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Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

Series: School of Mathematics Colloquium

An introduction for non-experts on real and finite Euler sums, also known as multiple zeta values.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.

Series: School of Mathematics Colloquium

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.