Speakers:
Steven Karp (University of Quebec at Montreal)
Jake Levinson (University of Washington)
Claudiu Raicu (Notre Dame)
Martha Yip (University of Kentucky)
Schedule:
Coffee and pastries 9-9:30am (Math Lounge, Altgeld Hall)
1st Talk: Steven Karp (UQaM) 9:30-10:25am
Title: Topology of totally positive spaces
Abstract:
The classical example of a totally positive space is the set of n x n
totally positive matrices, formed by matrices whose every submatrix has
positive determinant. Total positivity has been studied extensively over
the past century, and has seen a renewed interest in the past 30 years,
initiated by work of Lusztig and of Fomin and Zelevinsky. My talk will
focus on the topology of such spaces. Historically, the motivation for
studying the topology of totally positive spaces is that they have cell
decompositions which realize interesting posets in combinatorics,
related to Bruhat orders. A new motivation comes from recent work in
theoretical physics, which calls for understanding convex polytopes
generalized from affine space into the Grassmannian, where the notion of
convexity is replaced by total positivity. I will present new
techniques, developed in joint work with Pavel Galashin and Thomas Lam,
for establishing the homeomorphism type of totally positive spaces and
their compactifications. In particular, we prove that the totally
nonnegative part of a partial flag variety forms a regular CW complex,
confirming conjectures of Postnikov and of Williams.
2nd Talk: Martha Yip (U Kentucky) 11:00am-11:55am
Title: Chromatic symmetric homology for graphs: some new developments.
Abstract:
In his study of the four colour problem, Birkhoff showed that the
number of ways to colour a graph with k colours is a polynomial chi(k),
which he called the chromatic polynomial. Later, Stanley defined
the chromatic symmetric function X(x_1, x_2, ... ), which is a
multivariable lift of the chromatic polynomial so that when the first k
variables are set to 1, it recovers chi(k). This can be further
lifted to a homological setting; we can construct a chain complex of
graded S_n-modules whose homology has a bigraded Frobenius
characteristic that recovers X upon setting q=t=1.
In
this talk, we will explain the construction of the homology, discuss
some new results regarding the strength of the homology as a graph
invariant, and state some surprising conjectures regarding integral
symmetric homology for graphs.
This is based on joint work with Chandler, Sazdanovic, and Stella.
Lunch (see below for details) 12:00-2:00pm
3rd Talk: Jake Levinson (U Washington) 2:00-2:55pm
Title: A topological proof of the Shapiro--Shapiro Conjecture
Abstract:
Consider a rational curve, described by a map f : P^1 \to P^n. The
Shapiro--Shapiro conjecture says the following: if all the inflection
points of the curve (roots of the Wronskian of f) are real, then the
curve itself is defined by real polynomials, up to change of
coordinates. Equivalently, certain real Schubert varieties in the
Grassmannian intersect transversely — a fact with broad combinatorial
and topological consequences. The conjecture, made in the 90s, was
proven by Mukhin--Tarasov--Varchenko in '05/'09 using methods from
quantum mechanics.
I
will present a generalization of the Shapiro--Shapiro conjecture, joint
with Kevin Purbhoo, where we allow the Wronskian to have complex
conjugate pairs of roots. We decompose the real Schubert cell according
to the number of such roots and define an orientation of each connected
component. For each part of the decomposition, we prove that the
topological degree of the restricted Wronski map is given by a symmetric
group character. In the case where all the roots are real, this implies
that the restricted Wronski map is a topologically trivial covering
map; in particular, this gives a new proof of the Shapiro-Shapiro
conjecture.
4th Talk: Claudiu Raicu (Notre Dame) 3:30-4:30pm
Title: Regularity of S_n-invariant monomial ideals
Abstract:
Consider a polynomial ring in n variables, together with the action of
the symmetric group S_n by coordinate permutations. I will describe a
combinatorial formula for computing the Castelnuovo-Mumford regularity
of arbitrary S_n-invariant monomial ideals. This allows one to
characterize which of these ideals have a linear minimal free resolution
or which ones are Cohen-Macaulay, and also provides a concrete
description of the asymptotic behavior of regularity.
Poster session and informal discussions: 4:30-5:45pm
Shawn Nevalainen: Classifying ribbon categories that satisfy the sp(4) fusion rules
Jianping Pan and Wencin Poh: A Crystal on decreasing factorizations in the 0-Hecke monoid
Faqruddin Ali Azam: On the rational generating functions for intervals of partitions
Joseph Cummings: Phylogenetic networks
Byeongsu Yu: Monomial ideals in an affine semigroup ring
Sean Griffin: Ordered set partitions, Garsia-Procesi modules, and rank varieties
Colleen Robichaux: CM regularity and Kazhdan-Lusztig varieties
Josh Kiers: A Conjectural transfer principle arising from geometric Satake
Minyoung Jeon: Multiplicities of Schubert varieties in the symplectic flag varieties
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Lunch: There are numerous lunch options within walking distance,
the closest being the ????
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Dinner: ~6pm
We will walk over from Altgeld Hall around 6:00pm, just after the poster session ends.
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Getting to Urbana:Parking:
Lodging:
A) The
IIllini Union (574-277-6500)
Childcare: Parents attending the conference and looking for childcare may
find
care.com a useful reference.
