Department of

Mathematics

Seminar Calendar
for events the day of Thursday, February 19, 2015.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, February 19, 2015

11:00 am in 347 Altgeld Hall,Thursday, February 19, 2015

The character of the total power operation

Nathaniel Stapleton   [email] (MPI Bonn)

Abstract: The total power operation in Morava E-theory is a mysterious map. It is multiplicative but not additive. It has deep algebro-geometric properties that have been studied by Ando, Hopkins, Rezk, and Strickland. The Morava E-theory of finite groups admits a character theory which approximates E(BG) by a form of generalized class functions" on the group. In this talk we construct a total power operation on generalized class functions that is compatible with the total power operation for Morava E-theory through the character maps of Hopkins, Kuhn, and Ravenel. This takes advantage of an intriguing connection between the Drinfeld ring of full level structures on a height n formal group and the conjugacy classes of commuting n-tuples in the wreath product of a finite group with a symmetric group.

11:00 am in 241 Altgeld Hall,Thursday, February 19, 2015

The multiplicative orders of certain Gauss factorials

Karl Dilcher   [email] (Dalhousie University)

Abstract: This talk deals with the multiplicative orders of $\left(\frac{n-1}{M}\right)_n! \pmod{n}$ for odd prime powers $n=p^\alpha$, $p\equiv 1\pmod{M}$, where the Gauss factorial $N_n!$ denotes the product of all integers up to $N$ that are relatively prime to $n$. Considering the connection between the orders for $p^\alpha$ and for $p^{\alpha+1}$, we obtain new criteria for exceptions to a general pattern, with particular emphasis on the cases $M=3$, $M=4$ and $M=6$. In the process we also obtain some results of independent interest. Most results are based on generalizations of binomial coefficient congruences of Gauss, Jacobi, and Hudson and Williams. (Joint work with John B. Cosgrave).

1:00 pm in Altgeld Hall 243,Thursday, February 19, 2015

The Tits alternative for the automorphism group of a free product

Camille Horbez (Rennes)

Abstract: A group $G$ is said to satisfy the Tits alternative if every subgroup of $G$ either contains a nonabelian free subgroup, or is virtually solvable. The talk will aim at presenting a version of this alternative for the automorphism group of a free product of groups. A classical theorem of Grushko states that every finitely generated group $G$ splits as a free product of the form $G_1*...*G_k*F_N$, where $F_N$ is a finitely generated free group, and all $G_i$ are nontrivial, non isomorphic to $Z$, and freely indecomposable. In this situation, I prove that if all groups $G_i$ and $Out(G_i)$ satisfy the Tits alternative, then so does the group $Out(G)$ of outer automorphisms of G. I will present applications to proving the Tits alternative for outer automorphism groups of right-angled Artin groups, or of some classes of relatively hyperbolic groups. I will then present a proof of this theorem, in parallel to a new proof of the Tits alternative for mapping class groups of compact surfaces. The proof relies on a study of the actions of some subgroups of $Out(G)$ on a version of the outer space for free products, and on a hyperbolic simplicial graph. View talk at http://youtu.be/5AP1g7_8MCs

2:00 pm in 243 Altgeld Hall,Thursday, February 19, 2015

Martingales in vector and Banach lattices

Abstract: The theory of abstract martingales in a vector lattice is a generalization of the classical martingale theory. A filtration is replaced with a sequence of positive projections, while a martingale is a sequence in the lattice adapted to the filtration. We will discuss the structure of the space of martingales and its connections to the structure of the original lattice. We will also discuss regular martingales, i.e., differences of two positive martingales, and their connections to the theory of regular operators.

2:00 pm in 140 Henry Administration Bldg,Thursday, February 19, 2015

Arithmetic of Apollonian Circle Packings, part I

Elena Fuchs (UIUC Math)

Abstract: We will explore some theorems about the arithmetic of ancient objects called Apollonian Circle Packings. The tools we will use have applications far beyond these packings, and we will give some idea of what such applications are, as well.

3:00 pm in 243 Altgeld Hall,Thursday, February 19, 2015

Homological Properties of Determinantal Arrangements

Arnold Yim (Purdue University)

Abstract: An important aspect of a divisor $Y$ on a complex analytic manifold $X$ is its singular locus. Depending on the ultimate goal, one can view the divisor $Y$ as "well-behaved" if the singular locus is small (at best consists of isolated points), or if the singular locus is not very complicated (but fairly large). In this talk we take the latter position: the best possible case in this view is that of "normal crossings" where the singular locus looks locally like a union of coordinate hyperplanes. By studying the logarithmic flows on $X$ that stabilize (are tangent to) $Y$, one can understand just how complex the singular locus is. In particular, if the collection of these logarithmic flows form a free module, then the singular locus is simple and we say that the divisor is a free divisor. Free divisors show up naturally in many different settings. For example, many of the classically arising hyperplane arrangements (such as braid arrangements and all Coxeter arrangements) are free. Though much is known for hyperplane arrangements, things become more difficult when we consider arrangements of more general hypersurfaces. In this talk, we explore freeness for determinantal arrangements (arrangements defined by minors of a generic matrix) and generalize some of the classical results in this new setting.

4:00 pm in 245 Altgeld Hall,Thursday, February 19, 2015

The mathematics of geometric clustering

Rachel Ward (UT Austin)

Abstract: It is often said that "clustering is difficult only when it does not matter". Using tools from convex optimization and non-asymptotic random matrix theory, we aim to make this statement mathematically precise. Our focus will be on the k-means objective, arguably the most popular unsupervised clustering objective; it is NP hard to optimize in the worst case, but implemented heuristically as a pre- and post-processing step a myriad of machine learning algorithms. We will introduce a convex relaxation of the k-means objective, and provide nontrivial geometric conditions on a set of clusters for this convex relaxation is tight. These conditions are then used to derive near-optimal rates for distinguishing clusters in random point cloud models. It is important that in the same regime, heuristic algorithms such as Lloyd's method and kmeans++ will get stuck in local optima. We conclude by discussing open problems suggested by this work related to weighted kernel k-means and spectral clustering. This is joint work with P. Awasthi, A. Bandeira, M. Charikar, R. Krishnaswamy, and S. Villar.