Title: Some applications of logic to additive number theory

Abstract: I will review the Loeb measure construction; I will

assume some exposure to nonstandard analysis, or at least 1st order logic,

comparable to the review I gave last semester in my seminars on fixed

points. Time permitting I will give the Loeb-measure proof of Szemeredi’s

Theorem.

Logic seminar: David Ross

Title: Some applications of logic to additive number theory (cont.)

Room: Keller 404.

Abstract:

I will continue with some examples of results about sets of positive upper Banach density proved using Loeb measures.

Speaker: Evan Gawlik (UCSD)

Title: Numerical Methods for Partial Differential Equations on Evolving Domains

Abstract: Many important and challenging problems in computational science and

engineering involve partial differential equations with a high level

of geometric complexity. Examples include moving-boundary problems,

where the domain on which a PDE is posed evolves with time in a

prescribed fashion; free-boundary problems, where the domain is one of

the unknowns in and of itself; and geometric evolution equations,

where the domain is an evolving Riemannian manifold. Such problems are

inherently challenging to solve numerically, owing not only to the

difficulty of discretizing functions defined on evolving geometries,

but also to the coupling, if any, between the geometry’s evolution and

the underlying PDE. Similar difficulties, which are in some sense dual

to those just mentioned, are faced when the goal is to numerically

approximate functions taking values in a manifold. This talk will

focus on tackling these unique challenges that lie at the intersection

of numerical analysis, PDEs, and geometry.

Speaker: Tam Nguyen Phan (Binghamton U.)

Title: Examples of negatively curved and nonpositively curved manifolds

Abstract: Let M be a noncompact, complete, Riemannian manifold. Gromov proved that if the sectional curvature of M negative and bounded, and if the volume of M is finite, then M is homeomorphic to the interior of a compact manifold overline{M} with boundary B. In other words, M has finitely many ends, and each end of M is topologically a product of a closed manifold C with a ray. A natural question is how the geometry (i.e. in terms of the curvature) of M controls the topology of C. The same question is interesting in nonpositive curvature settings. I will discuss what topological restrictions there are on each end and give old and new constructions of such manifolds.

Title: Matrix algebra dimensions

Abstract: What is the dimension of a triply generated commutative matrix algebra? It seems that not much is known, but we’ll discuss some relevant ideas. For example, an old result, often called Gerstenhaber’s Theorem, states that the algebra of polynomials in two commuting nxn matrices has dimension at most n. Here we discuss the possibility of extending this result to algebras generated by three commuting matrices. Related questions concern the reducibility of the variety of commuting triples and the question of “approximate simultaneous diagonalizability”. We present some experimental results based on the Weyr canonical form (an under–appreciated alternative to the JCF).

Speaker: Farbod Shokrieh (Cornell U.)

Title:

Metric graphs, potential theory, and algebraic geometry

Abstract:

A metric graph can be viewed, in many respects, as an analogue of an

algebraic curve. For example, there is a notion of “Jacobian” for

graphs.

More classically, metric graphs can be viewed as electrical networks.

I will discuss the interplay between these two points of view, as well

as some recent applications to problems in algebraic geometry.