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Seminar Calendar
for events the day of Thursday, October 6, 2016.

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Thursday, October 6, 2016

11:00 am in 241 Altgeld Hall,Thursday, October 6, 2016

A proof of $M(x) = o(x)$ for Beurling generalized numbers

Harold Diamond (Illinois Math)

Abstract: In classical prime number theory there are several asymptotic formulas said to be ``equivalent'' to the Prime Number Theorem. One of these assertions is that $M(x)$, the summatory function of the Moebius function, is $o(x)$. Implications between these formulas are different for Beurling generalized numbers ($g$-numbers). We deduce the $g$-number version of $M(x) = o(x)$ using the PNT and a crude $O$-bound on the distribution of $g$-integers.

12:00 pm in 243 Altgeld Hall,Thursday, October 6, 2016

Hodge theory and its applications in Teichmuller dynamics

Simion Filip (Harvard)

Abstract: The moduli space of Riemann surfaces equipped with a holomophic 1-form carries an interesting action of the group SL(2,R) which shares some features with locally homogeneous spaces. Understanding this action provides insight into understanding dynamics on individual surfaces. Hodge theory, in particular techniques from variations of Hodge structures, play a role in understanding the dynamics in moduli space. I will introduce the basic objects in the story and explain how arithmetic concepts such as real multiplication or torsion points on Jacobians come into play. Time permitting, I will discuss questions in Hodge theory motivated by dynamics, in particular the concept of Lyapunov exponents associated to a variation of Hodge structures.

1:00 pm in 347 Altgeld Hall,Thursday, October 6, 2016

The Combinatorics of RNA Branching

Christine Heitsch (Mathematics, Georgia Tech)

Abstract: Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly, especially for sequences on the scale of viral genomes. However, results from enumerative, probabilistic, analytic, and geometric combinatorics yield insights into RNA structure formation, and suggest new directions in viral capsid assembly.

2:00 pm in 347 Altgeld Hall ,Thursday, October 6, 2016

C*-Algebras and Numerical Analysis

Chris Linden (UIUC Math)

Abstract: Following Hagen, Roch, and Silbermann, I will explain how C*-algebra techniques can be applied to questions about convergence and stability in numerical analysis. First, I will show how such questions can be reformulated in terms of invertibilty in a Banach algebra, and then give evidence that C*-algebras are an appropriate framework for answering them. No prior knowledge of numerical analysis or operator algebras will be assumed.

3:00 pm in 243 Altgeld Hall,Thursday, October 6, 2016

Uniform Artin-Rees bounds for Syzygies

Ian Aberbach (University of Missouri at Columbia)

Abstract: Let $(R,m)$ be a Noetherian local ring of dimension $d$, $M$ a finitely generated $R$-module, and $F_\bullet$ a finite free resolution of $M$. Let $Z_i$ be the $i$th syzygy module in the resolution. Eisenbud and Huneke raised the following uniform Artin-Rees style question: Given an ideal $I$, is there a uniform $k$ such that for all $i>0$ and for all $n \gg 0$ the containment $Z_i \cap I^n F_{i-1} \subseteq I^{n-k}Z_i$? They were able to show that this is true if $M$ has both constant rank and finite projective dimension on the punctured spectrum. We show, in fact, that not only is the general answer yes, but that the uniformity is true to a much greater extent. While the ``beginning’’ of the resolution may be quite badly behaved, once we get to the $d$th syzygy, we get uniform behavior for all modules and all ideals in $R$. Specifically, we show that there exists $k \ge 0$ such that for all finitely generated modules $M$, for all ideals $I$, and for $i> d$, $Z_i \cap I^n F_{i-1} \subseteq I^{n-k}Z_i$. The proof relies on Huneke’s uniform Artin-Rees theorem, and can also be viewed as an extension of that theorem.

4:00 pm in 245 Altgeld Hall,Thursday, October 6, 2016

Ramanujan, K-theory, and modularity

Frank Calegari (University of Chicago)

Abstract: The Rogers-Ramanujan identity: $$ 1 + \frac{q}{(1-q)} + \frac{q^4}{(1-q)(1-q^2)} + \frac{q^9}{(1-q)(1-q^2)(1-q^3)} + \ldots = \frac{1}{(1-q)(1-q^4)(1-q^6)(1 - q^9) \ldots}$$ says that a certain $q$-hypergeometric function (the left hand side) is equal to a modular form (the right hand side). To what extent can one classify all $q$-hypergeometric functions which are modular? We discuss this question and its relation to conjectures in knot theory and K-theory. This is joint work with Stavros Garoufalidis and Don Zagier.