Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, November 11, 2014.

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Tuesday, November 11, 2014

11:00 am in 243 Altgeld Hall,Tuesday, November 11, 2014

The higher Morita category of E_n algebras

Rune Haugseng (MPI Bonn)

Abstract: I will discuss a construction of a higher category of E_n-algebras and iterated bimodules, generalizing the classical bicategory of algebras and bimodules. This leads to generalizations of the Picard and Brauer groups, which have been studied in stable homotopy theory as interesting invariants of ring spectra, and should also lead to an "algebraic" construction of factorization homology as an extended topological quantum field theory.

11:00 am in 241 Altgeld Hall,Tuesday, November 11, 2014

Chunks of the zeta function and Euler-Maclaurin summation

Yevgenya Movshovich (Eastern Illinois University)

Abstract: This work began with the rather simple task of estimating the difference between two consecutive "chunks" of the harmonic series. The result, achieved by calculus, suggested turning our focus to asymptotic expansions for chunks of zeta-like functions. We find such asymptotic expansions first using the classical Euler-Maclaurin summation and also with techniques specifically adapted to functions $f(N)$ that have asymptotic series in powers of $1/N.$ Since asymptotic expansions are unique, finding two apparently different such expansions leads to a sequence of new and old identities for Bernoulli polynomials. In the process of working through this phase, we were forced to come to grips with the variety of definitions and conventions for both the Bernoulli numbers and their related Bernoulli polynomials. One of our conclusions is that they all have their place and should probably all get reasonably equal treatment, as long as care is taken to come up with the right notation and to determine how they all relate to one another. Part of this presentation, then, is our contribution to the centuries old effort to "get this right." Naturally, we think we have done that, but so have many others before us! This is joint work with Gregory Galperin and Peter Andrews.

1:00 pm in 345 Altgeld Hall,Tuesday, November 11, 2014

NIP for the Asymptotic Couple of Logarithmic Transseries

Allen Gehret (UIUC)

Abstract: In this talk I will outline a proof for NIP for the theory T_log. The method of proof involves determining all simple extensions (=1-types since T_log has a universal axiomatization), and observing that the number of simple extensions ($=g_T(\kappa)$) is bounded by $\mathrm{ded}(\kappa)^{\aleph_0}$.

1:00 pm in Altgeld Hall 243,Tuesday, November 11, 2014

Hyperbolic extensions of free groups

Spencer Dowdall (UIUC Math)

Abstract: Every subgroup G of the outer automorphism group of a finite-rank free group F naturally determines a free group extension E_G (that is, an extension of F). In this talk, I will discuss geometric conditions on the subgroup G that imply the corresponding extension E_G is hyperbolic. Specifically, E_G is hyperbolic provided G is purely atoroidal and the orbit map of G into the free factor complex is a quasi-isometric embedding. This allows one to easily build new examples of hyperbolic free group extensions. I will also discuss interesting consequences regarding the orbit of such a group in the Outer space of F. This is joint work with Samuel Taylor. View talk at http://youtu.be/dVOxlPWZyV8

2:00 pm in 347 Altgeld Hall,Tuesday, November 11, 2014

Large deviations and variational representations for infinite dimensional stochastic systems

Paul Dupuis   [email] (Brown University)

Abstract: We discuss how certain variational representations can be used to simplify large deviations and related analyses, especially in the infinite dimensional setting. We first review the statement of the representation for Gaussian noise and its use in a simple setting, and then describe the form of the representation and present an application in the infinite dimensional setting. If time permits we discuss some of the new features present when there is also Poisson noise.

3:00 pm in 241 Altgeld Hall,Tuesday, November 11, 2014

he number of maximal sum-free subsets of integers

Hong Liu   [email] (UIUC Math)

Abstract: Cameron and Erdős raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal sum-free sets in $\{1, \dots , n\}$. Our proof makes use of container and removal lemmas of Green as well as a result of Deshouillers, Freiman, Sós and Temkin on the structure of sum-free sets. Joint work with József Balogh, Maryam Sharifzadeh and Andrew Treglown.

3:00 pm in 243 Altgeld Hall,Tuesday, November 11, 2014

Castelnuovo-Mumford Regularity in Approximation Theory

Michael DiPasquale (UIUC Math)

Abstract: Piecewise polynomial functions (splines) on a planar polytopal complex are of interest in many areas of pure and applied mathematics, from computer-aided geometric design to equivariant cohomology of toric varieties. Splines of degree at most d form a vector space whose dimension is given by a polynomial in d when d is large. The point at which these dimensions become polynomial (the postulation number) is governed by an algebro-geometric invariant, the Castelnuovo-Mumford regularity. We give bounds on the Castelnuovo-Mumford regularity of the algebra of piecewise polynomial functions (splines) on a planar polytopal complex. As an application, we can determine when known polynomials (due to McDonald-Schenck in the polytopal case and Alfeld-Schumaker in the simplicial case) give the correct dimension of the spline space. In the simplicial case the derived bounds recover results of Hong and Ibrahim-Schumaker from the analytic perspective.