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for events the day of Tuesday, October 4, 2016.

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Tuesday, October 4, 2016

11:00 am in 345 Altgeld Hall,Tuesday, October 4, 2016

Nilpotence and Periodicity over Spec(C)

Bogdan Gheorghe (Wayne State University)

Abstract: The setting of the talk is stable motivic homotopy theory over Spec C. It is well-known that the algebraic cobordism spectrum MGL fails to detect nilpotence, for example eta is non-nilpotent but not detected by MGL. More periodicity is also observed, for example by the computation of the eta-periodic sphere by Andrews-Miller and Guillou-Isaksen. It is apparent that a bigger spectrum is needed to detect nilpotence, and such a spectrum might also define more Morava K-theories than the K(n)'s. In this talk we will show how to construct the usual motivic Morava K-theories and the Brown-Peterson spectrum from the ground up, without referring to algebraic cobordism. We will then tweak this construction to produce new orthogonal Morava K-theory fields K(w_n) as well as a Brown-Peterson spectrum wBP and a hypothetical cobordism spectrum wMU. If time permits we will show some computations in progress which are trying to connect these spectra with the usual theory of formal group laws and nilpotence and periodicity.

1:00 pm in 345 Altgeld Hall,Tuesday, October 4, 2016

Types and Algorithmic Lower Bounds

Siddharth Bhaskar (Dept. of Math, Indiana University)

Abstract: There are various models of computability over arbitrary first order structures. Two of the most important ones are computability by recursive programs and computability by tail recursive programs. The former accommodates general recursive definitions, whereas the latter is restricted to “iteration” or “looping.” One fundamental question we might want to know about a structure is all of its recursive functions are already tail recursive. In this talk, we will give two simple and one sophisticated example of how counting the quantifier-free types in a structure helps us separate these two classes. At the end of the talk we will ask whether there might be a division of structures into tame or wild on the basis of their recursion theory (as opposed to first-order definability).

2:00 pm in 241 Altgeld Hall,Tuesday, October 4, 2016

Vinogradov's mean value theorem

George Shakan (UIUC )

Abstract: Recently, Bourgain, Demeter and Guth showed that Vinogradov's mean value theorem deserves its name of "theorem." The conjecture followed from their general decoupling inequality. I'll consider the 2 dimensional case of their proof and explain some of the key inputs. In this case, their theorem asserts that the $L^6$ norm of the exponential sum $\sum_{j < N} e(j x + j^2 y) $ is $\ll N^{1/2 + \epsilon}$. Some of the key inputs are the translation invariance (in j), induction on scales, and a Kakeya-type estimate. This last estimate eventually follows from the elementary geometric fact that the intersection of two same-sized parallelograms of significantly different slopes have small intersection.

3:00 pm in 241 Altgeld Hall,Tuesday, October 4, 2016

Measurable combinatorics and the Lovász Local Lemma

Anton Bernshteyn (Illinois Math)

Abstract: For $k \in \mathbb{N}$, a $k$-coloring of a graph $G$ is a partition $V(G) = V_1 \cup \ldots \cup V_k$ of its vertex set into $k$ independent sets (i.e., sets with no edges between their elements). Given a graph $G$, we might want to know if it admits a $k$-coloring---and this question leads to a great body of work in graph theory. To complicate the matters further, we can require the sets $V_1, \ldots, V_k$ to be ``nice'' in some sense. For instance, if the vertex set of $G$ is the unit interval $[0;1]$, can the sets $V_1, \ldots, V_k$ be Lebesgue-measurable (or, even better, Borel)? The short answer is, not always. For example, one can show that every Borel graph with maximum degree $d$ is Borel $(d+1)$-colorable, yet there exists a Borel $2$-regular forest with no measurable $2$-coloring. Understanding the extent to which classical combinatorial results can be extended to Borel and measurable settings is the premise of the recently emerged field of Borel combinatorics, which has many applications in descriptive set theory, ergodic theory, and probability theory, to name a few. In this talk I will try to give a brief introduction to this fascinating area and to explain how the Lovász Local Lemma, a classical tool in probabilistic combinatorics, can be used to obtain results in the measurable framework.

3:00 pm in Altgeld Hall,Tuesday, October 4, 2016

Categorical Plucker formula and homological projective dual

Conan Leung (Chinese University of Hong Kong)

Abstract: A generalised Plucker formula describes changes of intersection numbers of complex Lagrangian under Mukai flop. In a recent joint work with Jiang and Xie, we generalise this to the level of derived category of coherent sheaves.

4:00 pm in 245 Altgeld Hall,Tuesday, October 4, 2016

Equation-free techniques for infectious disease data

Josh Proctor (Institute for Disease Modeling)

Abstract: Equation-free methods have become increasingly prevalent in the analysis of high-dimensional, complex systems. These methods have a number of advantageous characteristics for facing the modern challenges of complex systems including the ability to handle high-dimensional measurement data, discover reduced-order models, and analyze systems that do not have a set of well-defined governing equations. As a motivating example, equation-free methods can be applied to data collected by public health surveillance systems focused around the eradication of infectious diseases. The increased awareness for gathering high-quality data and the advent of new monitoring tools is beginning to generate large sets of data describing the spread of infectious disease. In this presentation, I will discuss how data-driven, equation-free methods, such as Koopman operator theory, Dynamic Mode Decomposition (DMD), and Sparse Identification of Nonlinear Dynamics (SINDy), can help analyze time-series data. This presentation will also include a discussion on how these methods can be theoretically generalized to handle data related to inputs and control of the system, e.g. vaccination rates and bednet usage, and to help in the design of surveillance programs.