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<start_of_turn>user

summarize the following text:

 Thank you very much to the organizers for inviting me to speak, which I guess came a
  long time ago.
  But I'm glad that we all made it here eventually.
  Yeah, so I'm sorry about two things, the first is to break a long streak of very nice chalk
  talk.
  I'm afraid during COVID I got very lazy, so I just give talks on my iPad now.
  The other thing I'm a bit sorry for is there won't really be chart degenerations in this
  So yeah, I'll say kind of a couple of words, but really it's a hint to talk to other experts
  in the audience about some chart degenerations arriving from these hospice centers.
  So that said, what am I going to talk about?
  Well, it's some joint work with Tasha Golashin, Tamek Lam, and David Speier.
  And it's up on the archives, so the details are available.
  Okay, so what is it?
  Well, I'm going to talk about some varieties, type A braid varieties, and a particular combinatorial
  structure on their coordinate rings, which is called a cluster structure.
  So Nathan talked a little bit about what this means yesterday, I'll kind of refresh.
  Okay, so it's a combinatorial structure, so there should be some combinatorial objects
  in the background that give you this information, and those combinatorial objects we call 3D
  playbit graphs.
  It's little pictures that look like this.
  Okay, and I guess I should say also, slides from my talk or some version of these slides
  are on my website, so you don't have to worry about taking notes.
  And okay, before I get going, I'll just say, so I'll say this a couple of times.
  If you have a cluster structure on coordinate ring beer variety, you can, under certain
  assumptions, produce chart degenerations.
  And those chart degenerations, they really depend on this combinatorial gadget that you
  cook up to give you the cluster structure.
  So that's sort of the, if you maybe don't care about cluster alpha growths by themselves,
  that's fine.
  If you wanted to get your hands on any of these chart degenerations, you do need this
  combinatorial information.
  Okay, so first, let me just tell you about type A braid varieties.
  So type A talk, so my G is SLN, and B is the subgroup of upper triangular matrices.
  And as usual, I identified a quotient G mod B with the complete flag variety.
  Okay, just to be really concrete, everyone has different conventions.
  How do I go from a coset G B to a complete flag?
  Well, the one dimensional subspace in my flag, that could be spanned by the first column
  of G, so left most column.
  The second, so the dimension two subspace, that's the left most two columns of G, etc.
  Okay.
  So to define these braid varieties.
  I'm going to need the notion of relative position of two flags, which is a permutation.
  So it's easier to describe when two flags are in relative position SI, so a simple transposition.
  So F and F prime are in SI position if they differ in exactly the I subspace.
  Great, so this transposition SI is telling you that subspace you have to change to get
  from one to the other.
  If you don't like thinking about flags, and instead you like thinking about cosets, well,
  I think you're a little weird, that's okay.
  There's also a way to phrase this implicit subspace.
  Okay, so what about if two flags are in position W, where W is just a permutation, not necessarily
  a simple transposition?
  Well, there's this same coset condition, or the way I prefer to think about it, what this
  means is, well, pick any reduced word for the permutation W that you want, then there's
  a unique sequence of flags, which will interpolate.
  So I interpolate between F and F prime, where, well, to go from F to the first flag in the
  sequence, I change the I1 subspace, so as dictated by my choice of reduced word, and
  then to go from F1 to F2, I change the I2 subspace, et cetera, et cetera, all the way
  until I get to F prime.
  So that's what being W related means.
  Okay, so that said, let me say what type A grade varieties are.
  Okay, so these are indexed by a pair.
  The first element in my pair, I'm going to denote by beta underlined, and this is just
  some word in simple transpositions, not necessarily reduced, so any word you want, and the other
  element in my pair, U, is going to be a permutation that lives inside of this word somewhere.
  So in this word, beta underlined, I have to be able to find a reduced expression for U.
  Okay, so given that pair, here is my grade variety, X beta underlined U.
  Okay, so a tuple of flags, which fit into a diagram like this.
  So I like to think about these tuples of flags as walks.
  So they're walks that start at the standard flag, and then in the first step of my walk,
  I take, well, I take a step in direction SI1, so I change the I1 subspace, and then in the
  second step, I change the I2 subspace, et cetera, all the way until I get to the last
  one.
  So this data of beta underlined is telling me the direction my walk has to move in.
  And then this data of U is going to constrain where I end up.
  So the exact way it constrains it is, my final flag in this walk, it has to be in relative
  position W, not U, with the anti-standard flag.
  Okay, this feels kind of weird, it's like, I will admit there are many more kind of natural
  ways to use the data of U to constrain the final flag.
  I'm making this choice to make connections with ridges and varieties.
  I should say, okay, so another way to phrase this, which maybe uses words heard before,
  well, if I just think about this space of walks where direction is dictated by beta
  underlined, that's an open part of a Bot-Samuelson variety.
  So in the Bot-Samuelson, in my walk, I could kind of like trip, or maybe I could be lazy.
  So in my walk, I could not take a step at all.
  So that would be an option in the full Bot-Samuelson, but I'm demanding that at each step of my
  walk, I really change the flag.
  So this is some open part.
  This is the open part of Bot-Samuelson, and then there's a natural map from this open
  part of Bot-Samuelson down to the flag variety, where I just look at the last flag in my walk.
  And my grade variety is just the fiber over a particular opposite Schubert cell in the
  flag variety of this map U projecting to the last flag.
  So those are my, those are my grade varieties.
  They were, well, not exactly these, but something quite close to these were studied by Laura
  Escobar under the name of Brick Manifolds.
  They're also kind of studied by Melick, who didn't call them anything.
  And then Cathal, Gorstie, Gorstie, and Simonsall called them Brick Manifolds.
  Any questions about the definition?
  Oh, very good question.
  The next point of the next slide.
  Yeah.
  So if you have two of these words.
  So beta underline and gamma underline, and they're related by grade moves.
  So either you're commuting like SI and SJ where I and J are far apart, or you're doing
  the grade move.
  So if they're related by grade moves, then the grade varieties are isomorphic.
  So in fact, what I'm going to do is instead of indexing my grade varieties by the pair
  of beta underline U, I'm just going to index it by beta U where beta is the element of
  positive grade monoid represented by beta underline.
  Positive grade monoid.
  You don't know what that is.
  It's not really important.
  It's just, it's just like the symmetric group, except you throw away that SI squared equals
  that.
  So yeah.
  So really what I'm trying to say is I have kind of a large isomorphism class of these
  varieties and I really just kind of, well, yeah, I kind of want to think about them all.
  So some, some things we know about them.
  So some things we know about these varieties, well, they're smooth, they're irreducible,
  they're affine or maybe quasi-affine if you want to be picky, and their dimension is
  the length of beta minus the length of beta.
  So there definitely are units.
  There, there, there really are units on these, on these varieties.
  So if you, if you feel a little bit like, okay, I mean, this is fine, but why are we
  looking at these?
  Let me give you a couple of examples, some nice examples, which I guess if you don't
  know about the nice examples in advance, they won't be helpful, but if you do, hopefully
  it's a motivation for examining.
  So the, really the nicest case to think about is when your, or the kind of nicest special
  case to think about is when your word beta underline is actually a reduced word.
  So it's a reduced expression for some permutation B. Okay.
  In that situation, having a walk like this between B and your final flag is actually
  just saying that B and your final, final flag are in position B with each other.
  That's all that it's saying.
  So all of the information, like all of these flags, okay, and moreover, all of these flags
  in the middle of the walk, they're actually completely determined by this flag FL plus
  So in the situation where beta underline is reduced, you may as well just pay attention
  to the final flag in your walk.
  And so when you do that, you get an isomorphism from this kind of breed variety, just to a
  subset of the full flag variety.
  And what's the subset?
  Well, it's just all the flag F that are in position B with the standard flag and position
  W not U with the end of standard flag.
  I mean, if you think about this a little bit harder, this is just an interesting example
  of an intersection of a Schubert's cell and an opposite Schubert's cell inside the full
  flag.
  This is called an open Richardson's variety.
  So I think they kind of come up in a lot of contexts, including total positivity and
  work at loosing and breach.
  Another interesting thing about them is their point counts over finite fields give you R
  polynomials, which are important for computing Cauchon-Lustig polynomials.
  So an even more special case of this nice special case.
  Is if your permutation V has a unique descent, so beta underline is reduced expression for
  this special kind of permutation V. In that situation, your grade variety is actually
  isomorphic to a subset of a Grasmanian, so even better than being a subset of the full
  It's isomorphic to what's called an open polythroid variety.
  So these were, that name was coined and these were defined by Kutz and Lam and Spire, very
  inspired by work of Kostnikov on polythroidism.
  And I should also mention that Reach previously did work on Richardson's projected to Grasmanians
  and other carbonyl flag varieties.
  So I can't actually give you any real information about these open polythroid varieties.
  What I will say is that there's a lot of really lovely combinatorics here.
  And these open polythroid varieties, I think, are fairly well understood and worth of study
  in their own right.
  I'm just going to say, oh, they're a special piece of this thing that I'm going to talk
  about.
  And then kind of the very nicest open polythroid variety, well, or at least the biggest one,
  is a particular subset of GRKN that I'm going to call GRKN-Circ.
  And that's the subset of the Grasmanian where the cyclically consecutive Kluger coordinates
  are known zero.
  This came up in Nathan's talk yesterday.
  So this is a special, very special example of one of these grades.
  So those are the varieties I'll be dealing with.
  And now let me say...
  So that one there is the complementary divisor.
  Mm-hmm.
  But that's not true in general, I bet.
  Yeah, it is.
  So let me...
  So...
  Yes.
  That's right.
  But we're always going to be thinking about complementary divisors.
  So that's what all the varieties will be.
  Or inside the brick manifold, it's the complement of an anticononical divisor.
  And what about here?
  That is here.
  So the...
  And it also seems to be sort of...
  Is it a sort of monomial divisor, I think, there, in the...
  What's the question?
  Super embedding, and...
  So it's the coordinates.
  We looked at it.
  It's got P on them.
  I'm not...
  He says that in general.
  I wonder whether it's in general as well.
  So, I mean, you have to say what coordinates you're choosing here, but if you choose the
  right coordinates, then I guess I'll tell you, yes.
  Exactly.
  Your coordinates are not always going to be...
  They're not always going to be...
  But I will...
  I mean, the coordinates will be some frozen variable that across the structure...
  Sorry.
  I didn't get what you said.
  It's a meta-theory keyword.
  We're all going the same way.
  Indeed.
  So those are the varieties.
  Now let me tell you what I'm going to tell you about them.
  So what I'm going to do is...
  So I pick a...
  I fix a braid variety of interest.
  So I activate a U.
  And moreover, I pick, actually, a braid word for this braid.
  So I pick a beta underline.
  And from this...

  <end_of_turn>
<start_of_turn>model

This text describes the speaker's talk about graded varieties and their relationship to flag variety in mathematics using an example of type A braids with cluster structures on coordinate rings (variety X). The key points covered include :



* **Cluster structure:** If a braid has this particular combinatorial object called "cluster sigma" which determines its chart degeneration, then that specific kind  of variation can produce the same degenerations.




- Two flags are in relative position W if they have similar subspaces spanned by columns of G as dictated through their reduced words and simple transpositions (SI) for flag variety indexing or coset representatives from a particular subgroup B within group F, which is equivalent to saying that two Flags lie on the same Schubert cell.




- Grade varieties are indexed pairs beta underlined U where there exists an expression  of permutation V in terms of this word such as "beta underline" being reduced and therefore minimizing steps needed for transformation from standard flag position W into final Flag F determined by their graded variety indexing pair (which is isomorphic to the fiber over a particular opposite Schubert cell).




- Open polythroid varieties are special cases where beta underlined comes with unique descent. These have nice combinatorial properties including being subsets of Grasmannian and possessing R polynomials as point counts which make them important for computing Cauchon - Lustig polynomial calculations, among other applications in mathematics research.. [end of text]

llama_print_timings:        load time =    2323.54 ms
llama_print_timings:      sample time =     249.35 ms /   285 runs   (    0.87 ms per token,  1142.98 tokens per second)
llama_print_timings: prompt eval time =  372637.11 ms /  3405 tokens (  109.44 ms per token,     9.14 tokens per second)
llama_print_timings:        eval time =  208740.72 ms /   284 runs   (  735.00 ms per token,     1.36 tokens per second)
llama_print_timings:       total time =  581974.90 ms /  3689 tokens
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