Mathematics

Relatively Random First-Order Structures

Henry Towsner (UPenn Math)

Abstract: 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 $M$.

Gromov-Hausdorff -Prohorov convergence of vertex cut-trees of n-leaf Galton-Watson trees

Hui He (Beijing Normal University)

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.

The strong chromatic index of graphs with maximum degree four is at most 21

Gexin Yu (College of William & Mary)

Abstract: 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.

Bernstein-Sato polynomials for maximal minors

Andras Lorincz (Purdue University)

Abstract: 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.

A variant of Gromov's H\"older equivalence problem for small step Carnot groups

Derek Jung   [email] (UIUC Math)

Abstract: 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.

Optimum Thresholding for Semimartingales using mean and conditional mean square error

Dr. Jose Figueroa-Lopez (Washington University, St. Louis)

Abstract: We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps of possibly infinite activity. General conditions are known for the nonparametric threshold estimator of the integrated variance proposed in Mancini (2009) to be asymptotically consistent for the integrated volatility. However, the finite sample properties of the estimator can significantly depend on the specific choice of the threshold. In this work, we aim at optimally selecting the threshold by minimizing either the estimation mean square error (MSE) or its conditional mean square error (cMSE). In both cases a parsimonious characterization of the optimum is established, which allows us to show that the optimal threshold sequence is proportional to the modulus of continuity of the underlying Brownian motion. Moreover, minimizing the cMSE enables us to propose a novel implementation scheme for the optimal threshold sequence. Monte Carlo simulations illustrate the superior performance of the proposed method.