Seminar Calendar
for events the day of Tuesday, March 6, 2012.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 6, 2012

Ergodic Theory
11:00 am   in 347 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by jathreya.
 Ilya Gekhtman (University of Chicago)Dynamics of Convex Cocompact Subgroups of Mapping Class GroupsAbstract: Convex cocompact subgroups of mapping class groups are subgroups of the mapping class group whose orbits in Teichmueller space are quasi-convex. We develop an analogue of Patterson-Sullivan theory for the action of subgroups G of Mod(S) on Teichmuller space and its boundary the space of projective measured foliations and use it to compute multiplicative asymptotics for the number of orbit points of G in a ball of radius R in Teichmueller space and the number of pseudo-Anosovs in G with dilatation at most R.

Topology Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by franklan.
 Ayelet Lindenstrauss (Indiana University)K-theory of formal power seriesAbstract: (Joint with Randy McCarthy.) We study the algebraic K-theory of parametrized endomorphisms of a unital ring R with coefficients in a simplicial R-bimodule M, and compare it with the algebraic K-theory of the ring of formal power series in M over R. Waldhausen defined an equivalence from the suspension of the reduced Nil K-theory of R with coefficients in M to the reduced algebraic K-theory of the tensor algebra TR(M). Extending Waldhausen's map from nilpotent endomorphisms to all endomorphisms, our map has to land in the ring of formal power series rather than in the tensor algebra, and is no longer in general an equivalence (it is an equivalence when the bimodule M is connected). Nevertheless, the map shows a close connection between its source and its target: it induces an equivalence on the Goodwillie Taylor towers of the two (as functors of M, with R fixed), and allows us to give a formula for the suspension of the invariant W(R;M) (which can be thought of as Witt vectors with coefficients in M, and is what the Goodwillie Taylor tower of the source functor converges to) as the inverse limit, as n goes to infinity, of the reduced algebraic K-theory of TR(M)/ (Mn).

Harmonic analysis and differential equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by vzh.
 Taras Lakoba, (University of Vermont, Math)Unusual properties of numerical instability of the split-step method applied to NLS solitonAbstract: The split-step method (SSM) is widely used for numerical solution of nonlinear evolution equations. Its idea and implementation are simple. Namely, it is common that the evolution of variable $u$ is governed by: $u_t = A(u,t) + B(u,t)$ where both individual'' evolutions $u_t = A(u,t) \qquad \mbox{and} \qquad u_t = B(u,t)$ can be solved exactly (or at least easily''). Then the numerical approximation of the full solution is sought in steps that alternatingly solve each equation. The SSM has long been used to simulate the NLS: $$i \, u_t - \beta u_{xx} + \gamma u|u|^2 = 0$$ ( so here $A=-\beta u_{xx}$ and $B=\gamma u|u|^2$). However, until recently, a possible development of numerical instability of the SSM has been studied only in one simplest case, which does not include the soliton or multi-soliton solutions of the NLS. In this talk I will present recent results concerning the development of the numerical instability of the SSM when it is used to simulated a near-soliton solution of NLS. Properties of this instability are stunningly different from instability properties of most other numerical schemes. I will not assume prior familiarity of the audience with instabilities of numerical methods and therefore will first review a couple of basic examples of such instabilities. This will set a benchmark for the subsequent exposition of the instability properties of the SSM. I will show how those properties, and --- more importantly --- their analysis, are different from the instability properties and analysis for most other numerical schemes.

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by ssolecki.
 Lou van den Dries (Department of Mathematics, University of Illinois at Urbana-Champaign)The structure of approximate groups according to Breuillard, Green, TaoAbstract: Roughly speaking, an approximate group is a finite symmetric subset A of a group such that AA can be covered by a small number of left-translates of A. Last year the authors mentioned in the title established a conjecture of H. Helfgott and E. Lindenstrauss to the effect that approximate groups are finite-by-nilpotent''. This may be viewed as a sweeping generalisation of both the Freiman-Ruzsa theorem on sets of small doubling in the additive group of integers, and of Gromov's characterization of groups of polynomial growth. Among the applications of the main result are a finitary refinement of Gromov's theorem and a generalized Margulis lemma conjectured by Gromov. Prior work by Hrushovski on approximate groups is fundamental in the approach taken by the authors. They were able to reduce the role of logic to elementary arguments with ultra products. The point is that an ultraproduct of approximate groups can be modeled in a useful way by a neighborhood of the identity in a Lie group. This allows arguments by induction on the dimension of the Lie group. I will give two talks: the one on Tuesday will describe the main results, and the sequel on Friday will try to give a rough idea of the proofs.

Probability Seminar
2:00 pm   in Altgeld Hall 347,  Tuesday, March 6, 2012
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Submitted by kkirkpat.
 Jonathon Peterson   [email] (Purdue)Large deviations and slowdown asymptotics for excited random walksAbstract: Excited random walks (also called cookie random walks) are self-interacting random walks where the transition probabilities depend on the number of previous visits to the current location. Although the models are quite different, many of the known results for one-dimensional excited random walks have turned out to be remarkably similar to the corresponding results for random walks in random environments. For instance, one can have transience with sub-linear speed and limiting distributions that are non-Gaussian. In this talk I will prove a large deviation principle for excited random walks. The main tool used will be what is known as the "backwards branching process" associated with the excited random walk, thus reducing the problem to proving a large deviation principle for the empirical mean of a Markov chain (a much simpler task). While we do not obtain an explicit formula for the large deviation rate function, we will be able to give a good qualitative description of the rate function. While many features of the rate function are similar to the corresponding rate function for RWRE, there are some interesting differences that highlight the major difference between RWRE and excited random walks.

Graph Theory and Combinatorics
3:00 pm   in 241 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by west.
 Andrew Treglown (Charles University, Prague)Embedding spanning bipartite graphs of small bandwidthAbstract: A graph $H$ on $n$ vertices has bandwidth at most $b$ if there exists a labelling of the vertices of $H$ by the numbers $1,\ldots,n$ such that $|i-j|\le b$ for every edge $ij$ of $H$. Boettcher, Schacht, and Taraz gave a condition on the minimum degree of a graph $G$ on $n$ vertices to ensure that $G$ contains every $r$-chromatic graph $H$ on $n$ vertices having bounded degree and bandwidth $o(n)$, thereby proving a conjecture of Bollobás and Komlós. We strengthen this result in the case where $H$ is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph $G$ on $n$ vertices that forces $G$ to contain every bipartite graph $H$ on $n$ vertices having bounded degree and bandwidth $o(n)$. This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on $G$ is relaxed to a certain robust expansion property. (Joint work with Fiachra Knox.)

Mathematical Biology
3:00 pm   in 345 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by zrapti.
 Spencer Hall   [email] (Indiana University, Department of Biology )Five reasons why resources matter for diseaseAbstract: We could produce more powerful theory to predict disease outbreaks if we took an approach rooted in community ecology. I want to argue this point by focusing on resources of hosts and a case study of fungal disease in a planktonic grazer (Daphnia). Using this system and a combination of observations of epidemics in lakes, experiments, and mathematical (differential equation) models, I will show five reasons why resources matter for disease. (1) Key epidemiological traits (think transmission rate, or yield of parasites from infected host) vary plastically with resources - and how hosts acquire and use them. (2) If we embrace this plasticity, we can better predict variation in disease in time and space. I'll illustrate with a case study of potassium as the resource. A model will also reveal some counter-intuitive predictions that stem from interactions of disease with a dynamic resource. (3) Resources can strongly influence how other species (predators, competitors) inhibit or fuel epidemics. For example, I'll show how a predator might spread disease through a trophic cascade. (4) Variation in feeding rate (i.e., resource acquisition) among clonal genotypes of hosts can create key tradeoffs in life history vs. epidemiological traits (e.g., transmission rate vs. fecundity). (5) This tradeoff can then help us understand how hosts might evolve to become more resistant or more susceptible to their parasites during epidemics of different sizes. All of these ecological and evolutionary outcomes for disease hinge on explicitly thinking about resources of hosts. As a result, host-resource interactions should play a much more prominent role in rapidly growing theory for disease ecology.

Mathematics Colloquium --Trjitzinsky Memorial Lectures
4:00 pm   in 314 Altgeld Hall,  Tuesday, March 6, 2012
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Submitted by kapovich.
 Robert Ghrist (University of Pennsylvania)Sheaves and the Global Topology of Data, Lecture IAbstract: This lecture series concerns Applied Mathematics -- the taming and tuning of mathematical structures to the service of problems in the sciences. The Mathematics to be harnessed comes from algebraic topology -- specifically, sheaf theory, the study of local-to-global data. The applications to be surveyed are in the engineering sciences, but are not fundamentally restricted to such. Beginning with a gentle introduction to algebraic topology and its modern applications, the series will focus on sheaves and their recent utility in sensing, coding, optimization, and inference. No prior exposure to sheaves required. Robert Ghrist is the Andrea Mitchell Penn Integrating Knowledge Professor in the Departments of Mathematics and Electrical/Systems Engineering at the University of Pennsylvania. A reception will be held in AH 314 immediately following the lecture.