Postdoc Seminars

IMA Postdoc Seminars are given weekly throughout the fall and spring semesters. Postdocs present on a variety of mathematical topics that may be unrelated to the current annual program theme. IMA visitors and University of Minnesota faculty are also invited to present on subjects of interest.

Zach Hamaker of the University of Minnesota will be organizing the 2014-2015 seminar series.

Titles, abstracts, and speakers for each seminar will be posted as available. All seminars are from 2:00pm - 3:00pm unless otherwise noted.

• ## The Unconditional Case of the S-Inequality for Exponential Measure

Piotr Nayar, University of Minnesota Twin Cities
September 23, 2014
Lind 305 [Map]

### Abstract

Let m be the exponential distribution, i.e., the density of m is equal to 1/2^n exp(-(|x_1|+...+|x_n|)). Let K be an unconditional convex set and let P be a strip of the form [-p,p] times R^{n-1} with m(K)=m(P). We prove that m(tK) geq m(tP) for t>1.

This is a joint work with Tomasz Tkocz (University of Warwick).
• ## Three Coloring the Discrete Torus or 3-States Anti-Ferromagnetic Potts Model in Zero Temperature

Ohad Feldheim, University of Minnesota Twin Cities
October 7, 2014
Lind 305 [Map]

### Abstract

We prove that a uniformly chosen proper three coloring of Z_{2n}^d has a very rigid structure when the dimension d is sufficiently high. In particular the coloring asymptotically almost surely takes one color on almost all of either the even or the odd sub-lattice. This implies for example that one color appears on nearly half of the lattice sites.

This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in statistical mechanics. The result improves an independent bound due to Galvin, Kahn, Randall and Sorkin. The proof, however, is quite different: using combinatorial methods which follow an algebraic-topological intuition, results of Peled about homomorphism height functions are extended to a new setting.

Joint work with Ron Peled.
• ## Perfect Matchings in Dense Uniform Hypergraphs

Jie Han, Georgia State University
October 14, 2014
Lind 409 [Map]

### Abstract

In graph/hypergraph theory, perfect matchings are fundamental objects of study. Unlike the graph case, perfect matchings in hypergraphs have not been well understood yet. It is quite natural and desirable to extend the classical theory on perfect matchings from graphs to hypergraphs, as many important problems can be phrased in this framework, such as Ryser's conjecture on transversals in Latin squares and the Existence Conjecture for block designs. I will focus on Dirac-type conditions (minimum degree conditions) in uniform hypergraphs and discuss some recent progresses. In particular, we determine the minimum codegree threshold for the existence of a near perfect matching in hypergraphs, which confirms a conjecture of Rödl, Ruciński and Szemerédi, and we show that there is a polynomial-time algorithm that can determine whether a k-uniform hypergraph with minimum codegree n/k has a perfect matching, which solves a problem of Karpiński, Ruciński and Szymańska completely.
• ## Finite Point Configurations and Fourier Analysis

Krystal Taylor, University of Minnesota Twin Cities
October 21, 2014
Lind 305 [Map]

### Abstract

We study the existence of certain geometric congurations in sub- sets of Euclidean space. In particular, we establish that a set with sufficient structure" contains an arbitrarily long chain with vertices in the set and preassigned admissible gaps. A $$k$$-chain in $$E \subset \mathbb{R}^d$$ with gaps $$\left\{ t_i \right\}^k_{i=1}$$ is a sequence $$\left\{ x^1, x^2, \dots, x^{k+1}, x^j, \subset E ; \left| x^{i+1} - x^{i} \right| = t_i; 1 \leq i \leq k \right\} .$$ In order to prove this result, we establish $$L^p(\mu) \rightarrow L^q(\nu)$$ mapping properties of the convolution operator $$T_\lambda f(x) = \lambda * (f\mu)(x)$$; where $$\lambda$$ is a tempered distribution, and $$\mu$$ and $$\nu$$ are compactly supported measures satisfying the growth bounds $$\mu(B(x, r)) \leq Cr^{s_\mu}$$ and $$\nu(B(x,r)) \leq Cr^{s_\nu}$$
• ## The Brunn-Minkowski Inequality - Its Refinements and Extensions

Arnaud Marsiglietti, University of Minnesota, Twin Cities
October 28, 2014

### Abstract

The Brunn-Minkowski inequality, which states that for every non-empty compact sets $A,B$ in $\R^n$ and every $\lambda \in [0,1]$ one has $$|(1-\lambda)A + \lambda B|^{1/n} \geq (1-\lambda)|A|^{1/n} + \lambda |B|^{1/n},$$ where $|.|$ denotes the volume (Lebesgue measure), is a fundamental inequality in mathematics. The aim of this talk is to present this inequality together with its consequences, refinements and extensions.
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