Colloquia 2012-2013

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Revision as of 14:32, 9 February 2011 by Shamgar (talk | contribs) (Spring 2011)
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Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Spring 2011

date speaker title host(s)
jan 21 Emanuele Macri (University of Bonn) Stability conditions and Bogomolov-type inequalities in higher dimension Andrei Caldararu
jan 28 Marcus Roper (Berkeley) Modeling microbial cooperation Paul Milewski
jan 31, 2:30pm, room 901 Ana-Maria Castravet (Arizona) Hypertrees and moduli spaces of stable rational curves Andrei
feb 4 Xinyi Yuan (Columbia University) Equidistribution in algebraic dynamics Tonghai
feb 11 Christian Schnell (U. Illinois at Chicago) On the locus of Hodge classes and its generalizations Andrei
feb 25 Omri Sarig (Penn State and Weizmann Institute) Measure rigidity for dynamical systems on (very) non-compact spaces Shamgar
mar 4 Jeff Weiss (Colorado) Nonequilibrium Statistical Mechanics and Climate Variability Jean-Luc
mar 11 Roger Howe (Yale) TBA Shamgar
mar 25 Pham Huu Tiep (Arizona) TBA Martin Isaacs
apr 1 Amy Ellis (Madison) Do algebra students need a reality check? How quantitative reasoning can support function understanding. Steffen
apr 8 Alan Weinstein (Berkeley) TBA Yong-Geun
apr 15 Max Gunzburger (Florida State) TBA James Rossm.
apr 22 Jane Hawkins (U. North Carolina) TBA WIMAW (Diane Holcomb)
apr 29 Jaroslaw Wlodarczyk (Purdue) TBA Laurentiu

Abstracts

Emanuele Macri (University of Bonn)

Stability conditions and Bogomolov-type inequalities in higher dimension

Stability conditions on a derived category were originally introduced by Bridgeland to give a mathematical foundation for the notion of \Pi-stability in string theory, in particular in Douglas’ work. Recently, the theory has been further developed by Kontsevich and Soibelman, in relation to their theory of motivic Donaldson-Thomas invariants for Calabi-Yau categories. However, no example of stability condition on a projective Calabi-Yau threefold has yet been constructed.

In this talk, we will present an approach to the construction of stability conditions on the derived category of any smooth projective threefold. The main ingredient is a generalization to complexes of the classical Bogomolov-Gieseker inequality for stable sheaves. We will also discuss geometric applications of this result.

This is based on joint work with A. Bayer, A. Bertram, and Y. Toda.

Marcus Roper (Berkeley)

Modeling microbial cooperation

Although the common conception of microbes is of isolated and individualistic cells, cells can also cooperate to grow, feed or disperse in challenging physical environments. I'll show how new mathematical models reveal the benefits and developmental and stability barriers to inter-cellular cooperation, by highlighting three paradigmatic examples: 1. The primitive multicellularity of our closest-related protozoan cousins, Salpingoeca rosetta. 2. The cooperative hydrodynamics that enhance spore dispersal in the devastating crop pathogen Sclerotinia sclerotiorum. 3. The architectural adaptations that allow filamentous fungi to thrive in many different ecological niches by harboring mixed communities of genotypically different nuclei.

Ana-Maria Castravet (Arizona)

Hypertrees and moduli spaces of stable rational curves

The Grothendieck-Knudsen moduli space \bar M_{0,n} of stable rational curves with n marked points is a building block towards many moduli spaces (stable curves, Kontsevich stable maps). Although there are several explicit descriptions of \bar M_{0,n}, its geometry is far from being understood. In this talk, I will introduce new combinatorial structures called hypertrees and use them to approach the string of open problems about effective cycles on \bar M_{0,n}. In particular, we will construct new exceptional divisors - which lead to new birational models of \bar M_{0,n} - and rigid curves - which are important for the Faber-Fulton conjecture on the structure of the Mori cone of \bar M_{0,n}. This is based on joint work with Jenia Tevelev.

Xinyi Yuan (Columbia)

Equidistribution in algebraic dynamics

Abstract: Algebraic dynamics studies iterations of algebraic endomorphisms of algebraic varieties. One problem is to study distribution of points with special dynamical properties. In this talk, I will describe some an equidistribution result of small points in algebraic dynamics. Its proof lies in arakelov geometry.

Christian Schnell (UIC)

On the locus of Hodge classes and its generalizations

The Hodge conjecture is one of several conjectures that attempt to relate algebraic cycles on a complex algebraic variety to the cohomology groups of the variety. Among other things, the conjectures predict that certain loci, a priori defined by holomorphic equations, can actually be defined by polynomials. Although we do not know how to prove the conjectures, those predictions have now all been verified. In the talk, I will survey this story, starting from the famous 1995 paper by Cattani-Deligne-Kaplan, and ending with recent work by Brosnan-Pearlstein, Kato-Nakayama-Usui, and myself. I promise to make the talk accessible to everyone.

Omri Sarig (Penn State)

Measure rigidity for dynamical systems on (very) non-compact spaces

The long term statistical behavior of the orbits of a dynamical system T:X --> X can be studied by analyzing the invariant measures of T. Different measures correspond to different statistics of meandering in space. Usually, there are many different invariant measures, and listing all possible types of statistical behavior is intractable. But sometimes there are few invariant measures and this is possible. In such a case we speak of "measure rigidity".

There are many classical examples of measure rigidity where X is compact.The phenomenology is by now understood. But the situation in the non-compact case is not clear. I will survey some examples (mostly horocycle flows on infinite genus hyperbolic surfaces) which indicate the exotic phenomena which may appear in seriously non-compact situations.



Jeff Weiss (Colorado)

Nonequilibrium Statistical Mechanics and Climate Variability

The natural variability of climate phenomena has significant human impacts but is difficult to model and predict. Natural climate variability self-organizes into well-defined patterns that are poorly understood. Recent theoretical developments in nonequilibrium statistical mechanics cover a class of simple stochastic models that are often used to model climate phenomena: linear Gaussian models which have linear deterministic dynamics and additive Gaussian white noise. The theory for entropy production is developed for linear Gaussian models and applied to observed tropical sea surface temperatures (SST). The results show that tropical SST variability is approximately consistent with fluctuations about a nonequilibrium steady-state. The presence of fluctuations with negative entropy production indicates that tropical SST dynamics can, on a seasonal timescale, be considered as small and fast in a thermodynamic sense. This work demonstrates that nonequilibrium statistical mechanics can address climate-scale phenomena and suggests that other climate phenomena could be similarly addressed by nonequilibrium statistical mechanics.

Amy Ellis (UW-Madison)

Do algebra students need a reality check? How quantitative reasoning can support function understanding.

Understanding functions is a critical aspect of algebraic reasoning, and yet research investigating students? abilities in algebra suggests that students experience severe difficulty in correctly creating, representing, and analyzing functions. I will share data from two studies describing how reasoning directly with real-world quantities can support middle-school students? developing understanding of linear and quadratic functions. These studies shed new light on the role of concrete problem contexts for abstract ideas, and suggest an alternate approach to helping students understand functional relationships in beginning algebra.