Thursday, October 23, 2014 - 11:00 , Location: Skiles 005 , Professor Igor Pritsker , Oklahoma State University , Organizer: Martin Short
The area was essentially originated by the general question: How many zeros of a random polynomials are real? Kac showed that the expected number of real zeros for a polynomial with i.i.d. Gaussian coefficients is logarithmic in terms of the degree. Later, it was found that most of zeros of random polynomials are asymptotically uniformly distributed near the unit circumference (with probability one) under mild assumptions on the coefficients. Thus two main directions of research are related to the almost sure limits of the zero counting measures, and to the quantitative results on the expected number of zeros in various sets. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by various bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures for associated sets in the plane, and quantify this convergence. Random coefficients may be dependent and need not have identical distributions in our results.
Friday, September 19, 2014 - 11:00 , Location: Skiles 005 , Professor Alberto Bressan , Penn State University , Organizer: Martin Short
The talk will survey the main definitions and properties of patchy vector fields and patchy feedbacks, with applications to asymptotic feedback stabilization and nearly optimal feedback control design. Stability properties for discontinuous ODEs and robustness of patchy feedbacks will also be discussed.
Wednesday, June 11, 2014 - 15:30 , Location: Skiles 006 , Ravi Vakil , Stanford University , firstname.lastname@example.org , Organizer: Joseph Rabinoff
Given some class of "geometric spaces", we can make a ring as follows. (i) (additive structure) When U is an open subset of such a space X, [X] = [U] + [(X \ U)] (ii) (multiplicative structure) [X x Y] = [X] [Y].In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. (This talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood.
Thursday, May 1, 2014 - 11:00 , Location: Skiles 006 , Gunnar Carlsson , Stanford University , Organizer: Joseph Rabinoff
The general problem of extracting knowledge from large and complex data sets is a fundamental one across all areas of the natural and social sciences, as well as in most areas of commerce and government. Much progress has been made on methods for capturing and storing such data, but the problem of translating it into knowledge is more difficult. I will discuss one approach to this problem, via the study of the shape of the data sets, suitably defined. The use of shape as an organizing problems permits one to bring to bear the methods of topology, which is the mathematical field which deals with shape. We will discuss some different topological methods, with examples.
Thursday, April 10, 2014 - 11:00 , Location: Skiles 006 , Jan Draisma , Eindhoven University of Technology , Organizer: Joseph Rabinoff
In this age of high-dimensional data, many challenging questions take the following shape: can you check whether the data has a certain desired property by checking that property for many, but low-dimensional data fragments? In recent years, such questions have inspired new, exciting research in algebra, especially relevant when the property is highly symmetric and expressible through systems of polynomial equations. I will discuss three concrete questions of this kind that we have settled in the affirmative: Gaussian factor analysis from an algebraic perspective, high-dimensional tensors of bounded rank, and higher secant varieties of Grassmannians. The theory developed for these examples deals with group actions on infinite-dimensional algebraic varieties, and applies to problems from many areas. In particular, I will sketch its (potential) relation to the fantastic Matroid Minor Theorem.
Thursday, March 27, 2014 - 11:05 , Location: Skiles 006 , Don Zagier , MPI Bonn and College de France , Organizer: Stavros Garoufalidis
An introduction for non-experts on real and finite Euler sums, also known as multiple zeta values.
Thursday, March 6, 2014 - 11:00 , Location: Skiles 006 , Annette Werner , Johann Wolfgang Goethe-Universität (Frankfurt) , Organizer: Joseph Rabinoff
Drinfeld's upper half-spaces over non-archimedean local fields are the founding examples of the theory of period domains. In this talk we consider analogs of Drinfeld's upper half-spaces over finite fields. They are open subvarieties of a projective space. We show that their automorphism group is the group of automorphisms of the ambient projective space. This is a problem in birational geometry, which we solve using tools in non-archimedean analytic geometry.
Thursday, February 20, 2014 - 11:00 , Location: Skyles 006 , Jan Medlock , Oregon State University , Organizer: Karim Lounici
The emergence of the 2009 H1N1 influenza A strain and delays in production of vaccine against it illustrate the importance of optimizing vaccine allocation. We have developed computational optimization models to determine optimal vaccination strategies with regard to multiple objective functions: e.g.~deaths, years of life lost, economic costs. Looking at single objectives, we have found that vaccinating children, who transmit most, is robustly selected as the optimal allocation. I will discuss ongoing extensions to this work to incorporate multiple objectives and uncertainty.
Thursday, February 13, 2014 - 11:00 , Location: Skiles 006 , Dmitry Panchenko , Texas A&M University , Organizer: Prasad Tetali
Abstract: I will talk about two types of random processes -- the classical Sherrington-Kirkpatrick (SK) model of spin glasses and its diluted version. One of the main goals in these models is to find a formula for the maximum of the process, or the free energy, in the limit when the size of the system is getting large. The answer depends on understanding the structure of the Gibbs measure in a certain sense, and this structure is expected to be described by the so called Parisi solution in the SK model and Mézard-Parisi solution in the diluted SK model. I will explain what these are and mention some results in this direction.