Colloquia 2012-2013

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)	TBA	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)	TBA	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.

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.

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.