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Seminar Calendar
for events the day of Tuesday, September 22, 2015.

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Tuesday, September 22, 2015

11:00 am in 243 Altgeld Hall,Tuesday, September 22, 2015

Real Johnson-Wilson Theories and Computations

Vitaly Lorman (Johns Hopkins)

Abstract: Complex cobordism and its relatives, the Johnson-Wilson theories, E(n), carry an action of C_2 by complex conjugation. Taking fixed points of the latter yields Real Johnson-Wilson theories, ER(n). These can be seen as generalizations of real K-theory and are similarly amenable to computations. We will outline their properties, describe a generalization of the \eta-fibration, and discuss recent computations of the ER(n)-cohomology of some well-known spaces, including CP^\infty.

12:00 pm in 345 Altgeld Hall,Tuesday, September 22, 2015

4-manifolds can be surface bundles over surfaces in many ways

Nick Salter (University of Chicago)

Abstract: An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the situation is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is “finitely many”? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on an asymptotic version of the “how many?” question. Time permitting, we will discuss some connections with the homology of the Torelli group, (non)-realization problems a la Nielsen and Morita, and symplectic topology.

1:00 pm in 241 Altgeld Hall,Tuesday, September 22, 2015

IGL Project

Abstract: John D'Angelo and Ming Xiao will meet with their IGL group to discuss CR Complexity.

1:00 pm in 345 Altgeld Hall,Tuesday, September 22, 2015

Weak axioms of determinacy and consistency strength

Sherwood Hachtman   [email] (UIC Math)

Abstract: At many natural stopping points in the hierarchy of determinacy axioms, we find that winning strategies contain the same information as "standard" models of some fragment of ordinary mathematics (e.g. second order arithmetic, or weak fragments of set theory, or large cardinals). In this talk we examine this phenomenon in low levels of the Borel hierarchy, paying particular attention to $\Sigma^0_3$ (or $G_{\delta \sigma}$) games. Philip Welch has recently discovered an intimate connection between these games and a transfinite-time generalization of Turing computability; we give a further, "reverse math" characterization of this determinacy in terms of models satisfying an axiom of monotone induction.

2:00 pm in 347 Altgeld Hall,Tuesday, September 22, 2015

Scaling limits of critical random graph models

Shankar Bhamidi (UNC Chapel Hill)

Abstract: Over the last few years, one major theme in the study of random discrete structures, is the notion of scaling limits; these discrete objects properly rescaled converge to continuum random objects in the large size limit. Concurrently, motivated by the presence of empirical data on a wide array of real world networks, there has been an explosion in the number of network models proposed to explain various functionals observed in real world systems including power law degree distribution and small world phenomenon. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t_c which specifies when a giant connected component (containing a positive faction of the vertices) emerges. A major open problem in this area, since the time of Erdos and Renyi is an understanding of properties of components in the critical regime namely at t_c. In the last decade, based on a number of computational studies, there is mounting evidence to support that for a wide class of models, the nature of this emergence is universal in the sense that: (a) If the degree distribution of the random graph model has finite third moments then maximal component sizes in the critical regime scale like n^{2/3} whilst these components scaled by n^{-1/3} converge to random fractals. (b) If the degree distribution has finite second but infinite third moments then the sizes of maximal components scale like n^a(\tau) while these components rescaled by n^{-b(\tau)} converge to continuum random limiting objects. Here a(\tau), b(\tau) are explicit functions of the degree exponent. It is also conjectured that a number of fundamental objects in probabilistic combinatorics including the Minimal spanning tree in the supercritical regime obey the exact same scaling. We report on recent progress in proving these conjectures. Joint work with Nicolas Broutin, Remco van der Hofstad, Sanchayan Sen and Xuan Wang.

2:00 pm in 241 Altgeld Hall,Tuesday, September 22, 2015

The limit shape of random permutations with polynomially growing cycle weights

Dirk Zeindler (Lancaster University)

Abstract: We study the limit shape of random permutations endowed with the generalised Ewens measure with algebraically growing weights. Furthermore we consider also the fluctuations at different points of the limit shape and study their joint behaviour.

3:00 pm in 241 Altgeld Hall,Tuesday, September 22, 2015

Stability results on cycles and paths

Alexandr Kostochka (UIUC Math)

Abstract: The classical Erdos-Gallai theorems from 1959 on the most edges in $n$-vertex graphs not containing paths or cycles with $k$ edges were sharpened later by Faudree and Schelp, Woodall, and Kopylov. For $n\geq 5k/4$ the strongest result was: if $t\geq 2$, $k=2t+1$, $n \geq \frac{5t-3}{2}$, and $G$ is an $n$-vertex $2$-connected graph with at least $h(n,k,t) = {k-t \choose 2} + t(n -k+ t)$ edges, then $G$ contains a cycle of length at least $k$ unless $G = H_{n,k,t} := K_n - E(K_{n - t})$. We prove stability versions of these results. In particular, if $n \geq 3t > 3$, $k=2t+1$ and the number of edges in an $n$-vertex $2$-connected graph $G$ with no cycle of length at least $k$ is greater than $h(n,k,t-1)={k-t+1 \choose 2} + (t-1)(n -k+ t-1)$, then $G$ is a subgraph of $H_{n,k,t}$. The lower bound on $e(G)$ is tight. This is joint result with Z. Furedi and J. Verstraete.

3:00 pm in 243 Altgeld Hall,Tuesday, September 22, 2015

Quantization, reduction mod p, and automorphisms of the Weyl algebra

Chris Dodd (Perimeter Institute)

Abstract: The Weyl algebra of polynomial differential operators is a basic object which appears in algebraic geometry, representation theory, and mathematical physics. In this talk, I will discuss some conjectures of A. Belov-Kanel and M. Kontsevich concerning the structure of the automorphism group of the Weyl algebra. The question turns out to be related to defining an appropriate notion of "support cycle" for a differential equation, which, in turn, involves techniques from positive characteristic. In particular, we shall explain a "quantization correspondence" which is based on reducing differential equations to finite characteristic.

4:00 pm in 245 Altgeld Hall,Tuesday, September 22, 2015

Probabilistic models and real world networks

Shankar Bhamidi (University of North Carolina)

Abstract: Owing to the availability of data on a wide array of real world networks, there has been a concerted effort on researchers in a number of fields including math, statistical physics, computer science and statistics to develop probabilistic models to explain the evolution of various features in these real world systems. For applied probabilists, this has resulted in wonderful playground of interesting mathematical problems also resulting in connections in a number of fields. In this talk I will describe a number of such connections including simple probabilistic models in Network Tomography and Phylogenetics; Models for social networks such as Twitter or change point detection and continuous time branching processes; coagulation models in chemistry and continuum scaling limits of dynamic models which incorporate limited choice in their evolution.