Department of

Mathematics


Seminar Calendar
for events the day of Thursday, November 7, 2013.

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Thursday, November 7, 2013

11:00 am in 241 Altgeld Hall,Thursday, November 7, 2013

A series identity, possibly connected with a divisor problem, in Ramanujan's Lost Notebook

Atul Dixit (Tulane Univ. Math)

Abstract: On page 336 in his lost notebook, S. Ramanujan proposes an identity that may have been devised to attack a divisor problem. Unfortunately, the identity is vitiated by a divergent series appearing in it. We prove here a corrected version of Ramanujan's identity. We also show its connection to an infinite series involving the modified Bessel function $K_{s}(x)$. We will also discuss a related series identity due to Ramanujan and S. Wigert along with its new generalization. This is joint work with Bruce C. Berndt, Arindam Roy and Alexandru Zaharescu.

11:00 am in 345 Altgeld,Thursday, November 7, 2013

Cofibration categories and quasicategories

Karol Szumilo (Uni Bonn)

Abstract: Note non-standard day and location. Classically, homotopy theories are described using homotopical algebra, e.g. as model categories or (co)fibration categories. Nowadays, they are often formalized as higher categories, e.g. as quasicategories or complete Segal spaces. These two types of approaches highlight different aspects of abstract homotopy theory and are useful for different purposes. Thus it is an interesting question whether homotopical algebra and higher category theory are in some precise sense equivalent. In this talk I will concentrate on cofibration categories and quasicategories. I will discuss some basic features of both notions building up to a result that the homotopy theory of cofibration categories is indeed equivalent to the homotopy theory of cocomplete quasicategories.

1:00 pm in Altgeld Hall 347,Thursday, November 7, 2013

Lone Axes in Outer Space

Catherine Pfaff (Marseille)

Abstract: As with SL(2,R) acting on hyperbolic space, a central method for studying a mapping class group is to study its action on its Teichmuller space and a central method for studying an outer automorphism group of a free group Out(F_n) is to study its action on its Culler-Vogtmann outer space CV_n. Each of these groups also have elements acting in some sense hyperbolically (pseudo-Anosov elements of mapping class groups and fully irreducible outer automorphisms of free groups). However, the analogy breaks down when one wants to study the invariant axis for a fully irreducible. It appears the correct object to study is actually a collection of axes, an "axis bundle." By proving when the axis bundle for a fully irreducible is just a single axis, we have highlighted the setting where a fully irreducible also behaves in this regard like a pseudo-Anosov or hyperbolic element. In fact, we have identified a setting where one can actually quite easily identify when two fully irreducibles are conjugate. This talk is based on joint work with Lee Mosher.

2:00 pm in 243 Altgeld Hall,Thursday, November 7, 2013

Multiscale analysis of 1-rectifiable measures: necessary conditions

Matthew Badger (SUNY Stony Brook)

Abstract: A fundamental concept in geometric measure theory is the division of sets and measures into rectifiable ("regular") and purely unrectifiable ("irregular") pieces. The qualitative theory of rectifiable sets and absolutely continuous rectifiable measures in Euclidean space developed across the last century, beginning with the seminal work of Besicovitch (1928, 1938) and later generalized and improved upon in a series of papers by Morse and Randolph (1944), Moore (1950), Marstrand (1964), Mattila (1975) and Preiss (1987). In particular, in the presence of absolute continuity, these investigations revealed a deep connection between the rectifiability of a measure and the asymptotic behavior of the measure on small balls. A quantitative counterpart to this theory emerged in the 1990s, with major contributions including the work of Jones (1990), David and Semmes (1991, 1993), Okikiolu (1992) and Pajot (1996, 1997). In this talk, I will discuss recent joint work with Raanan Schul. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in Euclidean space. In particular, we give new necessary conditions for a measure to give full mass to a countable family of finite length curves. A novelty of our main result is that no assumption is made on the upper Hausdorff density of the measure. Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to Hausdorff measure.

3:00 pm in 347 Altgeld Hall,Thursday, November 7, 2013

How good are mathematical models of genetic signaling networks?

Kresimir Josic   [email] (University of Houston, Mathematics)

Abstract: Synthetic biology holds the promise of allowing us to engineer living beings. I will start by reviewing some examples where mathematical models have been successful in synthetic biology. One such example is a synthetic gene oscillator in Escherichia coli that exhibits robust temperature compensation -- it maintains a constant period over a range of ambient temperatures. A mathematical model predicted and experiments confirmed the particular mechanisms that lead to temperature compensation despite Arrhenius scaling of the biochemical reaction rates. Such successes are encouraging. But how far can our theoretical models take us? I will argue that our models are still fairly coarse, and do not adequately describe all the important properties of genetic signaling networks. For instance, "transcriptional delay" - the delay between the start of protein production and the time a mature protein finds a downstream target - can have a significant impact on the dynamics of gene circuits. Such delay can inhibit transitions between states of bistable genetic networks, as well as destabilize steady states in other networks. I will show how these effects can be described by reduced, non-Markovian models that are quite different from established models. I will also discuss work with experimental collaborators to characterize the distribution of this delay.

3:00 pm in 243 Altgeld Hall,Thursday, November 7, 2013

Intersection Multiplicity of Serre in the Unramified Case

Chris Skalit (University of Chicago)

Abstract: Let $A$ be a regular local ring whose completion is a power series ring over a DVR. For properly-meeting subschemes of complimentary dimension, $Y, Z \subseteq \operatorname{Spec} A$, we show that the Serre intersection multiplicity, $\chi(\mathcal{O}_{Y},\mathcal{O}_Z) = \sum{(-1)^i \ell (\operatorname{Tor}_i^A(\mathcal{O}_Y,\mathcal{O}_Z))}$, is bounded below by the product of the multiplicities of $Y$ and $Z$. For those cases in which this bound is achieved, we investigate the implications it has for $Y, Z$, and their strict transforms on the blowup.

4:00 pm in 245 Altgeld Hall,Thursday, November 7, 2013

Intrinsic Diophantine approximation on spheres

Dmitry Kleinbock (Brandeis)

Abstract: The goal of the talk is to quantify the density of rational points in the unit sphere $S^n$ using a connection between the distribution of rational points on $S^n$ and homogeneous dynamics. This improves on recent results of Ghosh, Gorodnik and Nevo. Joint work with Lior Fishman, Keith Merrill and David Simmons.