What is what in hyperbolic geometry

Michael Kapovich

Matroids and their applications

Andrew Berget

When can every homogeneous degree d polynomial be written as linear combination of products of linear forms selected from a given set? What if each linear form can only be used once in any product appearing? The answer, of course, depends on the linear forms, but it is not clear that it only depends on the linear algebraic properties of the forms. I will give some examples of this problem and the answer, which will serve as motivation for studying matroids.

Applications of convex geometry

Amitabh Basu

I will outline the main research directions that I intend to pursue starting in the 2010-2011 session, in the framework of the VIGRE Research Focus Group program. There are three main themes that I want to investigate -

a) Problems from the Geometry of Numbers, which is a field which studies the interaction of convex sets with discrete lattices in Rⁿ.

b) Cutting Plane theory in the context of Integer Programming - a field which is receiving a lot of attention in the field of discrete optimization. It is a classic example of how polyhedral theory is applied in integer programming and optimization in general.

c) Applications of Algebraic Geometry in optimization and complexity. My main aim is to understand in depth the fields of 1) Convex Algebraic Geometry and 2) Geometric Complexity Theory - an approach to the P vs NP problem from computational complexity theory.

After giving a brief introduction to the topics, I will state the specific open problems in these areas that I hope to make progress on.

Permutation polytopes

Mohamed Omar

Permutation polytopes are convex hulls of real representations of finite groups. We shall introduce these objects and discuss current work on computing their Ehrhart polynomials. This is joint work with Jesus De Loera and Katherine Jones.

Generation and Propagation of Traffic Waves in a Simple Freeway Network: Macroscopic vs Microscopic Considerations

Martha Shott

Stop-and-go traffic waves are a frequent phenomenon observed in freeway congestion. Although these oscillations increase travel delay and contribute to greenhouse gas emissions, little is yet understood about the manner in which these waves occur and propagate (and thus how to mitigate their impact). In this talk, we examine the ability of several traffic flow models to exhibit characteristic oscillatory behavior in an on-ramp freeway network, and discuss how the information obtained from these models can be applied to traffic management.

Universality in random growth processes

Gregory Shinault

The Central Limit Theorem tells us that the Gaussian distribution describes fluctuations about the mean for a large sum of random variables. It seems the Airy process will analogously describe fluctuations in some random growth models, though we do not yet know how generally this will hold. I'll briefly introduce some growth models and describe some work I have been doing to understand asymptotics for the Airy process.

Markov chains and electrical networks

Anastasia Raymer

Any reversible Markov chain can be described in terms of an electrical network. I will introduce a probabilistic interpretation of voltage and briefly describe how I utilized this in acquiring a lower bound for the mixing time of the 15 Puzzle.

The compressible Euler equations on a fixed domain

Joseph Grimm

The compressible Euler equations are a good description of the behavior of a fluid without viscosity. I will present inequalities that sufficiently regular solutions to the Euler equations must satisfy. I will then discuss my current work to establish that there are such solutions.

Topological representations of matroid maps

Matthew Stamps

A matroid is a combinatorial object which captures the notion of independence in a vector space. A surprising result in the theory of matroids is the Topological Representation Theorem, which states that any matroid can be obtained from the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. In fact, for any given matroid, one can always construct such an arrangement. In this talk, I'll present one such construction and use it to show that the structure-preserving maps between matroids induce continuous maps between their topological representations. No prior knowledge of matroids will be assumed.