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for events the day of Thursday, March 28, 2019.

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Thursday, March 28, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 28, 2019

Core partitions, Numerical semigroups, and Polytopes

Hayan Nam (University of California at Irvine)

Abstract: A partition is an $a$-core partition if none of its hook lengths are divisible by $a$. It is well known that the number of $a$-core partitions is infinite and the number of simultaneous $(a, b)$-core partitions is a generalized Catalan number if $a$ and $b$ are relatively prime. Numerical semigroups are additive monoids that have finite complements, and they are closely related to core partitions. The first half of the talk, we will talk about an expression for the number of simultaneous $(a_1,a_2,\dots, a_k)$-core partitions. In the second half, we discuss the relationship between numerical semigroups and core partitions, along with how to count numerical semigroups with certain restrictions.

12:00 pm in 243 Altgeld Hall,Thursday, March 28, 2019

Classifying incompressible surfaces in hyperbolic mapping tori

Sunny Xiao (Brown U)

Abstract: One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path which satisfies certain combinatorial properties in the arc complex of the 4-punctured sphere. This extends and generalizes results of Floyd, Hatcher, and Thurston.

2:00 pm in 241 Altgeld Hall,Thursday, March 28, 2019

Joint Shapes of Quartic Fields and Their Cubic Resolvents

Piper Harron (University of Hawaii)

Abstract: In studying the (equi)distribution of shapes of quartic number fields, one relies heavily on Bhargava's parametrizations which brings with it a notion of resolvent ring. Maximal rings have unique resolvent rings so it is possible to live a long and healthy life without understanding what they are. The authors have decided, however, to forsake such bliss and look into what ever are these rings and what happens if we consider their shapes along with our initial number fields. What indeed! Please stay tuned. (Joint with Christelle Vincent)

3:00 pm in 347 Altgeld Hall,Thursday, March 28, 2019

Complexity, Combinatorial Positivity, and Newton Polytopes

Colleen Robichaux   [email] (UIUC)

Abstract: The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class ${\sf NP}\cap {\sf coNP}$ of problems with "good characterizations''. This suggests a new algebraic combinatorics viewpoint on complexity theory. This paper focuses on the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the $n\times n$ grid, together with a theorem of A. Fink, K. Meszaros, and A. St. Dizier (2018), which proved a conjecture of C. Monical, N. Tokcan, and A. Yong (2017). This is joint work with Anshul Adve and Alexander Yong.

4:00 pm in 245 Altgeld Hall,Thursday, March 28, 2019

Spherical conical metrics

Xuwen Zhu (University of California Berkeley)

Abstract: The problem of finding and classifying constant curvature metrics with conical singularities has a long history bringing together several different areas of mathematics. This talk will focus on the particularly difficult spherical case where many new phenomena appear. When some of the cone angles are bigger than $2\pi$, uniqueness fails and existence is not guaranteed; smooth deformation is not always possible and the moduli space is expected to have singular strata. I will give a survey of several recent results regarding this singular uniformization problem, connecting PDE techniques with complex analysis and synthetic geometry. Based on joint works with Rafe Mazzeo and Bin Xu.