bucket quite often scaling is still done manually with a scaling bar. Since far more injuries and fatalities from rockfalls occur during manual scaling we will focus here on the use of a scaling bar. Scaling bars are most often made from aluminum but occasionally they are fiberglass or steel. The basic design is simply a long shaft with a chisel point at one end. The chisel point may be straight or slightly bent usually about 15 to 30 degrees. As this miner scales notice how the angle of the bent chisel allows him to force the tip into a horizontal crack while maintaining his distance from the path of falling rock. If the angle were too large or the bar too short he would have to stand almost directly under the rock in order to get the point into such a crack. The length of the scaling bar should be about 2 feet shorter than the height or width of the working area. Well I've got a long and a short bar and the reason for that is I've got a tall back 12 to 14 feet and I'm narrow so I have a short bar so I can clean my ribs because the long bar is awkward to handle cleaning the ribs. The long bar I can reach the back and stand out of harm's way because it's long enough that I can touch the rock and stand in the clear as I'm barring down. As far as safety gear goes each mine usually has their own guidelines which must be followed. Some type of safety eyewear is generally recommended since many mining tasks present a potential hazard to the eyes. Many companies require miners to wear hand and arm protection while drilling and bolting and some require such protection for scaling as well. The bottom line is learn your mine's safety policy and abide by it. The process of scaling the newly blasted ground begins here under supported ground next to the blasted area. To protect the mucking crew most mines bar down even before mucking begins. The advantage of this is that you tackle the loose rock and you can work off the muck pile and be closer to your work. The drawback is that it's hard to get good footing when standing on the muck pile and where the pile is too high some areas may be difficult to access. Before you begin barring it is wise to closely examine the work area. Take a look at it from different angles. Assess the potential for loose rock and determine if there are any large blocks of loose rock which are keyed in by smaller rocks. This is a knack which takes considerable experience to master and is not an exact science. A good way to learn this is to work closely with an experienced miner and have them help you learn the nuances of the geology in your mine. Sounding is often used to determine if rock is loose. The following example demonstrates the ringing sound of solid rock followed by the more hollow sound often made by loose rock. Although it is an excellent technique it is best not to depend entirely on sounding but rather use both sight and sound together to determine if rock is loose. Before you bar down a piece of loose rock make sure you have good footing and a clear escape route in case more material comes down than you anticipated. Think about where the scaled rock will land and make sure you are positioned out of harm's way. A long bar held at a shallow angle will help you position yourself away from the falling rock. Sometimes sounding the rock is a jabbing or poking at the rock is a good motion to knock small rocks down it works well but when trying to get bigger rocks down it is very important to work the pry tip in behind the rock you are trying to bar keeping yourself in a safe position with the bar and prying away from it. When barring down a heading it is important to bar systematically from supported to unsupported ground develop the habit of starting under supported ground and using the bar to sound all ground ahead of you as you move forward. It is very important when scaling to make sure of any unseen hazards before you go after the obvious loose rock in the back always protect yourself first. In some minds the rock lies at a slight downward angle in the direction of the heading making it virtually impossible to visually assess the back or to bar down loose slabs. In this unique situation it is sometimes possible to bar your way along a rib usually the foot wall rib and work your way back toward the face from where you can then scale your way back toward supported ground. If you use this method it is critical that you sound the ground and bar all loose material above your path as you work your way to the face. If you cannot create a safe path to the face do not go there. Barricade the area and consult your supervisor. Okay we bring the muck through in there and we like to bar down real well because right after the blast there is a lot of loose rock and we like to bar down off the top of the muck pile so that we can reach the upper ribs in the back and we go ahead and muck. Get the round mucked out, we come back in we bar down again to remove all other hazards. If you've been scaling off the muck pile and are confident that there is no loose rock left it is time to muck out the pile. In some cases if the ground is too unstable to allow for mucking without the use of ground support it may be necessary to drill and bolt right off the pile before you begin mucking. If there is not enough room to do the bolting you may have to remove part of the pile first then stop the mucker to bar down, drill and bolt the back before continuing. Some lines encourage the workers to scale the lower rib areas with the mucker bucket. When this is done it is important to make sure that the operator is well protected by supported ground. In mines that use small LHDs or that have a steep hanging wall this practice is generally prohibited and the loose rock is typically all barred down by hand. In any case when mucking is complete it is wise to recheck the entire unsupported area first visually and then by sounding and barring down with a scaling bar. This is to ensure that no rock was loosened during the mucking process. When you're installing roof support the first thing you want to make sure of is that you stay under supported ground and then periodically while you are installing the new roof support you want to check the ground around you in case anything has loosened up. Every mine is unique. That's why it is always necessary to understand your roof control plan and support the roof as called out in the plan. The plan is designed based on geologic conditions and may sometimes even vary from one part of the mine to another. Take into account that the roof control plan only specifies the minimum level of support required. If conditions deteriorate then additional support is required to maintain stable ground and to comply with regulations. Remember that no matter what type of drill you use it is likely that some rock will shake loose as you drill even in an area that has already been scaled. This is especially true of jack leg drilling and many miners have been injured and killed by rock falls while operating a jack leg. When using a jack leg watch for rock loosening as you drill. When you see a potential loose area stop the drill and bar down all loose material. When drilling with a bolter some operators will use the drill to scale down loose material around the area where they are coloring the hole. When this is done it is important that the operator be located well away from the potential path of falling rock. In mines with unstable ground this is not done as much and is sometimes prohibited since the drill boom can be damaged by falling rock. I had an instance in Montana one time and we didn't have the luxury of this little rock guard on the boom and a big one came down and rolled right down the boom and knocked off all my controls in the operator's compartment so I try to be fairly careful. When the roof is supported according to the roof control plan it is now safe to work under it and then drilling for the next round. Before you start and as you drill watch the face for signs of rock loosening especially where the roof meets the rib. Even if bootlegs are not visible it is critical to know the drill patterns so you don't accidentally drill the next round the same as the previous one. More than one driller has been blinded by drilling into a live charge. While drilling remember that even when supported according to the roof control plan material can and will come loose. It is important to develop the habit of regularly checking your roof support for signs of deterioration and loosening of rock between the supports. Rock falls don't happen just in stoves. Anybody that goes underground needs to be aware of them. They can happen anywhere in an underground mine. Every person who enters an underground mine from engineers and geologists to infrequent visitors or contractors should be alert for unstable ground. Simply the change in moisture of the rock surface due to natural evaporation or movement of ground due to mining induced rock stresses can loosen rock over time. Watch for these telltale signs of unstable ground. Loose bolts loose material around bolt plates slabbing or crushing of ribs movement of fractures or slip faces bulges in the mesh cracks in shotcrete dry spots on roof or ribs and loose material on the floor. If you see any of these conditions in your mine, stop and investigate. If you are not able to remedy the situation it is your right and duty to barricade the area and report it to your supervisor. By doing the right thing and not letting such a situation go untended you could save someone's life and it might just be your own. When I was 12 years old I had never had any underground mining experience. First day on the job we got off on the level and I was assigned to an experienced miner. As we were heading to our working heading he pointed up and said somebody ought to bar down that slab. Well, I was listening to him but we kept going. The next day he said the same thing again somebody ought to bar down that slab. Well, it was two weeks later I moved to a different part of the mine and was working through the mine that a motorman had been crushed by a slab on that level that I began to work on initially. And later on that day I came to find out that he was hurt very badly by that slab that I should have barred down. When that old timer was saying someone should have barred that down he was talking about me or himself. I felt bad about that all my life. Most accidents are preventable. Learning and using the concepts we've covered is an effective way to protect yourself from injury due to rockfalls. Let's see how many of these concepts you can recall. After the blast, ventilate the area well. Do a visual assessment to detect any ground movement caused by the blast. Then wet down the entire heading and do a detailed inspection both visually and by sounding. All loose material must be barred down when barring keep these six steps in mind. Use a bar about two feet shorter than the height or width of your work area. Learn to identify loose material by both visual inspection and sounding. Make sure you have good footing to prevent slips and falls. Think about where the barred rock will fall when it comes down. Always lift upward on the bar or away from the path of falling rock. Scale systematically from supported toward unsupported ground. Before mucking, scale as much as possible to protect the mucker from falling material. After mucking, recheck the back and ribs. Sometimes rock will be loosened by the process itself. Even though you've mucked and barred down, putting in ground support can and will loosen rock. Know and follow your roof control plan. Expect rock to loosen and watch for it. Add extra support as needed to hold ground firmly. Stop and bar down all loose material. It is every miner's job to stay alert to changing ground conditions. Watch for signs such as these in all parts of your mind. Loose bolts or material around bolts, slabbing or crushing of ribs, movement of slip faces or fractures, cracks in shot treat or bulges in the meshing, dry spots in the back or ribs, or loose rock on the floor. Injuries and fatalities are not pretty. They have a devastating impact on miners and their families. Many of us have experienced a minor injury or near miss and have developed a false sense of security. This sense of security could easily lead to devastating injury or death. The best way to protect yourself from a potential rockfall is to use good judgment in your daily work and refuse to put yourself at risk. It's hard enough having to tell your fellow coworkers that their buddy is gone, but even more difficult is telling a spouse that they've lost a loved one. Many of us have already lost a friend due to a needless accident, and none of us wants to be next.  It might just save your life.


Thank you very much. I assume you can all hear me. Okay, so this session is a beginner session We're gonna walk through the design of a an audio application using micro python and the pie board So we'll be constructing a continuous listen repeat audio application Something like you might find in a toy you talk to it and it talks back to you This is for beginners. So I'm not going to assume anything in terms of programming and stuff like that We're using obviously micro python, but the pie board and its audio skin So micro python Unfortunately, there is another track at the moment which is introducing you to micro python So most of the experts are probably there But it's a lean implementation of python 3 optimized for micro controllers, but with numerous modules for hardware control The board itself is really quite small. I've I've got a copy here But luckily I've brought a bigger version So it's basically just a micro controller with a few additional bits and pieces like LEDs and switches It's very small in terms of capabilities and in terms of RAM and processing power But it does come with really cool bits of hardware and in python All you have to do is import pyb and you've got access to pretty much all the hardware quite simply Christine spindler is running a poster session in the Sponsors Hall so she can tell you a lot more about the hardware and its capabilities We also need an audio skin the ability to record and play back the board itself doesn't come with a microphone or a loudspeaker but You can basically attach an audio skin. I've got one here on an aluminium casing that gives you basically a Microphone a loudspeaker and the ability to add your own microphone if you want to But be prepared The audio skin comes in bits. So in order to use it you're going to need good soldering skills and a very good soldering iron What we're going to be doing is recording and playing back speech So I suppose we need to understand a little bit about what speech is I have a Spectrogram a frequency domain recording of the utterance six here and basically you've got an axis which is showing you frequency so there are your high frequency parts of your six and This axis is effectively time in terms of frames, which are about 10 milliseconds long So that's 350 milliseconds worth of speech if you want to record high quality speech You need to sample a very high frequency because human speech goes up to 14 kilohertz But over the phone you basically are restricted to something like three to four kilohertz because they discovered that's all you need to intelligently understand what's being spoken So how do we go from analog to digital? Well, we're going to use an ADC of course Which is on the board and we have to consider a few items and that is effectively the sampling frequency sampling frequency And there are basic standards here Nyquist Shannon tells us that in order to capture an analog signal accurately We have to sample at twice the speed of the maximum frequency Here we've got and if we don't what we've got here is sampling at one point five times the speed and you can see the sample points Basically give you an indication.  So you have to sample quite quickly the second consideration of course is The resolution how many bits you're going to store each individual sample and the pie board gives you devices which can Record at 8 bits or 12 bits a bear in mind that a compact disc the original formats were 16 bits at 44 point 1 kilohertz 44.1 kilohertz is an odd number, but they're very good reason for it. So you need to understand The frequency is going to be capturing the capture resolution and how much time you need to record speech So how do we record on a pie board? Well, we basically create an ADC object And then we can use one of two read methods We can call read timed or just read So we just import after we've imported my pie board we create our ADC object It's on a particular pin on your board. So you just connect it to pin x22 and once you've done that you can Hand your reading you're the process of capturing samples over to Micropython and it will effectively read a Block of data for you at a frequency you dictate so you can set up a buffer and say capture 6,000 samples per second that's okay But you've got no control over the real-time aspect of this it captures and you have to wait till it's finished So building a device that you want to be continuous And be interactive that's not the best thing to use so you in my particular case I'm forced to read the ADC samples manually and It's my responsibility then to record them at a designated rate So we can capture how do we replay well, that's using a digital to analog converter and Again, there's one of those on the board. It's quite simple to set up There are two parts to the DAC You need to set the volume and there's a potentiometer on the I squared C bus and then once you've set the volume You just provide the DAC with your data buffer and it will play it back out for you. So setting the volume is quite simple as Connecting to address 46 on the I squared C bus and once you've done that You can create a DAC object You define the bit resolution that you want to play back at you have a choice of 8 or 12 here And then you can use its right timed method provide a buffer provide a timer a built-in object in in micro python and Tell it how to play the audio in this particular case normal means just play the buffer till it's finished and then stopped I think they have a circular option So you can fill a buffer with a repeating pattern and just get it to play continuously So I have mechanisms to record and play back, how do I know that it's working? Sufficiently quickly it has to be real time because I have to capture the audio at a designated rate So I'm going to rely on hardware components to do this But when I I'm also going to have software to do the actual signal processing. I say signal processing there There's not a lot you can do on the pie board at this stage But there are two ways to do it. You can either use an oscilloscope Attach it to a pin on the board and you can you oops You can use a pie board pin and give it a symbolic name And you can just set that pin high or low and your oscilloscope you can see You can measure the Duration of the code that that that you're surrounded the pin Control with alternatively you can use timer objects. So you can create a timer to Calculate to counter one one microsecond rate and just read it so The advantage of the oscilloscope is it gives me the Opportunity to put a little picture up. So here's a screen grab of a The oscilloscope on the board that I've been using and this low period here This trough is effectively the duration of the capture function This is the method that is running at 8 kilohertz in this particular case. It was recording done at 6 kilohertz So this is all the time it takes to read from the ADC and put that sample into memory and do some crude Calculations with it as you can see it actually takes quite a bit of time 104 microseconds But you can do clever things with it with the pin control and discover that using a timer costs you 20 microseconds And actually doing the read itself is a relatively expensive 50 microseconds So that gives you a clear indication of how fast you could actually sample this at you're not going to get 20 kilohertz Because it's going to take you 50 microseconds just to get the the value into your board So initial setup is basically I need a buffer to write my data into I need a function to collect the data And I need a play play mechanism So a crude diagram here. Here's my capture function. Its responsibility is just to read from the ADC It's connected to a timer and I simply Create a timer and attach a callback function. So here I could have a sample frequency of 8 kilohertz So this function will get called at 8,000 times a second It simply does a read puts it in a buffer when that's finished my play function goes and provides that data to the DAC So I have some recordings that I've made Earlier, I wish I'd said something a little more impressive than one two three in Italian, but Hopefully this will work. I'll just play back an issue an initial recording at 8 kilohertz So that's uno due a three, but you'll you'll notice it's very very noisy And that's that's at 12 bits 8 kilohertz And when you analyze the noise you get really quite what disturbed and disappointed that of your To 4,000 samples 300 or so are just noise. It's a very noisy signal on this board for a number of reasons But I'm losing quite a bit of information Just in the noise and it's unpleasant unpleasant to to listen to so there are quick ways In order to reduce reduce the noise and the simple one I put together is something that just searches for periods of silence in the recorded speech and just sets them to zero So to do that I Basically record into my speech buffer as I call it and on a second pass. I go through this speech buffer knowing Roughly where zero is again, this has to adapt This is not a constant depending on your hardware zero might not be in the middle And I go through the buffer looking for periods with zero and mark their positions And then that gives me the opportunity to do two things I can then adjust my estimate for zero because it might have changed But I can also go now go through the buffer and set all of the noise to the noisy silence to to real silence and our playback if we hear the original and Then we apply this simple algorithm Yep, so a lot cleaner There's still noise in the speech signal and you can't simply just take away those bits because The speech in the noise is still valuable So I had recordings at a killers at 12 bits. So I decided well, there is so much noise So how about recording at 8 bits rather than 12? If I play that So 8 bits gives me an added advantage and I can store more speech on the board. I'm very limited in the Java heap I've probably got about a hundred k bytes Minus whatever the operating system and the Python modules take away from me So I can only record very short periods of speech, but at 8 8 8 bits I can get about 8 to 10 seconds and and there you'll probably heard this a lot of problems with 6 and 7 say and set a and that's probably because I'm recording at 8 kilohertz. So we are we're right on the edge of Bandwidth here. So a lot of the high frequency stuff just gets lost or disturbed. I Also in so in order to compensate for that I decided well How about increasing the capture rate from 8 kilohertz to 10 kilohertz? So I'll hear the 8 again and then I move up to 10 kilohertz Hopefully the say and set a will be a little better So I've made some improvements why can't I sample at 44.1 kilohertz like I have with the compact disks Well as we've seen it it takes me too long to read from the ADC and also the power of the device Sorry the memory of the device just won't let me capture much data at 44 kilohertz even if I could sample at that rate. I Wanted some application refinements So it's okay. Just recording and playing back for the purpose of this was to impress my nephew Look, here's a little device. You could put in your cuddly toy talk to it and it talks back to you So it has to have some sort of Automatic Wrong button Automatic speech detection so you don't have to press a button for it to listen So and I wanted to be continuously listening I wanted to be able to record to the SD card this device gives you the ability to plug in a fast SD card I wanted to play with the LEDs to give you some indication that it's listening. It's recording It's playing back and I wanted to be able the very least to disable the device by pressing the user button So it stopped listening So how do we can how do we interact with the user button? Well, there's a there are two switches a reset button on the board which reboots it obviously and a user switch And these are brought out as little brass buttons on your aluminum on your aluminium casing So in order to use a button all I have to do is provide a handler function the work I want to do when the buttons pressed I need a switch object Which represents the switch and then I just attach the handler via a callback method. So there's my work in this particular case I'm just setting a control variable, which just prevents the recording from taking place. I Create a switch object from the pie board module and then I invoke the callback method providing my Function so every time I press that button that function runs. That's quite straightforward Driving the LEDs. Well, that's pretty cool. We like things that light up and flash So it's got four LEDs and four colors So you can create much like the switch you can create an LED object and you can switch them on off toggle them and so on so again pie board LED 2 for for the green LED and I can switch it on I can switch it off I can toggle it LED 4 supports an intensity method which allows you to provide 256 levels of brightness they all support the intensity call But only the blue LED allows you to do the cool throbbing type of effect Automatic speech detection Okay, so this requires a little bit of thought the pie board doesn't give you this this is all part of my application domain code so Basically, I want to capture some speech continuously and Only start recording when I think something's being spoken. So this requires obviously the ability to determine whether there's speech or not and That's relatively Simple in this implementation So I have I have two buffers. I have my original Speech buffer that I'm recording to but I have a very small buffer that's treated like a circular Memory so as the capture function is running it's in two modes when it's listening for speech and to it's then recording So while it's listening it's writing to this circular buffer are doing some very crude analysis on the signal level to determine whether it's silence or Noise and because I've done some analysis of the noise I can understand roughly at what level these samples start to look like speech so once it's discovered that their speech it then switches to Straight record into this main buffer and then and then stops Writing to an SD card is useful for storing data in recordings The device does have some flash on board, but that's very very small and extremely slow so you have to insert insert an SD card and Discovering an SD card is my approach is really quite crude some Micro Python guru might say oh no, you don't want to do that. So I basically just look for SD in the system path if there's an SD there's an SD device and then I just do my Basic Python open right close so finally putting it all together I Have sketched this Application diagram the green parts are basically the MicroPath and hardware services the blue parts are the software modules I've provided and the red is basically just data So when we when we initialize the board the capture playback main loop Essentially set controls the capture function with some control variables So it will tell it right start listening if I've not pressed the please stop listening hold button the capture function then essentially Listen for speech it's capturing all the time once it's detected speech it writes into this long buffer It conveniently lights the amber LED for me to tell me it's recording when once that is done it then Unlights that LED the and sets a control variable So at least my outer capture loop now understands a recording has been made So it can set the blue Playback but playback LED and does one of a number of one of a number of things now Remember that I have a little bit of speech the speech detection buffer So I need to attach that to my original buffer because in my crew DAC method I can only give it the address of a buffer and play it and also that is circular so we don't know where the start of it is So the copy function unrolls this untangles this buffer puts it in here into there I then Once the copies done I then run my crude clever attenuation method which adjusts that buffer it adjusts the value of zero for the next capture etc etc and then Calls the play function once that's done it optionally dumps to the SD card if it exists and then the whole thing starts all over again and That effectively is the entire application the code is Is all Python and I've published it on to onto github So all I've got to say now is thank you very much for listening. I hope it's been useful You've got my contact details there and the address of the source code And I assume that effect. I'll probably publish these slides up to that address later on If I just play Thank you Alan for that wonderful talk So he ought to automate it away my job. That's not good Okay, yeah, thank you. I hope there are some questions are there, okay Hi, so you have a very small budget in terms of time to process Does that mean there are certain things that you shouldn't do like probably import modules How about calling functions and in general is there any way to make sure that no matter what pass your code takes? It would still be done within the budget to allocate it. Yeah indeed, so The overriding rule is inside your capture function you have to do as the absolute minimum that's required And my capture function essentially just calls the ADC Checks the sample value and puts it in the buffer all the processing that you would ever want to that do that signal I can't do in the buffer because I've measured the time the You're in a callback function apart from other interrupts that are going on the board you You probably are fairly safe in assuming you've got control of the CPU here your Python on metal So apart from other interrupts that are occurring you probably don't have a lot to do you could Yes, I don't know what would happen if you over if your capture function lasted too long You could probably detect that by setting some sort of like lock variable The one thing you can't do in a callback is you can't create any Python objects And you can't do floating point processing in Python because that in itself creates Python objects So you really are at the bottom level of code just moving bits and bytes around memory But that's why I had to put in that the pin function and the timers to to make sure there is some jitter You can see that when the board is running, but as long as you're within quite a percentage point of your A loud period you're probably quite safe Thank you. That was fascinating and you mentioned your nephew. I think at the at the beginning and have you used this with Children or young people in an educational context and if so, how did it work out? Not yet. The problem that the pie board has is that when I pull a cable there's no built-in battery backup So it has to be powered by something So I'm waiting for them to produce a battery module so I could put this into You know the the stuff toy to allow them to do that sort of thing. I mean these things are Incredibly popular now. They're all over the place. You know, you squeeze you toy you press that press the hand and talk to it But no, I haven't I haven't got back to him. Yeah, I've just released version one. So Thank you Some questions over there Have you done a testing with clear signals so you can tell which part of the noise that we're hearing comes from the recording And which one comes from the playback because usually those those Speakers are very low quality So No, I haven't but the the pie board the pie module does come with schematics and diagrams so one of the things you can do is Obviously not use their built-in microphone and there's an ability to provide your own signal that the path to the Analog components here is very short But that is the next step is to provide a reference signal on here to see where the noise is coming from It's not unusual to find noise in devices like this. They're not exactly professional quality Analog components. They need proper grounding and earth planes to separate them from the digital noise But no, that's the next step is to provide some sort of reference signal in at this point Okay, so first here Thanks for the talk. It seems like a really good way of not only getting into embedded but also Learning Python because I'm quite new to Python. So I'm going to play around with this I've got two quick questions. Hopefully firstly coming from an embedded background. Can you Do the ADC in the background? So for example in your in your code You were waiting for the ADC to get back to you and then you're buffering that data Can you just pull that and then have an interrupt come back to you? Or for example, use a double buffered approach like you would do with old audio Sorry and the question was do you know if there's anywhere in Rimini we can buy these whilst we're here You've got one here No That's a good point. The the the ADC read method obviously MicroPython's open source so we could go in there and have a look what's going on But the read time is is not very useful because you don't know where it is and you it's real time So at the end of the day you have to do whatever you want Once every 8,000 times a second if you can't guarantee that then there's a maximum length of time that you can record So the only way the only what I've done is is the absolute minimum Which is read from the ADC and essentially put it into memory, but I am I am checking the signal level I could I could connect two boards together Maybe connect this to something more powerful like a Raspberry Pi and do a lot more signal processing because you can some really clever things when you turn your continuous time domain signal Into a frequency domain because then you can walk into speech recognition and actually understand what people are saying So but no you really have you're very limited in what you can do But yeah, not as cheap as the pi zero and I I don't know whether MicroPython runs on the pi zero I would guess that it does but the beauty of this board is that it's got so much Hardware ready and you just have a microphone and a loud speaker Okay Yeah, so actually my question Related to what you said just now I wonder what's the advantage of using MicroPython on on pi board versus using one of the you know Bigel bone Raspberry Pi one of these boards that are more powerful and allow you to use the full blown Python You know and all the hardware that you can really work with so the main the main reason I looked at this device is its physical size It comes with headers, but you can buy the board without any headers So if you're going to attach this to a smaller component like a toy its size is important and and also it's the power consumption I think this is only drawing a Fraction, you know at tens of milliamps whereas the Raspberry Pi just consumes too much tower. It's too bulky So it's a trade-off between power In CPU and battery consumption and and and things like that So I mean this is about half the size of a Pi zero it's got but it's it's loaded with kind of quite a bit of hardware to play with So the reason why it's not cheap. I think it's 35 euros or something So it's not terribly cheap compared to the Pi zero, but it's got some really cool Hardware features and it's it's Python on metal. So That's cool and it makes noises Okay, so maybe I have a last question. So when is Alexa running on this thing and I can well, okay Yes, indeed. Well, as I say the problem is getting from the Continuous domain signal to the frequency domain. So you need to get to Something like that once you once you're able to get the data into that you can do your own speech recognition And that's I've done that in my past life, but Alexa they have an API so you can give it data So yeah, just needs time. Okay, so let's thank him again thank you and you


The last talk is by Byrne Goldrisch. It's a very important one at the conference. He's telling us about the Mathematics contributions of Craig Hineke. Of course, those contributions are too vast to manage in one talk, but he's giving us a sampler. Thank you very much. Thank you very much, Srinivasi, for the introduction. And also thank you very much for giving me the opportunity here to talk about Craig's work. Of course, it is a challenge, as he said, because there's so much of it, and to make a selection is difficult, so it really is just a sampler. I also want to say on a personal note I've met Craig for the first time over 41 years ago at a Snowden Winter Meeting of the AMS in Cincinnati. And then in the following fall, Craig joined Purdue University as a faculty member, where I was a postdoc at the time. And that's when we started working. And this was the beginning of a long-lasting collaboration and friendship. And I've always thought of Craig very much as an older brother, although he's not so much older than me. But it is, even if we don't meet for a long time, there's always an immediate connection mathematically and otherwise, so it's always great to be with him. Of course, Craig has had a tremendous influence on the field and the people working in the field. He has been a big supporter of young people and he has helped all of us through his work, of course, but also through the many questions he asked and the problems he shared very generously. And he certainly has changed the face of commutative algebra. There are lots of new developments where he was at the very beginning and initiated them or he built the ground floor on which other people then could keep constructing theorems and new theories. So this is a great contribution and we all have to be very grateful to him. Of course, it's sad that Mel cannot be here, even I think virtually he won't be here. On the other hand, this talk I'm giving is also very much inspired by a joint talk and a joint paper. We wrote Mel and I about Craig's work at the occasion of his 65th birthday. So, somehow he's there at least. So let me start. So again, this is a sampler, not more than a sampler. I'm talking about some themes of Craig's work. One of the themes is asymptotic behavior of ideals and that includes restrings, integral closures, symbolic powers, but also linkage, residual sections and similar things. I also talk about uniformity, which is a big theme in his work and according to him, it's a very big thing in commutative algebra, certainly it is a big theme in his work. I also talk about local chronology and R plus. I won't touch tight closure because Irena Swanson will talk about that on Thursday. So that's the work of Craig and unity and Craig and Mel about tight closure. So I'm sparing that out completely. Okay, so let's start then. So since Hilbert functions and multiplicities were introduced, the behavior of powers of ideals has been an important topic in commutative algebra. This has led to the study of restrings, associated graded rings and the relationship between powers and symbolic powers for instance. And Craig already in his thesis, written basically under the supervision of David Eisenbatt, came up with a very important notion in this area, that's a notion of de-sequences. So what's a de-sequence? Going the wrong way. What's a de-sequence? De-sequence is a vast generalization of a regular sequence. So let's take N elements A1 up to AN in a Nuthian ring, Nuthian is not so important here. And I, the ideal they generate, then these ideals form a de-sequence. If the following is true, well, what does regular sequence mean? It means that if one colons an element into the ideal generated by the previous elements, you would get the ideal generated by the previous elements. Now de-sequence is the same, but you have to also intersect with the ideal generated by the whole sequence. So this is a much weaker condition, of course. And it is satisfied in many instances, and that's one of the things Craig showed. And one of the applications of de-sequences is the following theorem, except the theorem doesn't want to come. Sometimes they don't, yeah, it's true. It's dark. Oh, mm-hmm, yeah. Oh, there it is, okay, no? That's it, that's it, okay. So that's a theorem that was proved independently by Craig and by Tito Valla. And it says that if you have an ideal generated by a de-sequence, then the natural map from the symmetric algebra onto the Ries algebra is an isomorphism, okay? And this is the first big result in this direction before this was known for regular sequences and so on, which is quite, quite, not so interesting. And why is that interesting? Well, it has for two reasons. First of all, if the symmetric algebra is a Ries algebra, then one has control over the associated primes of the symmetric algebra, which are usually a mess. So because the Ries algebra is torsion-free over other symmetric algebra is not in general. And on the other hand, for the Ries algebra, it has the benefit that one knows the defining equations of the Ries algebra because one knows the defining equations of the symmetric algebra. And that is a classical problem in elimination theory. So that's one of the advantages and this is already a very important result. Now, remember, regular sequences can be characterized in terms of the acyclicity of the Kozul complex. And there's a similar characterization for de-sequences. They can be characterized in terms of the acyclicity of another complex, which is called the M-complex. And that complex was introduced by Herzog, Siemens, and Vasconcelos. And what it is, is the following. I quickly want to mention the construction. So we start again with these elements in a new theorem ring. I see ideal degenerate. And then we look at the homology of the Kozul complex on these elements. So that's a graded model, if you want. And then one can construct a complex called the approximation complex. And what it is, is you take H and tensor it with a polynomial in N new variables, as many variables as there are generators of the ideal. And the differential is the Kozul differential, but not the Kozul differential on the A, is that would just be zero because I've taken the homology of the Kozul complex. Rather, one takes the Kozul differential on the T's on these new variables. And that's a bounded complex. And it turns out that it's zero's homology is always a symmetric algebra of i mod i squared, which maps onto the associated graded rings. So that's sort of the same picture we had before the symmetric algebra mapping on the least ring, just the same picture but tensed with i mod i. So that's what this gives us. And it turns out, and that was proof by Herzog-Cymus Vosconcelos, that if a ring is local with infinite residue field, then the acyclicity of the M complex is equivalent to the fact that i is generated by a d sequence. So this is a characterization, not of d sequence, but of ideals generated by d sequence. That's the difference, of course. So that's the homological characterization of d sequences and that connects this notion of d sequence to homological algebra, which can be used there effectively. And for instance, they prove also that if as a kormikoli ring and the height of i is positive and two important conditions are satisfied and they will appear over and over again in this talk. The first is the strong kormikoliness. i is strongly kormikoli if the causal homology is a kormikoli model, automatically a maximal kormikoli arm model. It's called strongly kormikoli because arm or i is h0. So strongly kormikoli in particular implies that arm or i is kormikoli, but it's much stronger, of course. In the second condition, that's a simpler condition. It's a condition on the local number of generators of the ideal. It simply says that locally at each prime, the number of generators of the ideal is at most as big as the dimension of the ambient ring.  This is a sliding condition and that can be easily checked in terms of height or fitting ideals. And what this does for you is that the strongly kormikoliness tells you that the M-complex is a bounded complex of maximal kormikoli models. And this other condition tells you that locally, since the number of generators is small, the M-complex can be replaced by a complex of a very short length. And these two facts together with the acyclicity lemma imply immediately that the M-complex is acyclic. So this is a sufficient condition for acyclicity of the M-complex. And once you have the M-complex as acyclic, then one can show in general, if the M-complex is acyclic, then this map up there from the symmetric algebra of I mod I squared onto the associated gradient ring is actually an isomorphism. And that isomorphism, again, by work of Valar lifts to an isomorphism between the symmetric algebra and the associated gradient ring. That's one way to see it. So acyclicity implies this isomorphism. And then the kormikoliness of this associated gradient ring, for instance, follows because now the M-complex is acyclic complex and all the models are maximal kormikoli models. Therefore, by the depth lemma, this associated gradient ring is kormikoli. And then, for instance, by work of Chungiketa, the restring also has to be kormikoli. So that's one way to say it. So this is really all one wants in some sense. It's very important to know that these algebras, the restring and the associated gradient ring are kormikoli. And it's also very important to know that the restring is isomorphic to the symmetric algebra. So this is somehow the desired result. Of course, there are drawbacks I'll talk about it a little bit later, okay? Now, Craig proved similar results, not using this homological machinery, but using what he calls kormikoli de-sequences. But he came up with very similar results with different methods, okay? Now, the question is, the condition here, as I said, there are two conditions. One is sort of harmless, this condition on the local number of generalists. The other is a subtle condition, strong kormikolness. And then the condition is, which ideas are strongly kormikoli? How do you check? Do you have good examples? And this is where Craig comes in again. And this is where linkage comes in. So when are ideas strongly kormikoli? And this needs to the notion of linkage. Linkage or liaison has been used since the 19th century as a method for classifying projective varieties in particular curves in P3, and then names are Max Nöter, Kayleigh, Hylphen, Gaeta, and many others. And here's the definition. Suppose R is a kormikoli ring and I and K are proper ideals. Then these ideals are linked, and that's the symbol one uses. If there exists a regular sequence, which has the property that K is the colon ideal, alpha colon I, and for symmetry, I is the colon ideal, alpha colon K. And from this definition, it follows immediately that alpha is in the intersection of I and K, and also that I and K both are mixed of the same co-dimension G, okay? So, and this relation is symmetric, but it's not reflexive or transitive, and therefore one passes to what's called the linkage class, the equivalence relation generated by this. Two ideals are in the same linkage class, or the same even linkage class. If for some N or some even N, I can be linked to K by a finite sequence of links, N of them, okay? And one says that's a particularly interesting case that I is leachy if it is in the linkage class of a complete intersection. So the leachy ideals are those which you can get from a complete intersection via iterated linkage, okay? And that's an important class because these ideals can be arbitrarily complicated. On the other hand, there are somewhat well behaved. They still remember that they come from a complete intersection a long time ago. So that's the advantage. And here are some examples. Perfect ideals of high two are leachy. This is basically the content of some of this classical work. Of course, they were talking about curves in P3, but and then the condition is that these curves are automatically called if and only if they're leachy. The same works for perfect ideals which are Gornstein of high three. And but it also works for other types of ideals. For instance, this ideal here, which is generated by the minus maximum minus of a generic matrix together with the entries of this product. So a vector times a matrix. So this is a baby case of a variety of complexes. And this is also linkage. This ideal is also the defining ideal by the way of the associated credit ring of a generic perfect ideal of high two. And those are examples of linkage. Leach ideas are many more and constructions how to get new leachy ideals from old ones. But here's the theorem of unity. This is amazing and they're interesting result. And it's really one of the first results that prove really non-trivial properties of leachy ideals. And it says that if we are in a Cormier-Cawley ring and INKI deals in the same even linkage class, even is important, even is closer than odd K because each ideal is evenly linked to itself but it's only linked to something totally different possibly. So then I is strongly Cormier-Cawley if and only if K is. So strong Cormier-Cawley is an invariant of the even linkage class. And in particular, since every completender section is strongly Cormier-Cawley and every completender section is linked to itself so even odd doesn't make a difference. It follows that every leachy ideal is strongly Cormier-Cawley. So all these ideals I mentioned before they all satisfy the assumptions in that theorem about associated graded rings and tree strings. So that's a very, very beautiful result and interesting result. Then in joint work we went on to study leachy ideals and not only leachy ideals to study other properties that are invariant under linkage in some sense. And one of those properties are the graded shifts in a minimal free resolution. So now take a homogeneous idea, which is perfect, call it I prime, in a polynomial ring, look at the minimal homogeneous resolution of this idea. Sorry, of this, what happened? Wrong button, okay, as usual. We look at the minimal free resolution of this idea. The resolution before the perfection has length G minus one where G is the co-dimension of the ideal and now localized at normal G is maximum ideal. And the statement is if the maximum of the last shift is at most the length of the resolution times the minimum of the first shift. So in other words, if the shifts don't grow very fast then for every ideal in the linkage class of I, even after localizing, and these ideals of course need not be coming from homogeneous ideal anymore. Any ideal, even in this localized ring one can say that the number of generators of this new ideal cannot be smaller than the number of generals of I and the type of R mod K cannot be smaller than the type of R mod I. So in other words, no matter what you do, you can never improve the ideal by even linkage. If this particular condition is satisfied, in particular, I cannot be leechy because if that condition is satisfied then I would have to be a completeness section but completeness sections are ruled out by these numerical conditions. So such an idea can never be leechy. One should think of this as a negative result in some sense a criteria, a necessary condition for being leechy. So one can use that to show something is not leechy and one can use it also to distinguish linkage classes. And in particular, one can get from this quite easily. The following, where we had seen already that there's only one linkage class of arithmetically corn macaury curves in P3 and this was known classically not in this language but this was known in the 19th century already. And also of arithmetically corn steam curves in P4 but using this previous result, one can easily show that there are infinitely many even smooth arithmetically corn macaury curves in P4 or arithmetically corn steam curves in P5 that belong to different linkage classes. So the behavior is completely wild if you step up one dimension and in fact these various linked curves they can even be obtained from each other by a linear automorphism of the projective space. So this way linkage is very, very fine. Because linkage is very fine, one wants to look at a more general notion and that's a notion of residual intersection, okay? And residual intersection is similar to linkage except that the two so to speak linked ideals need not have the same height. So you can have different ideals connected to each other. Otherwise the definition is similar. So our again is a new theorem ring, I is an ideal of height G, S is an integer at least G and then one considers a proper ideal K and such a proper ideal is an S, the residual intersection needs to keep track of this S of I if K is a colon ideal, J colon I for some S generated ideal J inside I and the height is at least S. So the crucial point here, the crucial condition that makes this somewhat strong and rigid is that the number of generators of the ideal one colons into is less or equal to the height of the colon, okay? You may be tempted to think that S is the largest possible height of this colon but that's not true. There is no cold altitude theorem for colon ideals. In fact, this height can be arbitrarily large. And it's actually interesting when the height cannot be arbitrarily large or what the height is because that's related to integral dependence of ideals is also related to dimensions of secant varieties and so on. So there are various interesting connections to that. Now assume for instance that I is Gornstein and I is unmixed that it's easy to see that G is cylinder section. So the case where S is smallest possible and M is G, the height of the ideal is simply linkage. So this is a vast generalization of linkage. But notice the situation is very different because if two ideals are linked and if you are in a Gornstein ring, then if Armot I is comicale, then also the link is comicale. That's a result proved by Peskin and Spiro. So in modern algebraic language, they reworked the classical theory of linkage in modern algebraic language and that's what they proved. They also observed that this is not true if the ambient ring is only comicale. So it doesn't work in a comicale ring. And then the crucial observation of Craig was that if you're in a comicale ring and you strengthen the condition of being comicale to being strongly comicale, then indeed the link is comicale, okay? That's a very crucial observation because of the following. It allowed him to prove a theorem which is a very surprising theorem because in general it's a theorem about the singular section. Of course in general the singular sections no matter if your ideal is comicale the singular sections need not be comicale anymore. They can be badly mixed and all kinds of strange things can happen. But nevertheless he can show the following. If we are in a comicale ring, i is an ideal and s is an s singular section so that familiar conditions, first of all, i is strongly comicale and second this g condition on the number of generators except now it's not g infinity. It's only what's called gs which means it's like g infinity but you don't require it for all primates. You only require this for primates up to height s minus one. So in particular if s is at most the dimension of the ring you'll never say anything about the global number of generators of the ideal. So this is weaker condition what we have seen before but then he shows that in this case indeed the height of k cannot be bigger than s so this is a cool intersection, cool altitude theorem for colon ideals and furthermore r mode k is comicale. So this is a rather surprising result and the way this observation comes in is one can reduce residual sections to links but on links on residual sections of smaller co-dimension but those will not be Gornstein anymore even if the original ambient ring is Gornstein but by induction they are comicale so one has to do linkage in comicale rings and what saves the day is the strong comicale assumption because even if you're only in a comicale ring the link will remain comicale if the original ideal is stronger than comicale namely strongly comicale and that was a key observation. There was a previous result by Art and Nagata but it was not correct and Greg observed that causal homology comes in or comes to help here. Okay so let's go back to these rings and establish a connection between these rings and residual sections. So let's look at an ideal, these are without any assumptions really. Let's look at this ideal. I is generated by n elements a1 up to an and as an ideal in R and R is of course a degree zero part of the symmetric algebra but you can think of I also as a degree one part of the symmetric algebra and then I denote the generators by a1 prime up to an prime and then one can form what's called the extended symmetric algebra and it's called SIMI, a joint T inverse this makes no sense it's just a symbol you can think of it as a T inverse but what it is it's just a symmetric algebra a joint available modulo all the relations that tell you that you can get from the a primes to the a's by multiplying with u so this is why the u works like a T inverse and this extended symmetric algebra maps on to the extended Ries ring which is R a joint I T comma T inverse and now the T inverse really does make sense in the Laurent polynomial ring and the u goes to T inverse there's always this surjection and now Craig and that's the crucial connection to the serial intersections proved that if I has positive height and is G infinity so this justice condition on the local number of genus no depth condition on causal homology or so or anything like this then this extended symmetric algebra is defined by an ideal K so we can write it as a polynomial ring modulo an ideal K which is an N residual section of the original ideal not quite of the original ideal together with this web so the defining ideal of the extended symmetric algebra is a residual intersection of the original ideal this gives a very tight connection between the serial intersections and Ries rings or extended symmetric algebras and so on and let me give an application of this which follows immediately from his work now assume again I score McCorley the height is positive and now throw in the stronger condition that I strongly call McCorley and G infinity then under these assumptions because of G infinity we know that the extended symmetric algebra is defined by a residual section by the previous theorem of Craig because we have strongly called McCorley and G infinity we know that residual sections are called McCorley so therefore the extended symmetric algebra is called McCorley since the extended algebra is called McCorley one can quite easily also prove that it is equal to the Ries algebra because one is enough to check this equality at the source at primes if one has called McCorley and one has control over source at primes so one gets this equality but then also the source at credit ring is called McCorley because simply the extended three string modulo principle ideal and from that again one also gets that the symmetric algebra is isomorphic to the Ries algebra and it is called McCorley okay this isomorphism of course the first isomorphism immediately implies the second isomorphism and the call McCorley simply comes again if you want from Chung Ikeda since the source at credit ring is called McCorley the Ries ring is called McCorley in this case so one can completely recover this result by Herzog-Cymus Vasconcelos simply from the cylinder sections and this sets up this connection between Ries rings and the cylinder sections in this setting where we have G infinity if we don't have G infinity this connection is less apparent but still is there and is used a lot so and that's a very important insight that came from Craig's work now so we've talked about Ries rings with the G infinity condition of the ideal of course the problem is that if the ring is local then G infinity implies in particular that the number of generators is bounded above by the dimension of the ring so that's of course a strong restriction it's a restriction on the number of generators that's something we don't want of course without that restriction you cannot expect that the symmetric algebra and the Ries algebra are isomorphic but one still wants that perhaps the Ries algebra is McCorley without being isomorphic to the symmetric algebra and Greg together with Sam Huckabar were the first to treat systematically the case of ideals that have arbitrary number of generators and are not integral over parameter ideals so that's a very huge step into a new direction of considering algebraic properties of Ries rings and as I said the problem is that the number of generators of the ideal I is too large so what you do you pass to an ideal which has smaller number of generators but still is very connected to the ideal and that's the minimal reduction of an ideal so we have to talk about reductions and integral dependence of ideals so one says that I is integral over J J is a sub ideal of I we are in a new theorem local ring with infinite residue field I is integral over J so I currently J is a reduction of I in some sense that's just different language if and only if the induced inclusion of Ries rings is an integral extension of rings in the usual sense and if you read this fact degree by degree you get this condition namely that the R plus first power of the larger ideal is obtained from the Rth power of the larger ideal times one power of the smaller ideal eventually for all R and the smallest such R where this works is called the reduction number of I with respect to J and it is written in this way so in some sense reduction is a simplification of the ideal and the reduction number measures how far the two ideals are apart from each other and then one wants to look at minimal reductions a reduction is called minimal if it is minimal with respect to inclusion they always exist by work of Northcott and Trees and another definition the analytic spread actually came up in Dale's talk already analytic spread even of filtration this is just analytic spread of Iatic filtration the analytic spread of I is the minimal number of generators of J for any minimal reduction of I so minimal reductions are highly non-unique but they all share the same minimal number of generators and that's because this invariant can also be defined as the Krull dimension of the special fiber ring the Ries ring tensed with K so that's a special fiber over the close point inside the blow up the brooch of the Ries ring the blow up of spec R along V of I and from this definition it's clear that the analytic spread is bounded above first of all by the dimension of R but also by the minimum number of genitals of I and it's bounded below by the height of I because I and the minimal reduction they have the same radical so therefore the height of I is the height of J and L of I is the number of genitals of J so simply use Krull's attitude here but now here we are, now we have a new ideal J which satisfies this condition we didn't have before namely that the number of generators is bounded above by the dimension of the ring so if I replace I by J then the number of generators is bounded above by the dimension of the ring R so one could maybe hope one has some information about the Ries ring of J and then goes back to R if reduction number is small enough and in fact Greg and Sam were able to implement that so write G for the height of I and L for the analytic spread and if the analytic spread is as I said is at least G if it's equal to G the case is kind of trivial, well understood but if not the first case to consider is the case where the difference between L and G is at most one or equal to one that's called analytic deviation one and in that case they prove what one would like to have again the assumptions are R is a Cormac-Corley ring let me go to the next, sorry R is a Cormac-Corley ring R mod I is almost Cormac-Corley means at either the depth is the dimension or at most one worse the height is at least one I is a completeness section local encode dimension G plus one and L that's the crucial assumption is at most G plus one and then the reduction number of I with respect to some minimal reduction is at most one and from that they get that the restring and the associated Cormac-Corley so but notice there's no condition on the number of generators of the LDI and they have a similar result for G plus two one has to strengthen the conditions a little bit but that's what they prove now I should say that meanwhile this has this this generate a lot of activities in the first half of the 90s many people worked on this and the eventually the case of Arbiter analytics spread was solved and the pools were different and so on but the main point is that again Greg and in this case with Sam they were at the foundation of this that they were the first ones who started with and maybe people never would have picked up this or they had to pick up this I mean their proof is very very complicated but things got much simpler as time went on but that's not the point I mean they were the point the ones that started this and of course in general one has to impose conditions on the reduction number one has to also impose some conditions on depths of powers or causal homology one just doesn't get away with these basic conditions but still this is the way it went in both case also interesting to observe that j-core and i is a residual intersection so there is again this connection still to residual intersection although the proofs don't go quite directly in terms of translation from one to the other okay let me now also talk briefly about the core I had mentioned that in general minimal reductions are highly non-unique and to remedy this one looks at the intersection of all reductions that's called the core of an ideal so we are in a Northean local ring infant residue field i is an ideal then the core of i is the intersection of all reductions is enough to look at all minimal reductions and this was introduced by Judy Sally and Reese, Reese and Sally and it also comes up naturally in the Pryon-San's Coda Theorem I'll get back to this later if i is regular then i raised to the power l of i the analytical spread is actually contained in the core so that's a lower bound for the core but in general the core is an up-rheor infinite intersection and it's very hard to compute or understand and the first again to find a formula very explicit formula for the core where Greg together with Irina Swanson this is now in a two-dimensional regular ring i is an integrally closed ideal so we have the Zariski theory of integrally closed ideals in regular rings and then they prove that the core of i is i times the second fitting ideal or j squared core i so the core which is an infinite up-rheor intersection of ideals well in this case not but in general it could be an infinite intersection then can be expressed in terms of a single reduction j is a single reduction j is a one minimal reduction again this was vastly generalized but this is really the ground floor so that's a very very it's also very different not easy proof at all so now let me pass to the other topic namely uniformity so uniformity in netherian rings so here's a quote from Greg Yonecki behind the obvious fineteness condition in netherian rings there lie many deeper and hidden types of fineteness which come to light in terms of uniform behavior so one way to think of uniform behavior is that it's that uniform means uniform for every ideal i in a given netherian ring R so I fix the ring R but I want uniform behavior for all ideals or study what uniform properties to all ideals have and a typical example and one that also inspired Greg as far as I understand is the following suppose we are in prime characteristic then a test element from tight closure theory has such a uniform property namely it multiplies the tight closure of any ideal into the ideal so in other words it's a uniform annihilator of the module i star modulo i regardless of what i is and that uniformity plays a very important role in tight closure theory here I will focus on Greg's work on uniformity related to the Artin-Ries theorem interclosure filtrations and symbolic power filtrations so let me explain this what is the Pryonson-Skoda theorem where here's the Pryonson-Skoda theorem proved by Pryonson-Skoda with analytic methods and the first algebraic proof and the general algebraic formulation is due to Liebman and Sartre so assume as a regular domain of dimension D finite then what the Pryonson-Skoda theorem does it compares the integral closure filtration with the iatic filtration in a uniform way and in a very tight way and what it says is that the integral closure of i n is contained in i n minus a constant namely minus D plus one so that's I'm adding or subtracting an integer and I get this contained so there's a very tight connection between the two filtrations but the point also is that this constant I'm adding or subtracting is independent of the i d i so it's a uniform constant so that's a uniform behavior now one way to think of this and that's the way they proved it actually is that this is simply a statement about the conductor of the extended res algebra it simply says that the element T to the one minus D is in the conductor of the extended res algebra and then the way the proof goes is when we just have to approximate the conductor from below one has to show that this particular element is in there and the way they do that is that they show very generally that Jacobian ideal is in the conductor and that's of course generalizing very classical results that's unknown algebraic number theory also in four rings for f n k-algebras by Amy Nerda and so on this was all known but of course they need this in much greater generality and that's what they prove and that's how the proof goes now one cannot expect this to be true if the ring is not regular because for instance if D is one in the one dimensional case this inclusion would say that all ideals are integrally closed but then the ring is regular so that doesn't work however in the one dimensional case you also have that you have the conductor ideal say if you're in a local case and then any non-zero ideal say if you're in a domain every non-zero ideal contains if you have an M primary ideal or an ideal which is not the whole ring then a power of that ideal is always contained in the conductor of the ring and from that alone you would get some kind of a uniform condition like this it wouldn't be the D minus one or one minus D but it would still be uniform so one could expect possibly that a uniform result of this type works and but of course before that one should see why in general such a result at all could work even if I fix the ideal I and it doesn't work in general that's a result of these but it does work if the ring is analytically unrhymified so if the ring is analytically unrhymified then for every ideal there exists a K so that I n bar is contained in I to the n minus K so that's simply the fact that in this case the integral closure of the extended ring is finitely generated over the extended ring and that's guaranteed by these if R is analytically unrhymified analytically unrhymified means the ring is local and the completion is reduced but he also showed that the converse holds if this inclusion holds for a single and primary ideal then the ring has to be analytically unrhymified so that's the weakest condition you can impose K but of course now the next step is assuming analytically unrhymified can you choose this K uniformly K and Greg calls this uniform bryonsonskoda uniform bryonsonskoda holds in R if that K can be chosen uniformly and he conjectures that uniform bryonsonskoda holds in any excellent ring of finite dimension so and that's a very vast conjecture and he essentially proved it so now this is related there's another uniformity type of result that's the uniform art in Ristium so let me quickly explain this so N in M is an inclusion of Mithian R modules then the art in Ristium says that there exists a K so that this intersection property holds so simply that says if I look at the iadic filtration on M and I restrict it to the submodel it's of course not an iadic filtration anymore but it's still at least it's i stable so that's the art in Ristium and which is used all over the place now what about, sorry this is a typo I shouldn't have said it uniform art in Rist of course what one wants is that the K doesn't depend on I but one wants not this equality one just wants the inclusion that I intersect, I N intersected, I M intersected N is contained in I K M intersected with N the equality one doesn't agree that's the containment that the left is in the right without the, this is not sorry what one wants is I, I should write sorry what one wants is that the left hand side is an I to the N minus K N, N that's it which of course holds so so that's the crucial content that's the one that's used and that's uniform art in Rist and here's the theorem which and then Craig also conjectured that uniform art in Rist holds for any excellent ring of finite cold dimension and he essentially proved this so that's the theorem, amazing theorem it says that uniform, Brinus von Skoda and uniform art in Rist hold in any reduced new theorem ring are satisfying either of these conditions either as essential finite type over an excellent local ring or over Z or R has prime characteristic and is F finite and the reduced and excellent condition is only used for the Brinus von Skoda theorem not for the art in Rist, uniform art in Rist and notice what separates this from the completely general case is that in the first bullet point the ring is supposed to be local if that wouldn't be there that would be the general case but it's very close to being general so let me quickly outline the proof just to give an idea of these things are related the proof of these two uniformity properties Brinus von Skoda and art in Rist are intertwined and use two other uniform properties and the first one I just call them uniform conductor elements because that should really remind you of the proof of the Brinus von Skoda theorem because it relates to conductors and what you want in the end is to find uniform conductor elements for extended Rist rings regardless of the ideal and that simply means there exists a K and an element C in R naught R naught this is a notation from tight closure theory it's a set of all elements which are not in any minimal prime so that fixed C times the integral closure of I to the n is contained in I to the n minus K for every ideal I again this should be uniform or equivalently there exists that's essentially what it says there exists an element in the conductor of the extended Rist ring of degree minus K and except one doesn't take the conductor with respect to the integral closure one just take the integral closure inside the Laurent polynorm ring because I is not assumed to be normal so that's this uniform conductor condition but the point is that this should hold for every ideal now there's another condition another uniformity condition that's uniform annihilation of homology and it says there exists an element D again in R naught that annihilates the positive degree homology for every bounded complex of finite free R models that satisfy the Buxbaum-Eisenbach acyclicity conditions with grade replaced by height and so such complexes of course are not acyclic because in the Buxbaum-Eisenbach criteria you need grade not height but those are the complexes that would be acyclic if the ring were cormacoli so this takes care of rings that are not cormacoli and well with that the proof Greg's proof works like this if you have these conditions A and B not just for R but R modulo P for every primate ideal then he shows that uniform Artyn Ries holds for the ring so the uniform Artyn Ries for the ring follows from these two conditions where the first condition A this uniform conductor condition very much smells like the Artyn Ries theorem smells like the Brinozon-Skoda theorem so that's the connection in some sense Brinozon-Skoda sort of implies Artyn Ries I mean very roughly speaking and the proof is a induction on the dimension of M modulo N it's rather involved but that's the connection and the other connection so how do we get uniform Brinozon-Skoda if we have condition A and uniform Artyn Ries then uniform Brinozon-Skoda holds and this is how everything is connected for uniformity properties are very much connected and that proof at least I can show this really very clear and easy so for every ideal we have that C times the integral closure of IN is in I to the N minus K that's simply by part A but this ideal is also in the principle ideal generated by C so therefore I can intersect with C R and this works by A and for every I without any restriction I uniformly but then I can use this is now a condition that looks like like uniform Artyn Ries this version here and we apply this to the case where M is the ring and N is the ideal generated by C and then we get that this intersection is in C times I to the N minus K minus some K prime and by uniform Artyn Ries and now cancel the C season on zeal divisor and you get uniform Brinozon-Skoda so this condition A together with uniform Artyn Ries implies uniform Brinozon-Skoda Okay. That's what I want to say about uniformity let me pass to symbolic powers again I've come back to uniformity at the end of that but of course symbolic powers that's also a case where we want to look at symbolic on uniform properties so either Nithyan ring is an ideal W is the complement of the union of the associated primes and then the end symbolic power is the extension of the Nth power to the localization contracted to the ring now with this definition the first symbolic power is the ideal itself of course there's a difference where one takes associate primes or minimum primes and it contains always the Nth power now symbolic powers have appeared since long in commutative algebra for instance they played a crucial role in Kuhl's original proof of the Kuhl altitude theorem they have a geometric meaning for instance if R is a polynomial over a field and I is radical then the Nth symbolic power is the intersection of M to the N where M is any maximum ideal in V of I and this of course is a set of all polynomials that vanish to order N at each closed point in general though symbolic powers are difficult to compute difficult to understand also difficult to understand how they relate to powers and of course the simplest question not the simplest but the most basic question one could ask the best one could hope for is when are symbolic powers equal to powers and there is a nice observation by Craig that if this whole say suppose they are no embedded associate primes then this implies that locally at every non-minimal prime the analytic spread is strictly less than the height of the prime so that means the analytic spread cannot be maximum the analytic spread is always less or equal to the height of the prime but the equality has to be strict so that's a very strong condition and interestingly and that's together with David Eisen but the converse holds if the associate graded thing is chronic quality so what this says is that if one knows that the associate grade drink is chronic quality then this condition on the analytic spread alone which is easy to check implies that the powers and the symbolic powers are the same and since Craig proved a lot of results about chronic quality of associate grade drinks he there but also proved a lot of results about the quality of symbolic powers and powers that's just this necessary condition you have to check and if it's satisfied the powers and the symbolic powers are the same now if Eisen excellent domain then he also observed that if one simply has this strong condition on the symbolic on the analytic spread alone one can conclude that the symbolic res ring which is the direct sum of all the symbolic powers is a new theorem ring so it's not necessarily equal to the res ring but it's at least a new theorem ring in fact it's even a new theorem model over the basic that's even stronger okay so that's one condition one gets and he in general he has a lot of work on when this symbolic res ring is called Macaulay without this strong condition on the analytic spread and that is interesting because for instance if we are in a new theorem local ring and either one dimension prime ideal then the new theorem is of the symbolic res ring implies that the ideal is a saturated completeness section and we had talks about this so for some time this was a very active subject and Craig contributed greatly to that now so that's this question when the symbolic algebras new theorem in general makes sense and what I said it implies saturated completeness section in the one dimension case now in general okay powers and symbolic powers are not the same the best one could hope for is a comparison and perhaps uniform comparison between ordinary and symbolic powers and there was a result by Ein, Larsersfeld and Smith which gave a very tight uniform comparison between powers and symbolic powers for reduced ideals in regular rings essentially a finite type over a field of characteristic zero and this result and they used asymptotic multiply ideals and this result was greatly generalized by Hawks and Unicard to the case of any regular ring containing a field for instance and they also treat the case where the ring is not regular and here's the statement so I as a new theorem ring containing a field I is an ideal and H is the maximal height of an associated prime of I if as regular then this symbolic power of I to the H and is contained in I to the N so again a very tight bound so it says in particular that the symbolic topology and the ideal addict topology are linearly equivalent but that this factor H is actually uniform it's only can be bounded for instance by the dimension of the ring if the dimension of the ring is finite and of course one has to take at least something like a linear function where the coefficient of N is not one because if as before you would just take N plus a constant that wouldn't work because it would imply the symbolic ring to be integral over the ring and then this would that's a very special case so in general one cannot expect that so this H has to be at least so N plus a constant would not work so it has to take at least a constant times N so that's the least of it that's the best one could expect for and if the ring is not necessarily regular but you're on a setting where you can talk about a Jacobian ideal then if you multiply the left hand side with the N plus first power of that Jacobian ideal you still have the containment and the proof is via reduction to prime characteristic and there they show that the Nth power of the Jacobian ideal times that symbolic power is contained not in I to the N but in the tight closure of I to the N and that immediately gives the case of the regular ring because in the regular ring the Jacobian ideal is the unit ideal and the tight closure of any ideal is the ideal so that does it and for two one uses that the Jacobian ideal can be generated by completely stable test elements which means that if we multiply this inclusion by J then the right hand side is actually contained in I to the N and that finishes the proof. Well maybe I skip this and go immediately to the uniformity questions so again this is the simplest version of the Unique Rockstar theorem as a regular ring containing a field of finite and the ring has dimension D which is finite then I to the symbolic power D N is contained and I N for every I so that's the uniformity so this D this coefficient only depends on the ring not on the ideal that's simply because whatever this H was in the previous theorem is always bounded by the dimension of the ring and the question then is can one generalize this to rings which are not regular and just forgetting even about uniformity one could ask is there such a constant K for a given ideal so that this inclusion holds so that is sort of the analogue of the Ries theorem but now for the symbolic power filtrations and indeed that's true by result of Swanson for instance if I is normal and excellent then that can be done. And then the question of course is when can one choose this K uniformly and this is a theorem of Unique cuts in Validasti and they prove indeed one can and the main condition here is that the ring has an isolated singularity that's the restricting condition so R is analytically irreducible and with the local in it has an isolated singularity and then two conditions that should remind you of something I said before R is essentially of type finite type over a field of characteristic zero or R has prime characteristic and is a finite and then indeed there exists a uniform K so that the KN symbolic power is contained the Nth power for every prime ideal at least. So the proof uses and that's why these assumptions look familiar uses the two uniformity results I mentioned before the uniform Priyansar Skoda and the uniform Artin Ries which hold in this case by the previous theorems but it also requires new uniform versions of Chevrolet's theorem and many other things and this is still an open problem in general but Craig and lately with Dan Katz they have done work they considered for instance a finite abelian extensions of regular rings and under suitable additional conditions there they can still do that. I think I had some more things but maybe it's better to stop at this point. The questions or comments? This one? Where the main thing is that you just need basically that the integral closure is a finite model that actually is a little bit overkill but that's what you need. Any other questions or comments from any of the people on Zoom? Just thank you, Berent, for such a nice talk. I shouldn't say nice since it's about my work. Sure, that's why it's nice. Oh, there you go. Good, good, good to see you. That was one of my great pleasures was working with Berent for so many years and as he said it's like we can pick up where we left off any time. It's a wonderful thing to have a collaborator. So thank you, Berent. Thank you, Greg, thank you. So many years of pleasurable work. Let's thank Bernt and Greg also for the comment.