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for events the day of Tuesday, September 17, 2019.

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Tuesday, September 17, 2019

12:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

Coloring invariants of knots and links are often intractable

Eric Sampterton (Illinois Math)

Abstract: I’ll give an overview of my result with Greg Kuperberg concerning the computational complexity of G-coloring invariants of knots, where G is a finite, simple group. We have a similar theorem for closed 3-manifolds. I’ll try to give a sense of the commonalities of the two proofs (e.g. “reversible computing with a combinatorial TQFT”), as well as where they differ (there’s some interesting algebraic topology that needed developing in the knot case). Time permitting, I’ll discuss the special case of hyperbolic knots and 3-manifolds.

1:00 pm in 345 Altgeld Hall,Tuesday, September 17, 2019

Conjugacy classes of automorphism groups of linearly ordered structures

Aleksandra Kwiatkowska (Universität Münster and Uniwersytet Wrocławski)

Abstract: In the talk, we will address the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a countable structure or of a Fraisse limit) that is extremely amenable and has ample generics. In fact, it is unknown if there exists a linearly ordered structure whose automorphism group has a comeager $2$-dimensional diagonal conjugacy class.
  We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. Moreover, we provide general conditions implying that there is no comeager conjugacy class or comeager $2$-dimensional diagonal conjugacy class in the automorphism group of an ordered Fraisse limit.
  This is joint work with Maciej Malicki.

1:00 pm in 347 Altgeld Hall,Tuesday, September 17, 2019

Finite Elements for Curvature

Kaibo Hu   [email] (University of Minnesota)

Abstract: We review the elasticity (linearized Calabi) complex, its cohomology and potential applications in differential geometry and continuum defect theory. We construct discrete finite element complexes. In particular, this leads to new finite element discretization for the 2D linearized curvature operator. Compared with classical discrete geometric approaches, e.g., the Regge calculus, the new finite elements are conforming. The construction is based on a Bernstein-Gelfand-Gelfand type diagram chase with various finite element de Rham complexes. This is a joint work with Snorre H. Christiansen.

2:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

Connected Fair Detachments of Hypergraphs

Amin Bahmanian (ISU Math)

Abstract: Let $\mathcal G$ be a hypergraph whose edges are colored. An $(\alpha,n)$-detachment of $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing the incident hinges and edges among the subvertices. A detachment is fair if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. We find necessary and sufficient conditions under which a $k$-edge-colored hypergraph $\mathcal G$ has a fair detachment in which each color class is connected. Previously, this was not even known for the case when $\mathcal G$ is an arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions, and completing partial regular combinatorial structures.

3:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

P=W, a strange identity for Hitchin systems

Zili Zhang (U Michigan)

Abstract: Start with a compact Riemann surface X with marked points and a complex reductive group G. According to Hitchin-Simpson’s nonabelian Hodge theory, the pair (X,G) comes with two new complex varieties: the character variety M_B and the Higgs moduli M_D. I will present some aspects of this story and discuss an identity P=W indexed by affine Dynkin diagrams – occurring in the singular cohomology groups of M_D and M_B, where P and W dwell. Based on joint work with Junliang Shen.