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.
• ## Frozen Random Walk

Laura Florescu, New York University
November 4, 2014
Lind 305 [Map]

### Abstract

We explore a variant of the symmetric random walk on $\mathbb{Z}$ where particles are frozen if they are on the extreme quarter on any of the two sides. The motivation of this process arises from a theoretical computer science result related to the algorithm giving a 2 coloring of an $n$ set with all discrepancies less than $6 \sqrt{n}$. We are interested in the maximum of the process and the mass distribution. Related to this is a deterministic process where we start with mass $1$ at the origin, and at each step we freeze $1/2$ of the extremal mass and we split the remaining mass at each discrete point equally among its neighbors. Under the assumption that the scaled distribution converges, we determine the distribution, and the maximum $q\sqrt{t}$ where $q$ satisfies $\frac{1}{2}q^2=\frac{qe^{-q^2/2}}{\sqrt{2\pi}(\Phi(q)-\Phi(-q))}$. We present various simulation results supporting the claims of the existence of a scaling limit as well as the connection between the random and the deterministic process. We emphasize that in the deterministic case, the limit shape of the mass distribution is not a truncated Gaussian at the expected endpoints. Joint work with Shirshendu Ganguly, Yuval Peres and Joel Spencer.
• ## Bijections for Reduced Decompositions

Zachary Hamaker, University of Minnesota, Twin Cities
November 18, 2014
Lind 305 [Map]

### Abstract

We discuss enumerative theory for reduced decompositions of permutations and signed permutations. Particular focus will be paid to the Little map and insertion algorithms. These bijections allow for a refined study of combinatorial statistics for reduced decompositions.
• ## Zykov's Symmetrization for Multiple Graphs with an Application to Erdos' Conjecture on Pentagonal Edges

Zeinab Maleki, Isfahan University of Technology
December 2, 2014
Lind 305 [Map]

• ## Winding of Planar Stationary Gaussian Processes

Naomi Feldheim, University of Minnesota, Twin Cities
March 24, 2015
Lind 305 [Map]

### Abstract

Consider the path of a shift-invariant random map from the real numbers to the complex plane whose finite marginal distributions are Gaussian. Such processes are used by physicists to model polymers and random flux lines of magnetic fields. We investigate the winding number of such paths around the origin. We give exact forumlae for the mean and variance of this quantity, and prove a Central limit theorem. In doing so, we give rigorous proofs to predictions by physicists, such as Le Doussal, Etzioni and Horovitz.
• ## Avoiding Repetitions on the Plane

Barbara Pilat, Warsaw University of Technology
March 31, 2015
Lind 305 [Map]

### Abstract

Inspired by classical Hadwiger-Nelson problem on chromatic number of R^2, we want to determine the number of colors required to avoid repetitions on R^2. Since it turns out that in the above problem we need infinitely many colors, we relax the problem. Namely, we show that 53 colors is enough to avoid repetitions on paths of collinear points. On the other hand, we show what can be avoided using only 2 colors. Joint work with M. Dębski, J. Grytczuk, U. Pastwa, J. Sokół, M. Tuczyński, P. Wenus, K. Węsek.
• ## The Frog Model on Trees

Tobias Johnson, University of Washington
April 7, 2015
Lind 305 [Map]

### Abstract

Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at some designated vertex wakes up and begins a simple random walk. When it lands on a vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model. I'll talk about a question posed by Serguei Popov in 2003: On an infinite d-ary tree, is the frog model recurrent or transient? That is, is each vertex visited infinitely or finitely often by frogs? Equivalently, do all frogs wake up eventually? The answer is that it depends on d: there's a phase transition between recurrence and transience as d grows. Furthermore, if the system starts with Poi(m) sleeping frogs on each vertex independently, there's a phase transition as m grows. This is joint work with Christopher Hoffman and Matthew Junge.
• ## On the Perimeter of a Convex Set

Galyna Livshyts, Kent State University
April 21, 2015
Lind 305 [Map]

### Abstract

The perimeter of a convex set in R^n with respect to a given measure is the measure's density averaged against the surface measure of the set. It was proved by Ball in 1993 that the perimeter of a convex set in R^n with respect to the standard Gaussian measure is asymptotically bounded from above by n^{1/4}. Nazarov in 2003 showed the sharpness of this bound. We are going to discuss the question of maximizing the perimeter of a convex set in R^n with respect to any log-concave rotation invariant probability measure. The latter asymptotic maximum is expressed in terms of the measure's natural parameters: the expectation and the variance of the absolute value of the random vector distributed with respect to the measure. We are also going to discuss some related questions on the geometry and isoperimetric properties of log-concave measures.
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