Mega generalization breakthrough today. I figured out how to write the union and product results without repetition. I think the definitions I have developed to do this may need to be tweaked a bit, but overall I'm pleased.
I got my funding requests in for BEST today, for both sources from BEST and from UNT. Lots of paperwork, but when you weigh it against the compensation its really not that big of a deal.
There was an RTG talk today. Chris spoke on some automatic continuity results for rings of complex-analytic functions.
Friday, April 29, 2016
Thursday, April 28, 2016
4/28/2016
A different kind of progress today. I had an epiphany and realized I could state the results for unions and products at least a but more generically (without explicitly referencing the reals or hyperfinite quotients). I think this will clean up the writing a bit. Meanwhile, I got my travel plans to New York sorted. On an unrelated note, I found out that my year and the year before are not receiving the tuition scholarship next year. This makes things a little more complicated financially.
Wednesday, April 27, 2016
4/27/2016
More write up work today. I finished the write up for products of smooth sets with cardinals. I also spent a non-trivial amount of time working on applications for travel funding and setting up travel plans. New York is only two weeks, I am excited. I need to to update my slides from the section meeting though. Right now research is an exercise in patience. The things I am writing up now may be a bit tedious, but I am going to need them when I start looking at the more fun directions.
Tuesday, April 26, 2016
4/25/2016
Lots of exciting things happening with my research. There are actually too many things to look at right now, I have a queue. I'm adding products to the write-up. The results are clearly true, but I have a couple of different ways I can try to write them up. I am trying to remove some tedious and repetitive elements from the write up.
My abstract was officially accepted for BEST. I now have to do the paperwork to apply for funding. Time to update that CV...
My abstract was officially accepted for BEST. I now have to do the paperwork to apply for funding. Time to update that CV...
Friday, April 22, 2016
4/22/2016
I am nearly done compiling the current results for E_0. At the same time, putting the full write-up together has put some of the results into a new a perspective, and I have some new proofs for slightly better results. I also have some weird new questions. There seems to be a possibility of developing more general large cardinal properties for quotients of the reals.
I am presenting at CUNY in about three weeks now, and I have been invited to speak at BEST, which is in San Diego in June. I am looking forward to both of these trips. This is the fun part of academia. Incidentally, it is also why you can't be antisocial and be a successful mathematician (unless you are ridiculously talented, and even then...). It is a somewhat of a pet peeve of mine when people say that math is just a left-brain logical activity. It is partly that, but it really is a creative process and a social process as well.
I am not graduating this year. I am graduating next year, unless something goes horribly wrong. Next year is year 7. I find this to be a ridiculous amount of time, and it is really bothering me. It's odd, because I've known this for essentially this entire year and it hasn't bothered me. Recently, however, the entire group of us who are going into our 7th year (6 people or so in total) were informed that we will no longer be teaching courses, and if we don't graduate next year we will probably lose funding. Hearing this, its hard not to feel like a total failure and like I am being punished for being such a failure. It's a totally ridiculous emotion. For one, I know that the majority of people who came into grad school with and didn't drop out or just get a masters are still here. I know that UNT requires us to teach in order to get funded. This in itself not unusual, but the amount of teaching they require us to do is at least twice as much as other schools and four times as much as big schools do. UNT also has a bit of a problem with the structure of the graduate program: they require us to take general knowledge courses more than other schools. I also know that I started working on my problem less than a year ago, and it is the first problem I have ever worked on. (This is not an attack on UNT, they are doing what they have to do with the funding they have).
There is tension here. Administration and societal expectations are that I should have graduated years ago. On the other hand, this time has not been wasted. I have been allowed to grow as a general mathematician and a general set theorist in ways that will be a major boon to me in the future. I have been allowed to teach essentially every low level math course up to and including calculus 3. I have graded for a proofs course and acted as extra instruction for the magnet high school inside of UNT. In other words, I definitely feel ready to teach whatever to whomever.
Logically, I am aware that this doesn't add up to failure, let alone total failure. Logically, not teaching means I have more time to focus on my dissertation and job applications. Emotionally, I feel like someone punched me in the gut. This is not a normal blog post, and has probably become more emotionally charged than looks good, so I'll wrap it up. I'm over grad school, and I want to be done. I'm ready for the next step.
I am presenting at CUNY in about three weeks now, and I have been invited to speak at BEST, which is in San Diego in June. I am looking forward to both of these trips. This is the fun part of academia. Incidentally, it is also why you can't be antisocial and be a successful mathematician (unless you are ridiculously talented, and even then...). It is a somewhat of a pet peeve of mine when people say that math is just a left-brain logical activity. It is partly that, but it really is a creative process and a social process as well.
I am not graduating this year. I am graduating next year, unless something goes horribly wrong. Next year is year 7. I find this to be a ridiculous amount of time, and it is really bothering me. It's odd, because I've known this for essentially this entire year and it hasn't bothered me. Recently, however, the entire group of us who are going into our 7th year (6 people or so in total) were informed that we will no longer be teaching courses, and if we don't graduate next year we will probably lose funding. Hearing this, its hard not to feel like a total failure and like I am being punished for being such a failure. It's a totally ridiculous emotion. For one, I know that the majority of people who came into grad school with and didn't drop out or just get a masters are still here. I know that UNT requires us to teach in order to get funded. This in itself not unusual, but the amount of teaching they require us to do is at least twice as much as other schools and four times as much as big schools do. UNT also has a bit of a problem with the structure of the graduate program: they require us to take general knowledge courses more than other schools. I also know that I started working on my problem less than a year ago, and it is the first problem I have ever worked on. (This is not an attack on UNT, they are doing what they have to do with the funding they have).
There is tension here. Administration and societal expectations are that I should have graduated years ago. On the other hand, this time has not been wasted. I have been allowed to grow as a general mathematician and a general set theorist in ways that will be a major boon to me in the future. I have been allowed to teach essentially every low level math course up to and including calculus 3. I have graded for a proofs course and acted as extra instruction for the magnet high school inside of UNT. In other words, I definitely feel ready to teach whatever to whomever.
Logically, I am aware that this doesn't add up to failure, let alone total failure. Logically, not teaching means I have more time to focus on my dissertation and job applications. Emotionally, I feel like someone punched me in the gut. This is not a normal blog post, and has probably become more emotionally charged than looks good, so I'll wrap it up. I'm over grad school, and I want to be done. I'm ready for the next step.
Sunday, April 10, 2016
4/10/2016
Day 2 of the Conference:
Solecki began the morning with an exposition on the Pseudoarc and projective Fraisse limits. In particular he realizes the Pseudoarc as a quotient of a portion of a projective Fraisee limit and uses the structure there to provide a proof of its homogeneity, I feel like I want to start a seminar on just Fraisse theory.
Next Martin Sabok talked about the topological conjugacy relation for Toeplitz subshifts. Lots of twists and turns with the results in this talk. Its also nice to see some of the more recent techniques of Jackon and Gao (and Seward and Khrone) being used by other people.
Bruno Braga talked about coarse and uniform embeddings. Banach spaces everywhere. The most exciting part of this talk for me was a clever retooling of the fact that you can extend maps continuously from closed sets to the whole space in the metric space setting. Its nice to see classics get respun.
Clinton Conley talked about one-ended subforests and treeability of groups. This and the previous were both chalk talks. There are the seeds of some very powerful techniques here.
I started running out paper here.
Steve Jackson then discussed the Borel combinatorics of Abelian group actions. This was a summary of recent work Jackson, Gao, and Khrone have put into this field. They have developed some pretty impressive/extendable tools for studying the combinatorics of these actions. There are a surprising number of complications even at the hyperfinite level.
Samuel Coskey finished the morning off with a proof of a continuous logic version of the Lopez Escobar theorem. Continuous logic is super cool, and I'm pretty sure I am going to try and give an expositroy talk on just it for the GLG.
We then had lunch, and I started getting nervous in anticipation of my talk.
Robin Tucker Drob opened up the afternoon section talking about weak containment versus strong containment for group actions in the measure setting. There was a nice generalization of some older work here, along with what appears a good multipurpose lemma.
Next up was Simon Thomas, who attempted to embarrass us all with an exposition of some basic questions that we seem to be unable to solve. I always enjoy it when Martin measures show up, and I think he did a good job of explaining why these problems are actually a bit hairy. Still, it is embarrassing to have simply statable, obvious-looking problems floating around without solutions.
I am now officially out of paper.
Peter Burton talked about Furstenburg entropy and its behavior under weak containment. There are some intriguing open problems here about what kinds of entropy a group can generate under different actions.
Andrew Marks defined a jump operation for Borel graphs in analogy with the jump operation for equivalence relations (not the Friedman-Stanley one). Cool result with some big hammers at play in the proof.
Cheng Chang showed that the problem of classifying continua is a universal orbit equivalence relation. The construction here, reducing the problem of classifying closed to classifying continua is quite clever.
Cody Dance then talked about the external ultrapower of HOD in L(R) via the club filter on omega_1. He is able to use this analysis to give a nice answer to a seemingly sticky combinatorial problem in L(R). Another nice interweaving of inner model theory and determinacy,
I ended the day with talk about my research. It seemed to go okay.
Then I got stuck at my layover...
Solecki began the morning with an exposition on the Pseudoarc and projective Fraisse limits. In particular he realizes the Pseudoarc as a quotient of a portion of a projective Fraisee limit and uses the structure there to provide a proof of its homogeneity, I feel like I want to start a seminar on just Fraisse theory.
Next Martin Sabok talked about the topological conjugacy relation for Toeplitz subshifts. Lots of twists and turns with the results in this talk. Its also nice to see some of the more recent techniques of Jackon and Gao (and Seward and Khrone) being used by other people.
Bruno Braga talked about coarse and uniform embeddings. Banach spaces everywhere. The most exciting part of this talk for me was a clever retooling of the fact that you can extend maps continuously from closed sets to the whole space in the metric space setting. Its nice to see classics get respun.
Clinton Conley talked about one-ended subforests and treeability of groups. This and the previous were both chalk talks. There are the seeds of some very powerful techniques here.
I started running out paper here.
Steve Jackson then discussed the Borel combinatorics of Abelian group actions. This was a summary of recent work Jackson, Gao, and Khrone have put into this field. They have developed some pretty impressive/extendable tools for studying the combinatorics of these actions. There are a surprising number of complications even at the hyperfinite level.
Samuel Coskey finished the morning off with a proof of a continuous logic version of the Lopez Escobar theorem. Continuous logic is super cool, and I'm pretty sure I am going to try and give an expositroy talk on just it for the GLG.
We then had lunch, and I started getting nervous in anticipation of my talk.
Robin Tucker Drob opened up the afternoon section talking about weak containment versus strong containment for group actions in the measure setting. There was a nice generalization of some older work here, along with what appears a good multipurpose lemma.
Next up was Simon Thomas, who attempted to embarrass us all with an exposition of some basic questions that we seem to be unable to solve. I always enjoy it when Martin measures show up, and I think he did a good job of explaining why these problems are actually a bit hairy. Still, it is embarrassing to have simply statable, obvious-looking problems floating around without solutions.
I am now officially out of paper.
Peter Burton talked about Furstenburg entropy and its behavior under weak containment. There are some intriguing open problems here about what kinds of entropy a group can generate under different actions.
Andrew Marks defined a jump operation for Borel graphs in analogy with the jump operation for equivalence relations (not the Friedman-Stanley one). Cool result with some big hammers at play in the proof.
Cheng Chang showed that the problem of classifying continua is a universal orbit equivalence relation. The construction here, reducing the problem of classifying closed to classifying continua is quite clever.
Cody Dance then talked about the external ultrapower of HOD in L(R) via the club filter on omega_1. He is able to use this analysis to give a nice answer to a seemingly sticky combinatorial problem in L(R). Another nice interweaving of inner model theory and determinacy,
I ended the day with talk about my research. It seemed to go okay.
Then I got stuck at my layover...
Saturday, April 9, 2016
4/09/2016
Day 1 of the conference. Lots of really good talks today, and an interesting diversity in material. Here are some thoughts on talks I got notes for.
Su Gao talked about non-archimedian Abelian Polish groups. Lots of really nice results and resolutions to some reduction problems.
Aristotelian Panagiotopolous talked about Menger compacta. There are some really interesting ideas here combining dimensional analysis with DST. More applications of Fraisse limits as well. I'm curious if these ideas can be extended into transfinite dimensions.
Kathryn Mann talked about automatic continuity for the homeomorphism group of a manifold. There were some tools presented here that I hadn't seen before. There is also a nice array of questions brought up by her results.
Richard Rast talked about tools for distinguishing isomorphism classes of countable models. There was some really creative uses of Scott sentences here, and the result feels like a breakthrough in this arena.
William Chan talked about an equivalence relation generated by admissible ordinals for reals. I wish he had more time. There was a lot of delicate set theory at play here. It was a very interesting synthesis of ideas.
Then there was lunch.
Joseph Zielinski talked about a proper he defined called locally Roelcke precompact. It is a natural generalization of a natural property with some interesting connections to locally compact spaces.
Martino Lupini talked about an object he has coined: the Lusky simplex. This one has a lot of functional analysis connections. Honestly this was significantly outside of my wheelhouse.
Konstantin Slutsky talked about different ways to categorize flows and how they interact with dimension. Allowing for the ignoring of very small sets, it turns out they are all essentially equivalent. I find this very unintuitive.
Konstantinos Beros explicitly defined some complete sets in the difference hierarchy. A really solid construction with tantalizing possibilities for generalization.
Aaron Hill closed the day out by applying the space of rank-1 transformations to reduce the complexity of problems in ergodic theory. It's been a trip watching him develop this theory over the years, and the tool he has created is kind of incredible. It will be interesting to see what else he has able to do in this direction.
Friday, April 8, 2016
4/08/2016
Day 0 of the AMS sectional complete. Salt lake city is surrounded by beautiful mountains, and filled with beautiful trees. Lots of walking today to scope out the campus.
Talks start at 8 am tomorrow. I'm nervous to be presenting, but excited to finally have something to contribute. There seem to be a lot of good talks lined up tomorrow, and walking two miles to the conference is better than coffee.
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