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BEGIN:VEVENT
SUMMARY:Dongkwan Kim (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20201001T170000Z
DTEND;VALUE=DATE-TIME:20201001T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/1
DESCRIPTION:Title: Robinson-Schensted correspondence for natural unit interval orders\nby Dongkwan Kim (University of Minnesota) as part of UCLA Combinatoric
s Forum\n\n\nAbstract\nStanley-Stembridge conjecture\, currently one of th
e most famous conjectures in algebraic combinatorics\, asks whether a cert
ain generating function with respect to a natural unit interval order is a
nonnegative linear combination of complete homogeneous symmetric function
s. There are many partial progress on this conjecture\, including its conn
ection with the geometry of Hessenberg varieties. \n\nIn this talk we stud
y the Schur positivity\, which is originally proved by Haiman and Gasharov
. We define an analogue of Knuth moves with respect to a natural unit inte
rval order and study its equivalence classes in terms of D graphs introduc
ed by Assaf. Then\, we show that if the given order avoids certain two sub
orders then an analogue of Robinson-Schensted correspondence is well-defin
ed\, which proves that the generating function attached to each equivalenc
e class is Schur positive. It is hoped that it proposes a new combinatoria
l aspect to investigate the Stanley-Stembridge conjectures and cohomology
of Hessenberg varieties. This work is joint with Pavlo Pylyavskyy.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oliver Pechenik (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20201008T170000Z
DTEND;VALUE=DATE-TIME:20201008T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/2
DESCRIPTION:Title: Gröbner geometry of Schubert polynomials through ice\nby Oliver
Pechenik (University of Waterloo) as part of UCLA Combinatorics Forum\n\n
\nAbstract\nKnutson and Miller (2005) showed that the equivariant cohomolo
gy class of a matrix Schubert variety $X_w$ is the corresponding double Sc
hubert polynomial $S_w$. Moreover\, after Gröbner degeneration with respe
ct to any antidiagonal term order\, the resulting irreducible components a
re naturally labeled by the pipe dreams for w. In later work with Yong (20
09)\, they used diagonal term orders to obtain alternative combinatorics f
or certain $X_w$. We present further results in this direction\, with conn
ections to a neglected Schubert polynomial formula of Lascoux (2002) in te
rms of the 6-vertex ice model (recently rediscovered by Lam\, Lee\, and Sh
imozono in the guise of “bumpless pipe dreams”).\n\nNote: the talk wil
l be accessible to the general audience.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Sherman-Bennett (Harvard University/UC Berkeley)
DTSTART;VALUE=DATE-TIME:20201015T170000Z
DTEND;VALUE=DATE-TIME:20201015T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/3
DESCRIPTION:Title: Plabic graphs and cluster structures on positroid varieties\nby
Melissa Sherman-Bennett (Harvard University/UC Berkeley) as part of UCLA C
ombinatorics Forum\n\n\nAbstract\nOpen positroid varieties are smooth irre
ducible subvarieties of the Grassmannian\, which can be naturally defined
using "positively realizable" matroids (positroids\, for short). They wer
e first introduced by Knutson\, Lam\, and Speyer\, motivated by work of Po
stnikov on the totally nonnegative (real) Grassmannian and positroid cells
. Open positroid varieties are indexed by a number of combinatorial object
s\, including families of plabic (i.e. planar bicolored) graphs. \n\nI wil
l discuss some algebraic information plabic graphs give us about open posi
troid varieties. Together with Serhiyenko and Williams\, we showed that pl
abic graphs for an open Schubert variety $V$ (a special case of open posit
roid varieties) give seeds for a cluster algebra structure on the homogene
ous coordinate ring of $V$. Among other things\, this implies that plabic
graphs give positivity tests for elements of $V$. \n\nOur work generalize
s a result of Scott on the Grassmannian\, and confirms a longstanding folk
lore conjecture on Schubert varieties\; it was later generalized to arbitr
ary positroid varieties by Galashin and Lam. I'll also discuss recent work
with Fraser\, in which we show that relabeled plabic graphs also give see
ds for a cluster algebra structure on coordinate rings of open positroid v
arieties\, uncovering another source for positivity tests. \n\nNo knowledg
e of cluster algebras will be assumed in the talk.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Swee Hong Chan (UCLA)
DTSTART;VALUE=DATE-TIME:20201022T170000Z
DTEND;VALUE=DATE-TIME:20201022T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/4
DESCRIPTION:Title: Sorting probability for Young diagrams\nby Swee Hong Chan (UCLA)
as part of UCLA Combinatorics Forum\n\n\nAbstract\nCan you always find tw
o elements $x$\, $y$ of a partially ordered set\, such that\, the probabil
ity that x is ordered before y when the poset is ordered randomly\, is bet
ween $1/3$ and $2/3$?\nThis is the celebrated $1/3 - 2/3$ Conjecture\, whi
ch has been called "one of the most intriguing problems in the combinatori
al theory of posets".\n\nWe will explore this conjecture for posets that a
rise from (skew-shaped) Young diagrams\, where total orderings of these po
sets correspond to standard Young tableaux. We will show that that these p
robabilities are arbitrarily close to $1/2$\, by using random walk estimat
es and the state-of-the-art hook-length formulas of Naruse. \n\nThis is a
joint work with Igor Pak and Greta Panova. This talk is aimed at a genera
l audience.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura Escobar (Washington University in St. Louis)
DTSTART;VALUE=DATE-TIME:20201029T170000Z
DTEND;VALUE=DATE-TIME:20201029T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/5
DESCRIPTION:Title: Which Schubert varieties are Hessenberg varieties?\nby Laura Esc
obar (Washington University in St. Louis) as part of UCLA Combinatorics Fo
rum\n\n\nAbstract\nSchubert varieties are subvarieties of the flag variety
parametrized by permutations\; they induce an important basis for the coh
omology of the flag variety. Hessenberg varieties are also subvarieties of
the flag variety with connections to both algebraic combinatorics and rep
resentation theory. I will discuss joint work with Martha Precup and John
Shareshian in which we investigate which Schubert varieties in the full fl
ag variety are Hessenberg varieties.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abdul Basit (Iowa State University)
DTSTART;VALUE=DATE-TIME:20201112T180000Z
DTEND;VALUE=DATE-TIME:20201112T190000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/7
DESCRIPTION:Title: Point-box incidences and logarithmic density of semilinear graphs\nby Abdul Basit (Iowa State University) as part of UCLA Combinatorics Fo
rum\n\n\nAbstract\nZarankiewicz's problem in extremal graph theory asks fo
r the maximum number of edges in a bipartite graph on $n$ vertices which d
oes not contain a copy of $K_{k\,k}$\, the complete bipartite with $k$ ver
tices in both classes. We will consider this question for incidence graphs
of geometric objects. Significantly better bounds are known in this setti
ng\, in particular when the geometric objects are defined by systems of al
gebraic inequalities. We show even stronger bounds under the additional co
nstraint that the defining inequalities are linear. We will also discuss c
onnections of these results to combinatorial geometry and model theory. \n
\nNo background is assumed\, and the talk will be accessible to non-expert
s. Joint work with Artёm Chernikov\, Sergei Starchenko\, Terence Tao\, an
d Chieu-Minh Tran.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonah Blasiak (Drexel University)
DTSTART;VALUE=DATE-TIME:20201119T180000Z
DTEND;VALUE=DATE-TIME:20201119T190000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/8
DESCRIPTION:Title: Crystal graphs\, katabolism\, and Schur positivity\nby Jonah Bla
siak (Drexel University) as part of UCLA Combinatorics Forum\n\n\nAbstract
\nKatabolism is a mysterious operation on tableaux which involves cutting
and reassembling the pieces\nusing Schensted insertion.\nIt is featured in
several Schur positivity conjectures related to\nk-Schur functions and Ha
ll-Littlewood polynomials.\nCrystal graphs are the combinatorial skeletons
of gl_n modules and are a powerful tool for connecting representation the
ory and combinatorics. \nFor instance\, they give a beautiful explanation
of the RSK correspondence.\nUsing crystal graphs\, we uncover the mystery
behind katabolism and resolve a Schur positivity conjecture of Shimozono a
nd Weyman.\nThis talk will include many pictures of crystals and tableaux.
\nThis is joint work with Jennifer Morse and Anna Pun.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Williams (UT Dallas)
DTSTART;VALUE=DATE-TIME:20201203T180000Z
DTEND;VALUE=DATE-TIME:20201203T190000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/10
DESCRIPTION:Title: Strange Expectations\nby Nathan Williams (UT Dallas) as part of
UCLA Combinatorics Forum\n\n\nAbstract\nWe extend our previous work on si
multaneous cores for affine Weyl groups. In type A\, our uniform formula r
ecovers Drew Armstrong's conjecture for the average number of boxes in a s
imultaneous core. This is joint work with Marko Thiel and Eric Stucky.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Bernardi (Brandeis University)
DTSTART;VALUE=DATE-TIME:20201210T180000Z
DTEND;VALUE=DATE-TIME:20201210T190000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/11
DESCRIPTION:Title: A Universal Tutte Polynomial\nby Olivier Bernardi (Brandeis Uni
versity) as part of UCLA Combinatorics Forum\n\n\nAbstract\nWouldn't it be
nice to have a polynomial expression parametrizing at once the Tutte poly
nomial of every matroid of a given size?\nIn this talk\, I will explain ho
w to achieve this goal. The solution involves extending the definition of
the Tutte polynomial from the setting of matroids to the setting of polyma
troids (this is akin to the generalization from graphs to hypergraphs)\, a
nd adopting a geometric point-counting perspective. On our way\, we will c
onnect several notions: the activity-counting invariants of Kalman and Pos
tnikov\, the point-counting invariants of Cameron and Fink\, and the class
ical corank-nullity definition of the Tutte polynomial of matroids.\nThis
is joint work with Tamas Kalman and Alex Postnikov.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Swee Hong Chan (UCLA)
DTSTART;VALUE=DATE-TIME:20210930T210000Z
DTEND;VALUE=DATE-TIME:20210930T220000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/12
DESCRIPTION:Title: Log-concave inequalities for posets\nby Swee Hong Chan (UCLA) a
s part of UCLA Combinatorics Forum\n\nLecture held in UCLA Math Sciences b
uilding\, room MS 3915A.\n\nAbstract\nThe study of log-concave inequalitie
s for combinatorial objects have seen much progress in recent years. One s
uch progress is the solution to the strongest form of Mason's conjecture (
independently by Anari et. al. and Brándën-Huh) that the f-vectors of ma
troid independence complex is ultra-log-concave. In this talk\, we discuss
a new proof of this result through linear algebra and discuss generalizat
ions to greedoids and posets. This is a joint work with Igor Pak.\n\nThe t
alk is aimed at a general audience.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Petrov (UVA and MSRI)
DTSTART;VALUE=DATE-TIME:20211014T210000Z
DTEND;VALUE=DATE-TIME:20211014T220000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/13
DESCRIPTION:Title: Schur rational functions\, vertex models\, and random domino tiling
s\nby Leonid Petrov (UVA and MSRI) as part of UCLA Combinatorics Forum
\n\nLecture held in UCLA Math Sciences Building\, room MS 3915A..\n\nAbstr
act\nIt is known that Schur symmetric polynomials admit a number of genera
lizations (Macdonald's 1992 variations) which retain determinantal structu
re - for example\, factorial and supersymmetric Schur functions. We descri
be an overarching family of Schur-like rational functions arising as parti
tion functions of fully inhomogeneous free fermion six vertex model. These
functions are indexed by partitions\, have as variables the pairs $(x_i\,
r_i)$\, $i=1\,...\,N$\, of horizontal rapidities and spin parameters\; and
\, moreover\, depend on vertical rapidities and spin parameters $(y_j\,s_j
)$\, $j>=1$. We establish determinantal formulas\, orthogonality\, Cauchy
identities\, and other properties of our functions. We also introduce rand
om domino tiling models based on the Schur rational functions (a la Schur
processes of Okounkov-Reshetikhin 2001)\, and obtain bulk (lattice) asympt
otics leading to a new deformation of the extended discrete sine kernel. B
ased on the joint project https://arxiv.org/abs/2109.06718 with A. Aggarwa
l\, A. Borodin\, and M. Wheeler.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yibo Gao (MIT)
DTSTART;VALUE=DATE-TIME:20211028T170000Z
DTEND;VALUE=DATE-TIME:20211028T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/14
DESCRIPTION:Title: The canonical bijection between pipe dreams and bumpless pipe dream
s\nby Yibo Gao (MIT) as part of UCLA Combinatorics Forum\n\nLecture he
ld in *Virtual only*.\n\nAbstract\nPipe dreams and bumpless pipe dreams ar
e two combinatorial objects that enumerate Schubert polynomials\, and it h
as been an open problem to find a weight-preserving bijection between thes
e two objects since bumpless pipe dreams were introduced by Lam\, Lee and
Shimozono. In this talk\, we present such a bijection and establish its ca
nonical nature by showing that it preserves Monk's rule. This is joint wor
k with Daoji Huang.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Terrence George (University of Michigan)
DTSTART;VALUE=DATE-TIME:20211104T210000Z
DTEND;VALUE=DATE-TIME:20211104T220000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/15
DESCRIPTION:Title: Electrical networks and Lagrangian Grassmannians\nby Terrence G
eorge (University of Michigan) as part of UCLA Combinatorics Forum\n\n\nAb
stract\nCactus networks were introduced by Thomas Lam as a generalization
of planar electrical networks.\nHe defined a map from these networks to a
Grassmannian and showed that the image of this map lies inside the totally
nonnegative part of this Grassmannian. We show that the image of Lam's ma
p consists of exactly the elements that are both totally nonnegative and i
sotropic for a particular skew-symmetric bilinear form. This is joint work
with Sunita Chepuri and David Speyer.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suho Oh (Texas State)
DTSTART;VALUE=DATE-TIME:20211118T220000Z
DTEND;VALUE=DATE-TIME:20211118T230000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/16
DESCRIPTION:Title: Extending Shellings\nby Suho Oh (Texas State) as part of UCLA C
ombinatorics Forum\n\n\nAbstract\nShellable complexes are simplicial compl
exes with the shelling property: their facets can be ordered nicely\, whic
h translates to interesting properties in algebra and combinatorics. Simon
in 1994 conjectured that any shellable complex can be extended to the k-s
keleton of a simplex while maintaining the shelling property. We go over v
arious tools and results related to this problem. In particular\, we will
be going over a recent joint work with M. Coleman\, A. Dochtermann and N.
Geist on proving this conjecture for a smaller class.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josh Swanson (USC)
DTSTART;VALUE=DATE-TIME:20211007T210000Z
DTEND;VALUE=DATE-TIME:20211007T220000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/17
DESCRIPTION:Title: Combinatorics of harmonic polynomial differential forms\nby Jos
h Swanson (USC) as part of UCLA Combinatorics Forum\n\nLecture held in UCL
A Math Sciences Building\, room MS 3915A..\n\nAbstract\nA recent conjectur
e of Zabrocki introduced super diagonal coinvariant algebras as a represen
tation-theoretic model for the Delta conjecture of Haglund--Remmel--Wilson
. Subsequent work with Wallach introduced a basis for the alternating comp
onent of the super coinvariant algebra consisting of explicit harmonic pol
ynomials in commuting and anti-commuting variables. We will discuss two fa
milies of relations involving these harmonics\, which are related to Tanis
aki ideals and which we call Tanisaki witness relations. This talk will fo
cus on the combinatorics of these objects rather than the underlying abstr
act motivation.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolle Gonzalez (UCLA)
DTSTART;VALUE=DATE-TIME:20211021T210000Z
DTEND;VALUE=DATE-TIME:20211021T220000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/18
DESCRIPTION:Title: From symmetric functions\, to knots\, and back again.\nby Nicol
le Gonzalez (UCLA) as part of UCLA Combinatorics Forum\n\nLecture held in
UCLA Math Sciences Building\, room MS 3915A.\n\nAbstract\nFor fourteen yea
rs the shuffle conjecture remained open. In essence\, it gave a combinator
ial formula for the Frobenius character of the space of diagonal harmonics
in terms of word parking functions\, which are certain symmetric function
s that can be indexed by lattice paths. Concretely\, the conjecture stated
that this combinatorial sum was equal to the action of the ubiquitous nab
la operator on the nth elementary symmetric polynomial. In a startling pap
er\, Carlsson and Mellit proved this conjecture by introducing a new algeb
ra\, closely related to the double affine Hecke algebra\, called $A_{q\,t}
$ and defining an important polynomial action for it. This algebra allowed
them to perform operations on symmetric functions by lifting the correspo
nding structures via raising and lowering operators to certain higher leve
l polynomial rings\, performing the computations there\, and then projecti
ng them back. Shortly thereafter\, using parabolic flag Hilbert schemes\,
the algebra and its representation was also realized geometrically by Carl
sson\, Gorsky\, and Mellit\, reaffirming its connection to the work Haiman
on the space of diagonal harmonics. I will introduce a topological interp
retation of this algebra and its representation. Namely\, I will describe
how we can realize symmetric functions and the $A_{q\,t}$ operators as bra
id diagrams on an annulus and how many of the complicated algebraic relati
ons using plethysms in the original formulation follow trivially from isot
opy of the diagrams. This paradigm not only eases many computations\, it a
lso informs us of new operators on symmetric functions that while natural
from a topological perspective might be very difficult to see algebraicall
y\, thus yielding new light on an already rich structure. Of particular in
terest is its ability to potentially explain the many conjectures relating
the homology of the toric links and q\,t combinatorics.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:GaYee Park (UMass Amherst)
DTSTART;VALUE=DATE-TIME:20211012T170000Z
DTEND;VALUE=DATE-TIME:20211012T180000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/19
DESCRIPTION:Title: Minimal semi-standard skew tableaux and the Hillman-Grassl correspo
ndence\nby GaYee Park (UMass Amherst) as part of UCLA Combinatorics Fo
rum\n\nLecture held in *Virtual only*.\n\nAbstract\nStandard tableaux of s
kew shape are fundamental objects in\nenumerative and algebraic combinator
ics and no product formula for the\nnumber is known. In 2014\, Naruse gave
a formula as a positive sum over\nexcited diagrams of products of hook-le
ngths. In 2018\, Morales\, Pak\, and\nPanova gave a $q$-analogue of Naruse
's formula for semi-standard tableaux\nof skew shapes. They also showed\,
partly algebraically\, that the\nHillman-Grassl map restricted to skew sha
pes gave their $q$-analogue. We\nstudy the problem of making this argument
completely bijective. For a skew\nshape\, we define a new set of semi-sta
ndard Young tableaux\, called the\nminimal SSYT\, that are equinumerous wi
th excited diagrams via a new\ndescription of the Hillan-Grassl bijection
and have a version of excited\nmoves. Lastly\, we relate the minimal skew
SSYT with the terms of the\nOkounkov-Olshanski formula for counting SYT of
skew shape. This is joint\nwork with Alejandro Morales and Greta Panova.\
n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melissa Sherman-Bennett (University of Michigan)
DTSTART;VALUE=DATE-TIME:20211116T220000Z
DTEND;VALUE=DATE-TIME:20211116T230000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/20
DESCRIPTION:Title: The hypersimplex and the m=2 amplituhedron\nby Melissa Sherman-
Bennett (University of Michigan) as part of UCLA Combinatorics Forum\n\nLe
cture held in MS 6627 (!).\n\nAbstract\nI'll discuss a curious corresponde
nce between the $m=2$ amplituhedron\, a $2k$-dimensional subset of $\\math
rm{Gr}(k\, k+2)$\, and the hypersimplex\, an $(n-1)$-dimensional polytope
in $\\mathbb R^n$. The amplituhedron and hypersimplex are both images of t
he totally nonnegative Grassmannian under some map (the amplituhedron map
and the moment map\, respectively)\, but are different dimensions and live
in very different ambient spaces. I'll talk about joint work with Matteo
Parisi and Lauren Williams in which we give a bijection between decomposit
ions of the amplituhedron and decompositions of the hypersimplex (original
ly conjectured by Lukowski--Parisi--Williams). Along the way\, we prove th
e sign-flip description of the $m=2$ amplituhedron conjectured by Arkani-H
amed--Thomas--Trnka and give a new decomposition of the $m=2$ amplituhedro
n into Eulerian-number-many chambers\, inspired by an analogous hypersimpl
ex decomposition.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marianna Russkikh (MIT)
DTSTART;VALUE=DATE-TIME:20211109T180000Z
DTEND;VALUE=DATE-TIME:20211109T190000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/21
DESCRIPTION:Title: Dimers and circle patterns\nby Marianna Russkikh (MIT) as part
of UCLA Combinatorics Forum\n\n\nAbstract\nThe dimer model is a model from
statistical mechanics corresponding to random perfect matchings on graphs
. Circle patterns are a class of embeddings of planar graphs such that eve
ry face admits a circumcircle. We describe how to construct a 't-embedding
' (or a circle pattern) of a dimer planar graph using its Kasteleyn weight
s\, and discuss algebro-geometric properties of these embeddings.\nThis ne
w class of embeddings is the key for studying Miquel dynamics\, a discrete
integrable system on circle patterns: we identify Miquel dynamics on the
space of square-grid circle patterns with the Goncharov-Kenyon dimer dynam
ics and deduce the integrability of the former one and show that the evolu
tion is governed by cluster algebra mutations.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hunter Spink (Stanford)
DTSTART;VALUE=DATE-TIME:20211202T220000Z
DTEND;VALUE=DATE-TIME:20211202T230000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/22
DESCRIPTION:Title: Anti-concentration of random walks on model-theoretic definable set
s\nby Hunter Spink (Stanford) as part of UCLA Combinatorics Forum\n\n\
nAbstract\nClassical anti-concentration results show that random walks in
$\\mathbb{R}^d$ with BIG independent steps can’t concentrate in balls mu
ch better than they can concentrate on individual points.\n\nModel-theoret
ic *definable sets* include Boolean combinations of subsets of $\\mathbb{R
}^d$ defined using equalities and inequalities of arbitrary compositions o
f polynomials\, $e^x$\, $\\ln(x)$ and analytic functions restricted to com
pact boxes. For example\, the intersection of $e^{\\sin(1/(1+(xyz)^2))+x^2
y}+zy \\geq0$ and $xyz=5$ in $\\mathbb{R}^3$.\n\nIn this talk\, I will dis
cuss recent results which show that random walks in $\\mathbb{R}^d$ with A
RBITRARY independent steps can’t concentrate in definable sets not conta
ining line segments much better than they can concentrate on individual po
ints. Time permitting\, I will discuss how these results extend to other g
roups like $\\mathrm{GL}_d(\\mathbb{R})$.\n\nJoint work with Jacob Fox and
Matthew Kwan.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allen Knutson (Cornell)
DTSTART;VALUE=DATE-TIME:20211204T000000Z
DTEND;VALUE=DATE-TIME:20211204T010000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/23
DESCRIPTION:Title: The commuting scheme and generic pipe dreams\nby Allen Knutson
(Cornell) as part of UCLA Combinatorics Forum\n\nLecture held in ! MS 5225
!.\n\nAbstract\nThe space of pairs of commuting matrices is more mysterio
us than you\nmight think -- in particular\, Hochster's 1984 conjecture tha
t it is\nreduced remains unresolved. I'll explain how to degenerate it to
one\ncomponent of the "lower-upper scheme" {(X\,Y) : XY lower triangular\,
\nYX upper triangular}\, a reduced complete intersection\, and how to\ncom
pute the degree of any component as a sum over "generic pipe dreams".\nAs
a consequence\, this recovers both the "pipe dream" and\n"bumpless pipe dr
eam" formulae for double Schubert polynomials.\nSome of this work is joint
with Paul Zinn-Justin.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Karpenkov (University of Liverpool)
DTSTART;VALUE=DATE-TIME:20220414T210000Z
DTEND;VALUE=DATE-TIME:20220414T220000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/24
DESCRIPTION:Title: Combinatorics and Geometry of Markov numbers\nby Oleg Karpenkov
(University of Liverpool) as part of UCLA Combinatorics Forum\n\nLecture
held in MS 7608.\n\nAbstract\nMarkov numbers are positive solutions to the
Markov Diophantine equation $x^2+y^2+z^2=3xyz$. The set of Markov numbers
can be very simple\, generated iteratively starting with the smallest sol
ution $(1\,1\,1)$\, which defines a natural binary tree structure on the s
et of all solutions. Markov numbers appear in the study of integer minima
of quadratic forms\, cluster algebra\, etc.\n\nIn this talk we introduce g
eneralized Markov numbers and extend the classical Markov theory. We show
that the principles hidden in Markov's theory are much broader and can be
substantively extended beyond the limits of Markov's theory.\n\nIn particu
lar we discuss recursive properties for these numbers and find correspondi
ng values in the Markov spectrum. Further we give a counterexample to the
generalized Markov uniqueness conjecture. The proposed generalization is b
ased on geometry of numbers\, it substantively uses lattice trigonometry a
nd geometric theory of continued numbers.\n\nThe talk is accessible to the
general audience.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugene Gorsky (UC Davis)
DTSTART;VALUE=DATE-TIME:20221006T210000Z
DTEND;VALUE=DATE-TIME:20221006T220000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/25
DESCRIPTION:Title: Cluster structures on braid varieties\nby Eugene Gorsky (UC Dav
is) as part of UCLA Combinatorics Forum\n\nLecture held in MS 7608.\n\nAbs
tract\nGiven a positive braid\, one can define a smooth affine algebraic v
ariety called the braid variety. Braid varieties generalize several import
ant varieties in Lie theory such as open Richardson and positroid varietie
s. I will construct a cluster structure on a braid variety of arbitrary ty
pe using combinatorial objects called weaves. This is a joint work with Ro
ger Casals\, Mikhail Gorsky\, Ian Le\, Linhui Shen and Jose Simental.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Be'eri Greenfeld (UCSD)
DTSTART;VALUE=DATE-TIME:20221027T233000Z
DTEND;VALUE=DATE-TIME:20221028T001000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/26
DESCRIPTION:Title: Growth of unbounded subsets in nilpotent groups and random mapping
statistics\nby Be'eri Greenfeld (UCSD) as part of UCLA Combinatorics F
orum\n\nLecture held in MS 6627.\n\nAbstract\nLet $G$ be an infinite group
. Let $g(k\,n)$ be the maximum number of length-$n$ words over an arbitrar
y $k$-letter subset of $G$. How does $g(k\,n)$ behave? Obviously\, $g(k\,n
)$ is at most $k^n$\, and Semple-Shalev proved that if $G$ is finitely gen
erated and residually finite then $g(k\,n)< k^n$ if and only if $G$ is vir
tually nilpotent. It is then natural to ask how far $g(k\,n)$ can get from
$k^n$\; for $k$ fixed and $n$ tending to infinity\, $g(k\,n)$ is polynomi
ally bounded.\n\nWe quantify the Semple-Shalev Theorem at the other extrem
e\, where $k=\\Theta(n)$. Specifically\, for a finitely generated residual
ly finite group $G$\, the ratio $g(k\,n)/k^n$ either tends to zero (if and
only if $G$ is virtually abelian)\, or is greater than or equal to an exp
licitly calculated optimal threshold. For higher- step free nilpotent grou
ps\, this ratio tends to 1.\n\nAlong the way\, we find the probability tha
t a random function $f:[n]\\to [n]$ can be recovered from a suitable "inve
rsion set"\, and geometrically interpret our results via random paths in $
\\mathbb{Z}^n$ and the areas of their projected polygons. Finally\, we pro
vide a model-theoretic characterization of suboptimality of $g(k\,n)$ by m
eans of free sub-models and polynomial identities\, which enables to gener
alize the discussion to various other classes of algebraic structures.\n\n
This is a joint work with Hagai Lavner.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Colleen Robichaux (UCLA)
DTSTART;VALUE=DATE-TIME:20221028T002000Z
DTEND;VALUE=DATE-TIME:20221028T010000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/27
DESCRIPTION:Title: Degrees of Grothendieck polynomials and Castelnuovo-Mumford regular
ity\nby Colleen Robichaux (UCLA) as part of UCLA Combinatorics Forum\n
\nLecture held in MS 6627.\n\nAbstract\nWe give an explicit formula for th
e degree of a vexillary Grothendieck polynomial. This generalizes a previo
us result of Rajchgot-Ren-Robichaux-St.Dizier-Weigandt for degrees of symm
etric Grothendieck polynomials. We apply these formulas to compute the Cas
telnuovo-Mumford regularity of certain Kazhdan-Lusztig varieties coming fr
om open patches of Grassmannians as well as the regularity of mixed one-si
ded ladder determinantal ideals. This is joint work with Jenna Rajchgot an
d Anna Weigandt.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daping Weng (UC Davis)
DTSTART;VALUE=DATE-TIME:20221028T011000Z
DTEND;VALUE=DATE-TIME:20221028T015000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/28
DESCRIPTION:Title: Grid plabic graphs\, Legendrian weaves\, and (quasi-)cluster struct
ures\nby Daping Weng (UC Davis) as part of UCLA Combinatorics Forum\n\
nLecture held in MS 6627.\n\nAbstract\nGiven a "grid" plabic graph on $\\m
athbb{R}^2$\, we can construct a Legendrian link\, which is a link in $\\m
athbb{R}^3$ satisfying certain tangential conditions. We study a moduli sp
ace problem associated with the Legendrian link\, and construct a natural
(quasi-)cluster structure on this moduli space using Legendrian weaves. In
particular\, we prove that any braid variety associated with $\\beta \\De
lta$ for a 3-strand braid $\\beta$ admits cluster structures with an expli
cit construction of initial seeds. We also construct Donaldson-Thomas tran
sformations for these moduli spaces.\n\nIn this talk\, I will introduce th
e theoretical background and describe the basic combinatorics for construc
ting Legendrian weaves and the (quasi-)cluster structures from a grid plab
ic graph. This is based on a joint work with Roger Casals (arXiv:2204.1324
4).\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Korotkikh (MIT)
DTSTART;VALUE=DATE-TIME:20221111T003000Z
DTEND;VALUE=DATE-TIME:20221111T011000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/29
DESCRIPTION:Title: Spin $q$-Whittaker symmetric functions and vertex models.\nby S
ergei Korotkikh (MIT) as part of UCLA Combinatorics Forum\n\nLecture held
in MS 6627.\n\nAbstract\nWe introduce a new family of symmetric functions
called spin $q$-Whittaker functions. We have found these functions using s
olvable vertex model from mathematical physics and they basically are defi
ned by taking a specific sum over path ensembles on a square grid. I will
describe this construction from scratch and will explain how it leads to t
he following two properties. First property is the generalization of Cauch
y summation identity: an important identity from algebraic combinatorics w
hich encapsulates orthogonality relations for symmetric functions. Second
property is a unique structure of the vanishing points of our functions wh
ich leads to a characterization in terms of an interpolation problem which
is similar to the work of Okounkov from 1997 about interpolation properti
es of symmetric functions. All necessary background on vertex models and s
ymmetric functions will be explained and\, time permitting\, I will also c
over connections to probability and quantum groups.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Greta Panova (USC)
DTSTART;VALUE=DATE-TIME:20221111T012000Z
DTEND;VALUE=DATE-TIME:20221111T020000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/31
DESCRIPTION:Title: The world of poset inequalities\nby Greta Panova (USC) as part
of UCLA Combinatorics Forum\n\nLecture held in MS 6627.\n\nAbstract\nParti
ally ordered sets are ubiquitous\, yet poorly understood structures in com
binatorics. Counting their linear extensions and order preserving maps do
not have nice closed formulas and thus we can only hope to understand them
qualitatively or asymptotically in greater generality. In this talk I wil
l show some inequalities relating linear extensions and order preserving m
aps for general posets. We will discuss various proofs\, problems and conj
ectures. Based on a series of joint papers with Swee Hong Chan and Igor Pa
k.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tianyi Yu (UCSD)
DTSTART;VALUE=DATE-TIME:20221130T003000Z
DTEND;VALUE=DATE-TIME:20221130T011000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/32
DESCRIPTION:Title: Top degree components of Grothendieck and Lascoux polynomials\n
by Tianyi Yu (UCSD) as part of UCLA Combinatorics Forum\n\nLecture held in
MS 6627.\n\nAbstract\nThe Schubert polynomials of the Symmetric group of
n form a basis of the space they span. This vector space is well-studied a
nd has dimension $n!$. Its Hilbert series is the $q$-analogue of $n!$. Ano
ther basis of this space is given by key polynomials\, which are character
s of the Demazure modules. Schubert and key polynomials are the ``bottom l
ayers'' of Grothendieck and Lascoux polynomials\, two inhomogeneous polyno
mials. In this talk\, we look at the space spanned by their ``top layers''
. We construct two bases involving the top layer of Grothendieck and the t
op layer of Lascoux polynomials. We then develop a diagrammatic way to com
pute the degrees of these polynomials. Finally\, we describe the Hilbert s
eries of this space involving a classical q-analogue of the Bell numbers.
\nThe talk does not assume knowledge of Grothendieck or Lascoux polynomial
s. This is a joint work with Jianping Pan.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cosmin Pohoata (IAS)
DTSTART;VALUE=DATE-TIME:20221130T012000Z
DTEND;VALUE=DATE-TIME:20221130T020000Z
DTSTAMP;VALUE=DATE-TIME:20221209T130553Z
UID:ucla_comb_sem/33
DESCRIPTION:Title: Convex polytopes from fewer points\nby Cosmin Pohoata (IAS) as
part of UCLA Combinatorics Forum\n\nLecture held in MS 6627.\n\nAbstract\n
Finding the smallest integer N=ES_d(n) such that in every configuration of
N points in R^d in general position there exist n points in convex positi
on is one of the most classical problems in extremal combinatorics\, known
as the Erdos-Szekeres problem. In 1935\, Erdos and Szekeres famously conj
ectured that ES_2(n)=2^{n−2}+1 holds\, which was nearly settled by Suk i
n 2016\, who showed that ES_2(n)≤2^{n+o(n)}. We discuss a recent proof t
hat ES_d(n)=2^{o(n)} holds for all d≥3. Joint work with Dmitrii Zakharov
.\n
LOCATION:https://researchseminars.org/talk/ucla_comb_sem/33/
END:VEVENT
END:VCALENDAR