Mathematics

MidWest Model Theory Day at UIC

Abstract: see http://homepages.math.uic.edu/~freitag/MWMT11

Pointwise bounds for the 3-dimensional wave equation and spectral multipliers

Michael Goldberg (U. Cincinnati)

Abstract: The sine propagator for the wave equation in three dimensions, $\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}$, has an integral kernel $K(t,x,y)$ with the property $\int_{\mathbb R} |K(t, x, y)|dt = (2\pi|x-y|)^{-1}$. Finiteness comes from the sharp Huygens principle and power-law decay comes from dispersion. Estimates of this type are useful for proving ``reversed Strichartz" inequalities that bound a solution in $L^p_x L^q_t$ for admissible pairs $(p,q)$. We examine the propagator $\frac{\sin(t\sqrt{H})}{\sqrt{H}}P_{ac}(H)$ for operators $H = -\Delta + V$ with the potential $V$ belonging to the Kato-norm closure of test functions. Assuming zero is not an eigenvalue or resonance, the bound $\int_{\mathbb R} |K(t,x,y) \leq C|x-y|^{-1}$ continues to be true. Combined with a Huygens principle for the perturbed wave equation, this estimate suggests pointwise bounds for spectral multipliers of fractional integral or H\"ormander-Mikhlin type. This is joint work with Marius Beceanu (SUNY - Albany).

Statistical physics of exponential random graphs

Mei Yin (University of Denver )

Abstract: The exponential random graph model has been a topic of continued research interest. The past few years especially has witnessed (exponentially) growing attention in exponential models and their variations. Emphasis has been made on the variational principle of the limiting free energy, concentration of the limiting probability distribution, phase transitions, and asymptotic structures. This talk with focus on the phenomenon of phase transitions in large exponential random graphs. The main techniques that we use are variants of statistical physics but the exciting new theory of graph limits, which has rich ties to many parts of mathematics and beyond, also plays an important role in the interdisciplinary inquiry. Some open problems and conjectures will be presented.

Rational points of generic curves and the section conjecture

Tatsunari Watanabe (Purdue University)

Abstract: The section conjecture comes from Grothendieck's anabelian philosophy where he predicts that if a variety is "anabelian", then its arithmetic fundamental group should control its geometry. In this talk, I will introduce the section conjecture and the generic curve of genus g >=4 with no marked points as an example where the conjecture holds. The primary tool used is called weighted completion of profinite groups developed by R Hain and M Matsumoto. It linearizes a profinite group such as arithmetic mapping class groups and is relatively computable since it is controlled by cohomology groups.

Orthogonal one-factorizations\\ of complete multipartite graphs

Mariusz Meszka (AGH University of Science and Technology, Kraków, Poland)

Abstract: A one-factor of a graph $G$ is a regular spanning subgraph of degree one. A one-factorization of $G$ is a set ${\cal F}=\{F_1,\,F_2,\ldots ,F_r\}$ of edge-disjoint one-factors such that $E(G)=\bigcup_{i=1}^r E(F_i)$. Two one-factorizations ${\cal F}=\{F_1,\,F_2,\ldots ,F_r\}$ and ${\cal F'}=\{F'_1,\,F'_2,\ldots ,F'_r\}$ are orthogonal if $|F_i\cap F'_j|\leq 1$ for all $1\leq i < j \leq r$. A set of $k$ one-factorizations $\{{\cal F}^1,{\cal F}^2,\ldots ,{\cal F}^k\}$ of $G$ is mutually orthogonal if, for every $1\leq i < j\leq k$, ${\cal F}^i$ and ${\cal F}^j$ are orthogonal. A pair of orthogonal one-factorizations of an $s$-regular graph $G$ on $2n$ vertices corresponds to the existence of a Howell design of type $(s,2n)$, for which a graph $G$ is called an underlying graph. Let $S$ be a set of $2n$ symbols. A Howell design $H(s,2n)$ on the symbol set $S$ is an $s\times s$ array that satisfies the following conditions: (1) every cell is either empty or contains an unordered pair of symbols from $S$, (2) every symbol of $S$ occurs exactly once in each row and exactly once in each column of $H$, (3) every unordered pair of symbols occurs in at most one cell of $H$. Necessary condition for the existence of Howell designs $H(s,2n)$ is $n\leq s\leq 2n-1$. A pair of orthogonal one-factorizations of a complete bipartite graph $K_{n,n}$ corresponds to a Howell design $H_k(n,2n)$, and moreover they are equivalent to a pair of mutually orthogonal latin squares of side $n$. In the other extreme case, an $H(2n-1,2n)$ is a Room square of side $2n-1$, which corresponds to two orthogonal one-factorizations of a complete graph $K_n$. An important question related to Howell designs concerns properties of graphs which are underlying graphs of Howell designs. While for $s=2n-1$ and $s=2n-2$ these graphs are unique (the complete graph $K_{2n}$ and the cocktail party graph $K_{2n}\setminus F$, respectively, where $F$ is a one-factor), determining these graphs in general seems to be hopeless. The goal of this talk is to show that balanced complete multipartite graphs are underlying graphs of Howell designs. The main result provides almost complete solution to the existence problem of two orthogonal one-factorizations of a complete balanced multipartite graph $K_{p\times q}$. Some infinite families of $k$ mutually orthogonal one-factorizations of $K_{p\times q}$ for $k\geq 3$ will be also presented.

The Waist Inequality and Quantitative Topology

Hadrian Quan   [email] (UIUC Math)

Abstract: If F is a continuous map from the unit n-Sphere to $\mathbb{R}^q$, then one of its fibers has (n-q)-measure at least that of an (n-q)-dimensional equator. This estimate joins other results like the Isoperimetric Inequality for being simple to state and much harder to prove than at first glance. We’ll discuss the history of this result and some of its relations to topology, geometry, and combinatorics. Time permitting, we shall also sketch a proof of Gromov’s with non-optimal constant.