Mathematics

Dual Ramsey theorem for trees

Slawek Solecki (UIUC)

Abstract: The classical Ramsey theorem was generalized in two major ways: to the dual Ramsey theorem by Graham and Rothschild (despite being a dual, this is a generalization) and to Ramsey theorems for trees initially by Deuber and Leeb. Bringing these two lines of thought together, I will present the dual Ramsey theorem for trees. The abstract approach to Ramsey theory, I developed earlier, is used in the proof.

Elliptic Actions on Teichmüller Space

Matthew Durham (UIC Math)

Abstract: Kerckhoff's solution to the Nielsen realization problem showed that the action of any finite subgroup of the mapping class group on Teichmüller space has a fixed point. The set of fixed points is a totally geodesic submanifold. We study the coarse geometry of the set of points which have bounded diameter orbits in the Teichmüller metric. We show that each such almost-fixed point is within a uniformly bounded distance of the fixed point set, but that the set of almost-fixed points is not quasiconvex. In addition, the orbit of any point is shown to have a fixed barycenter. In this talk, I will discuss the machinery and ideas used in the proofs of these theorems.

Quantum spin systems and graphical representations

Shannon Starr (UA Birmingham)

Abstract: Quantum spin systems are mathematical models for ferromagnetism. The quantum nature of the model is usually a difficulty. For some models one also has graphical representations, which can be used to turn equilibrium properties of a quantum model into dynamical properties of a stochastic system. This is especially useful when combined with Feynman-Kac. I will describe joint work with Nick Crawford and Stephen Ng where we obtained some new results this way.

Mantel's Theorem for Random Hypergraphs

Ping Hu   [email] (UIUC Math)

Abstract: A cornerstone result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of $K_n$ is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large $p$, every maximum triangle-free subgraph of $G(n,p)$ is w.h.p. bipartite. Recently, DeMarco and Kahn proved this for $p > K\sqrt{\log n/n}$ for some constant $K$, and apart from the value of the constant, this bound is best possible. We study an extremal problem of this type in random hypergraphs. Denote by $F_5$ the 3-uniform hypergraph with vertex set $\{a,b,c,d,e\}$ and edge set $\{abc, ade, bde\}$. Frankl and Füredi proved that the maximum 3-uniform hypergraph on $n$ vertices containing no copy of $F_5$ is tripartite for $n>3000$. A natural question is that for what $p$ is every maximum $F_5$-free subhypergraph of $G^3(n,p)$ w.h.p. tripartite. We show this holds for $p>K \log n/n$ for some constant $K$ and does not hold if $p=0.1\sqrt{\log n}/n$. Joint work with Jozsef Balogh, Jane Butterfield and John Lenz.

On Euler integral transforms and their inversions

Juliy Baryshnikov (UIUC Math & ECE)

Abstract: We (it is a joint work with Rob Ghrist and Dave Lipsky) consider topological Radon-type integral transforms on constructible functions using the Euler characteristic as a measure. After surveying Schapira's inversion I will describe a pseudo-inversion formulae for inverting annuli (sets of Euler measure zero).

MMP for deformed Hilbert scheme of points on projective plane

Chunyi Li (UIUC)

Abstract: The idea of running the minimal model program for the moduli space of sheaves via the wall-crossing of Bridgeland stability conditions is, as far as I know, first introduced by Toda. In the Hilb^n P2 case, a strong form conjecture, which is about the correspondence between the base locus decomposition walls for the effective cone of Hilb^n P2 and the destabilizing walls on the stability condition plane, is posed by Arcara, Bertram, Coskun and Huizenga. In this talk, I will introduce the stability condition on D^b(coh P2) and the birational geometry of (deformed)Hilb^n P2. Also I would state our theorem which proves ABCH's conjecture and generalizes the result to deformed Hilb^n P2 case.

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