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Series: School of Mathematics Colloquium

The problem of finding rational solutions to cubic equations is central in number theory, and goes back to Fermat. I will discuss why these equations are particularly interesting, and the modern theory of elliptic curves that has developed over the past century, including the Mordell-Weil theorem and the conjecture of Birch and Swinnerton-Dyer. I will end with a description of some recent results of Manjul Bhargava on the average rank.

Series: School of Mathematics Colloquium

The study of mechanical linkages is a very classical one, dating back to
the Industrial Revolution. In this talk we will discuss the geometry
of the configuration spaces in some simple idealized examples and, in
particular, their curvature and geometry. This leads to an interesting
quantitative description of their dynamical behaviour.

Series: School of Mathematics Colloquium

While this could be a lecture about our US Congress, it instead deals with problems that are asymptotically related to best-packing and best-covering. In particular, we discuss how to efficiently generate N points on a d-dimensional manifold that have the desirable qualities of well-separation and optimal order covering radius, while asymptotically having a prescribed distribution. Even for certain small numbers of points like N=5, optimal arrangements with regard to energy and polarization can be a challenging problem.

Series: School of Mathematics Colloquium

Front propagation in fluid flows arise in power generation of automobile
engines, forest fire spreading, and material interfaces of solidification
to name a few. In this talk, we
introduce the level set formulation and the resulting Hamilton-Jacobi
equation, known as
G-equation in turbulent combustion. When the fluid flow has enough
intensity, G-equation becomes non-coercive and non-linearity no longer
dominates.
When front curvature and flow stretching effects
are included, the extended G-equation is also non-convex. We discuss recent
progress in
analysis and computation of homogenization and large time front speeds in
cellular flows (two dimensional Hamiltonian flows) from both Lagrangian and
Eulerian perspectives, and the recovery of experimental observations from
the G-equations.

Series: School of Mathematics Colloquium

Given a d-dimensional convex body C containing the origin in
its interior and a real t>1, we seek to construct a polytope P with
as few vertices as possible such that P is contained in C and C is
contained in tP. I plan to present a construction which breaks some
long-held records and is nearly optimal for a wide range of parameters d
and t. The construction uses the maximum volume ellipsoid, the John
decomposition of the identity and its recent sparsification by Batson,
Spielman and Srivastava, Chebyshev polynomials, and some tensor
algebra.

Series: School of Mathematics Colloquium

The classical Weyl equidistribution theorem says that if v is a non-resonant
vector then the sequence v, 2v, 3v... is uniformly distributed on a
torus. In this
talk we discuss the rate of convergence to the uniform distribution.
This is a joint work with Bassam Fayad.

Series: School of Mathematics Colloquium

Statistical modeling amounts to specifying a set of candidates for what the probability distribution of an observed random quantity might be. Many models used in practice are of an algebraic nature in thatthey are defined in terms of a polynomial parametrization. The goal of this talk is to exemplify how techniques from computational algebraic geometry may be used to solve statistical problems thatconcern algebraic models. The focus will be on applications in hypothesis testing and parameter identification, for which we will survey some of the known results and open problems.

Series: School of Mathematics Colloquium

Random matrix theory is a fast developing topic with connections to so many areas of mathematics: probability, number theory, combinatorics, data analysis, mathematical physics, to mention a few.
The determinant is one of the most studied matrix functionals. In our talk, we are going to give a brief survey on the studies of this functional, dated back to Turan in the 1940s. The main focus will be on recent developments that establish the limiting law in various models.

Series: School of Mathematics Colloquium

Kickoff of the Tech Topology Conference from December 7-9, 2012.

This will be a Colloquium talk, aimed at a general audience. The topic is the curve complex, introduced by Harvey in 1974. It's a simplicial complex, and was introduced as a tool to study mapping class groups of surfaces. I will discuss recent joint work with Bill Menasco about new local pathology in the curve complex, namely that its geodesics can have dead ends and even double dead ends.

Series: School of Mathematics Colloquium

Populations, whether they be viral particles, bio-chemicals, plants or animals, are subject to intrinsic and extrinsic sources of stochasticity. This stochasticity in conjunction with nonlinear interactions between individuals determines to what extinct populations are able to persist in the long-term. Understanding the precise nature of these interactive effects is a central issue in population biology from theoretical, empirical, and applied perspectives.
For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable.
For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness
asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical work on Kansas prairies, acorn woodpecker populations, and microcosm experiments demonstrating these phenomena will be discussed.