Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, November 29, 2016.

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Tuesday, November 29, 2016

11:00 am in 345 Altgeld Hall,Tuesday, November 29, 2016

The Brauer group of the moduli stack of elliptic curves

Benjamin Antieau (University of Illinois at Chicago )

Abstract: Mumford proved that the Picard group of the moduli stack of elliptic curves is a finite group of order 12, generated by the Hodge bundle of the universal family of elliptic curves. After giving background on Brauer groups of ring spectra and our motivation, I will talk about recent work with Lennart Meier, motivated by elliptic cohomology, which considers the Brauer group of the moduli stack and shows that it vanishes. Non-6-torsion can be handled by general descent spectral sequence methods. To handle the 2-torsion, we study the moduli stack of elliptic curves with full level 2 structure, showing with geometric and arithmetic arguments that it is a 2-group. Then, by exhibiting 2-adic elliptic curves with certain discriminants, we show that the non-zero classes do not extend to the integral moduli stack.

12:00 pm in 243 Altgeld Hall,Tuesday, November 29, 2016

The Symplectic Geometry of Polygon Space and How to Use It

Clayton Shonkwiler (Colorado State University)

Abstract: In statistical physics, the basic (and highly idealized) model of a ring polymer is a closed random flight in 3-space with equal-length steps, often called a random equilateral polygon. In this talk, I will describe the moduli space of random equilateral polygons, giving a sense of how this fits into a larger symplectic and algebraic geometric story. In particular, the space of equilateral n-gons turns out to (almost) be a toric symplectic manifold, yielding a (nearly) global coordinate system. These coordinates are powerful tools both for proving theorems and for developing numerical techniques, some of which I will describe, including a very fast and surprisingly simple algorithm for directly sampling random polygons recently developed with Jason Cantarella (University of Georgia), Bertrand Duplantier (CEA/Saclay), and Erica Uehara (Ochanomizu University).

1:00 pm in 345 Altgeld Hall,Tuesday, November 29, 2016

Games orbits play

Aristotelis Panagiotopoulos (UIUC Math)

Abstract: Classification problems occur in all areas of mathematics. Descriptive set theory provides methods to assign complexity to such problems. Using a technique developed by Hjorth, Kechris and Sofronidis proved for example, that the problem of classifying all unitary operators $\mathcal{U}(\mathcal{H})$ of an infinite dimensional Hilbert space up to unitary equivalence $\simeq_U$ is strictly more difficult than classifying graph structures with domain $\mathbb{N}$ up to isomorphism. We will present a conceptual and unifying approach to several anti-classification results, including Hjorth's turbulence theorem. We will introduce a dynamical criterion for showing that an orbit equivalence relation is not Borel reducible to the orbit equivalence relation induced by a CLI group action; that is, a group which admits a complete left invariant metric (recall that, by a result of Hjorth and Solecki, solvable groups are CLI). We will finally deduce that $\simeq_U$ is not classifiable by CLI group actions.
This is a joint work with Martino Lupini.

1:00 pm in 7 Illini Hall,Tuesday, November 29, 2016

Dispersive Estimates for the 4D Schrodinger Equation

Ebru Toprak   [email] (UIUC Math)

Abstract: In this talk, I will present our latest improvement on the L^1 \rightarrow L^{\infty} dispersive estimate for the four dimensional Schrodingerís evolution when there is an obstruction at zero. I will introduce a technique in which we use oscillatory integral estimate methods and spectral properties of the Laplacian. I will explain what the obstructions at zero mean and how and why they effect the decay rate of the dispersive estimate.

2:00 pm in 347 Altgeld Hall,Tuesday, November 29, 2016

Particle representations for deterministic and stochastic reaction-diffusion equations

Louis Wai-Tong Fan   [email] (University of Wisconsin)

Abstract: Reaction diffusion equations (RDE) are an important and popular tool for modeling complex spatial-temporal patterns including Turing patterns, traveling waves and periodic switching. These models, however, ignore the stochasticity and discreteness of many complex systems in nature. Recognizing this drawback, scientists are developing individual-based models for model selection purposes. The latter models are sometimes studied under the framework of interacting particle systems (IPS) by mathematicians. In this talk, I will present some new scaling limits including stochastic partial differential equations (SPDE) on metric graphs and coupled SPDE. These SPDE not only interpolate between IPS and RDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and novel duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics. In particular, I will present rigorous results about the lineage dynamics of a biased voter model introduced by Hallatschek and Nelson (2007). Based on previous work with Rick Durrett and ongoing work with Ashley Baltes, Sebastien Roch, John Yin and Zhanpeng Zeng.

3:00 pm in 243 Altgeld Hall,Tuesday, November 29, 2016

Uniform Asymptotic Growth on Symbolic Powers of Ideals

Robert Walker (University of Michigan)

Abstract: Symbolic powers ($I^{(N)}$) in Noetherian commutative rings are mysterious objects from the perspective of an algebraist, while regular powers of ideals ($I^s$) are essentially intuitive. However, many geometers tend to like symbolic powers in the case of a radical ideal in an affine polynomial ring over an algebraically closed field in characteristic zero: the N-th symbolic power consists of polynomial functions "vanishing to order N" on the affine zero locus of that ideal. In this polynomial setting, and much more generally, a challenging problem is determining when, given a family of ideals (e.g., all prime ideals), you have a containment of type $I^{(N)} \subseteq I^s$ for all ideals in the family simultaneously. Following breakthrough results of Ein-Lazarsfeld-Smith (2001) and Hochster-Huneke (2002) for, e.g., coordinate rings of smooth affine varieties, there is a slowly growing body of "uniform linear equivalence" criteria for when, given a suitable family of ideals, these $I^{(N)} \subseteq I^s$ containments hold as long as N is bounded below by a linear function in s, whose slope is a positive integer that only depends on the structure of the variety or the ring you fancy. My thesis (arxiv.org/1510.02993, arxiv.org/1608.02320) contributes new entries to this body of criteria, using Weil divisor theory and toric algebraic geometry. After giving a "Symbolic powers for Geometers" survey, I'll shift to stating key results of my dissertation in a user-ready form, and give a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd from "Ant-Man," I hope this talk will be awesome.

3:00 pm in 241 Altgeld Hall,Tuesday, November 29, 2016

Dirac's theorem and DP-colorings

Anton Bernshteyn (Illinois Math)

Abstract: Correspondence coloring, or DP-coloring, is a generalization of list coloring introduced recently by Dvořák and Postle. I will talk about a version of Diracís theorem on the minimum number of edges in critical graphs in the framework of DP-colorings. A corollary of this result is a solution to the problem, posed by Kostochka and Stiebitz, of classifying list-critical graphs that satisfy Diracís bound with equality. This is joint work with Alexandr Kostochka.

4:00 pm in 314 Altgeld Hall,Tuesday, November 29, 2016

From Obtuse Triangles to DNA Models and Synthetic Polymers: The Geometry of Random Polygons

Clayton Shonkwiler (Colorado State University )

Abstract: In 1884 Lewis Carroll posed the following problem, purportedly solved in bed during a sleepless night: "Three Points are taken at random on an infinite Plane. Find the chance of their being the vertices of an obtuse-angled Triangle." At least 10 different answers have been proposed over the last 130 years, due to various interpretations of the phrase "at random". I would suggest that Carroll's question is really about random triangles, not random points in the plane, and that the more basic question is: what does it mean to choose a triangle "at random"? And, really, why stop at triangles: what does it mean to choose an n-gon at random? I will give a very concrete answer to this question which has the virtue of being highly symmetric, meaning that it is relatively easy to compute probabilities and averages (which is to say, to integrate), including the probability that a triangle is obtuse. Moreover, the construction generalizes not just to n-gons in the plane, but also to n-gons in 3-dimensional space that are used to model so-called ring polymers, which are polymers like bacterial DNA forming closed loops. In fact, some of the same ideas which give an answer to Lewis Carroll's question have been used in chemistry labs to create synthetic polymers with novel topologies.