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Seminar Calendar
for events the day of Monday, March 10, 2014.

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Monday, March 10, 2014

10:00 am in 243 Altgeld Hall,Monday, March 10, 2014

Galois symmetry and motivic homotopy

Jeremiah Heller (MIT)

Abstract: Motivic homotopy theory was introduced by V. Voevodsky as part of his work on the Milnor conjecture. It provides a way of using topological methods as a way to study algebraic varieties over a field. We discuss a new connection between motivic homotopy theory and the study of invariants of manifolds with symmetry. This builds on the classical Galois correspondence and in the case of a real closed field it leads to a surprising generalization of the Fundamental Theorem of Galois theory. This is joint work with K. Ormsby.

2:00 pm in 147 Altgeld Hall,Monday, March 10, 2014

Duality of Persistence Modules

Yuriy Mileyko   [email] (Hawaii Math)

Abstract: Topological data analysis (TDA), a new approach to handling high-dimensional data, gained a lot of attention lately. TDA focuses on qualitative rather than quantitative information supported by the data. A central concept in TDA is persistent homology, a topological invariant capturing structural changes in continuous objects reflected by the discrete point clouds. In particular, persistent homology allows one to determine the thresholds where new topological features emerge and where they are later destroyed, providing a so-called birth-death decomposition. Birth-death decompositions are often represented as collections of planar points called persistence diagrams, and they have been extensively used in a variety of applications. Often, continuous objects associated with a point cloud are sublevel sets of a function on a manifold. In such a case homological duality results may lead to nice relations between persistence diagrams in different homological dimensions, which doesn't only provide more insight into the theory of persistent homology, but can also be used in practice to reduce computational time. In this talk, I will review some of the existing duality results in persistent homology and show that by focusing not on persistence diagrams but on persistence modules, which are algebraic objects representing birth-death decompositions, one can obtain significantly more general duality results.

3:00 pm in 145 Altgeld Hall,Monday, March 10, 2014

Local Rigidity & Nash-Moser Methods

Roy Wang (Utrecht University Math)

Abstract: J. Conn used analytic methods to prove his theorem on the linearization of Poisson structures. For some time that proof was heuristically interpreted as a local rigidity result for linear, compact, semi-simple Poisson structures. In his thesis I. Marcut made this interpretation rigorous, which lead to surprising new results. In collaboration we aim to isolate the method and formulate a local rigidity theorem, which we apply to other geometrical structures. As an example I sketch a proof of the Newlander-Nirenberg theorem.

4:00 pm in 143 Altgeld Hall,Monday, March 10, 2014

Vector Multiplets and the moduli space of Seiberg-Witten theory

Sheldon Katz (Illinois Math)

Abstract: I begin by refining last week's discussion of chiral fields, for application to supersymmetric gauge theory. This leads to the notion of the vector multiplet of N=1 super Yang-Mills in dimension 4. We then consider N=2 super Yang-Mills in dimension 4, the N=2 vector multiplet, and the N=2 hypermultiplet. The vector multiplet and hypermultiplet moduli spaces are then discussed for the compactification of type II string theory on a Calabi-Yau threefold, and for Seiberg-Witten theory.

4:00 pm in Altgeld Hall,Monday, March 10, 2014

Indefinite Integral Quadratic Forms Beyond Classical Reduction Theory

Han Li (Yale)

Abstract: The classical reduction theory of integral quadratic forms was developed by Hermite, Minkowski, Siegel and many others. It is known that a non-degenerate integral quadratic form in n-variables is integrally equivalent to a form whose height (the maximum value of the coefficients) is less than its determinant (up to a multiple constant), and whose value at (1, 0,...0) is less than the n-th root of its determinant. However, for indefinite forms in at least 3 variables it turns out that neither of the estimates is optimal. In this talk we will discuss some classical results and recent effort in improving these estimates. This is a joint work with Prof. Margulis.

5:00 pm in 241 Altgeld,Monday, March 10, 2014

Free Monotone Transport, Part 2

Michael Brannan (UIUC Math)

Abstract: I will discuss the recent paper of A. Guionnet and D. Shlyakhtenko with the same title.