Seminar Calendar
for events the day of Tuesday, March 7, 2017.

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Tuesday, March 7, 2017

Topology Seminar
11:00 am   in 345 Altgeld Hall,  Tuesday, March 7, 2017
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Submitted by rezk.
 Guillaume Brunerie (IAS)Invariant homotopy theory in homotopy type theoryAbstract: This talk will be about homotopy type theory and in particular the branch of it known as invariant homotopy theory, or synthetic homotopy theory. The main idea is that homotopy type theory is a formal language which can be used to talk about "spaces-up-to-homotopy-equivalence". The basic objects can be thought of as spaces, but the language has the property that all the structures, properties, constructions and proofs that we can express are invariant under homotopy equivalence. One advantage is that every construction or proof done in this setting is expected to be automatically valid in every infinity-topos, not just in the infinity-topos of spaces, while still looking elementary. In this sense, we can see homotopy type theory as an internal language for infinity-topoi. Moreover, such proofs are also amenable to computer formalization, as homotopy type theory is strongly related to computer proof assistants. I will present the basic concepts and show what a few proofs and constructions look like in invariant homotopy theory. In particular, we will see the universal cover of the circle, the Hopf fibration, cohomology, and the Steenrod operations.

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, March 7, 2017
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Submitted by anush.
 Henry Towsner (UPenn Math)Relatively Random First-Order StructuresAbstract: The Aldous–Hoover Theorem gives a characterization of those random processes which generate "exchangeable" first-order structures. A random first-order structure on the natural numbers is exchangeable if, after any permutation of the natural numbers, it has the same distribution. The original proof of the full Aldous–Hoover Theorem used ultraproducts, and the topic remains intimately tied to the way probability measures behave in ultraproducts.  For some purposes, full exchangeability is too strong. We investigate "relative exchangeability", where we only require that the distribution be preserved by automorphisms of a fixed first-order structure $M$. A full Aldous–Hoover theorem is not always possible in this setting, and how much we recover turns out to depend on the amalgamation properties of $M$.

Probability Seminar
2:00 pm   in 347 Altgeld Hall,  Tuesday, March 7, 2017
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Submitted by wangjing.
 Hui He (Beijing Normal University)Gromov-Hausdorff -Prohorov convergence of vertex cut-trees of n-leaf Galton-Watson trees Abstract: We study the vertex cut-tree of Galton-Watson trees conditioned to have n leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have n vertices. Our main result is a joint Gromov-Hausdorff -Prohorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prohorov convergence to Gromov-Hausdorff-Prohorov remains open for their models conditioned to have n vertices. This is a joint work with Matthias Winkel.

Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Tuesday, March 7, 2017
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Submitted by rtramel.
 Andras Lorincz (Purdue University)Bernstein-Sato polynomials for maximal minorsAbstract: Initially introduced for hypersurfaces, Bernstein-Sato polynomials have been recently defined for arbitrary varieties by N. Budur, M. Mustata and M. Saito. Nevertheless, they are notoriously difficult to compute with very few explicit cases known. In this talk, after giving the necessary background, I will discuss some techniques that allow the computation of the Bernstein-Sato polynomial of the ideal of maximal minors of a generic matrix. Time permitting, I will also talk about connections to topological zeta functions and show the monodromy conjecture for this case.Initially introduced for hypersurfaces, Bernstein-Sato polynomials have been recently defined for arbitrary varieties by N. Budur, M. Mustata and M. Saito. Nevertheless, they are notoriously difficult to compute with very few explicit cases known. In this talk, after giving the necessary background, I will discuss some techniques that allow the computation of the Bernstein-Sato polynomial of the ideal of maximal minors of a generic matrix. Time permitting, I will also talk about connections to topological zeta functions and show the monodromy conjecture for this case.

Graph Theory and Combinatorics Seminar
3:00 pm   in 241 Altgeld Hall,  Tuesday, March 7, 2017
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Submitted by molla.
 Gexin Yu (College of William & Mary)The strong chromatic index of graphs with maximum degree four is at most 21Abstract: A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In 1985, Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has a strong edge-coloring using at most $\frac{5}{4}\Delta^2$ colors if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. This conjecture is widely unresolved with the only verified case being for $\Delta = 3$, due independently to Andersen as well as Horák, Qing, and Trotter. In this paper, we show that the strong chromatic index of graphs (where we allow for multiple edges) with maximum degree at most four is always at most 21. This improves a previous bound due to Cranston and moves closer to the conjectured upper bound of 20. This is joint work with Mingfang Huang and Michael Santana.

 Derek Jung   [email] (UIUC Math)A variant of Gromov's H\"older equivalence problem for small step Carnot groupsAbstract: This is the second part of a talk I gave last semester in the Graduate Geometry/Topology Seminar. A Carnot group is a Lie group that may be identified with its Lie algebra via the exponential map. This allows one to view a Carnot group as both a sub-Riemannian manifold and a geodesic metric space. It is then natural to ask the following general question: When are two Carnot groups equivalent? In this spirit, Gromov studied the problem of considering for which $k$ and $\alpha$ there exists a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^k\to G$. Very little is known about this problem, even for the Heisenberg groups. By tweaking the class of H\"older maps, I will discuss a variant of Gromov's problem for Carnot groups of step at most three. This talk is based on a recently submitted paper. Some knowledge of differential geometry and Lie groups will be helpful.