Mathematics

A remark on the well-posedness of the degenerated Zakharov system

Vanessa Oliveira (Federal University of Bahia and UIUC)

Abstract: We extend the local well-posedness theory for the Cauchy problem associated to a degenerated Zakharov system. The new main ingredients are the derivation of Strichartz and maximal function norm estimates for the linear solution of a Schr\"odinger type equation with missing dispersion in one direction. The result here improves the one obtained by Linares, Ponce and Saut.

Three-dimensional affine crystals

Bill Goldman (University of Maryland)

Abstract: Crystallographers in the 19th century developed the notion of a crystallographic group to express the geometric symmetries of crystals. The classical theorems of Bieberbach gives a qualitative classification of crystallographic groups in terms of finite groups of integer matrices. In modern terms, this led to to the classification of discrete groups of isometries of Euclidean space, or, equivalently, Riemannian manifolds of zero curvature. The more general theory in affine geometry is much richer and more interesting. In this talk I will describe the recent classification in dimension three, and the intricate relationships to hyperbolic non-Euclidean geometry as well as the geometry of special relativity. Watch on Youtube at http://youtu.be/Bfe47VugpX4

Metric Geometry in the O-minimal setting

Erik Walsberg (UCLA/UIUC)

Abstract: I will discuss metric spaces which are definable in o-minimal expansions of the real field. I will also discuss the Gromov-Hausdorff limits of sequences of elements of a definable family of metrics spaces.

Spectral method for half-space kinetic equation

Qin Li (Caltech)

Abstract: Kinetic equation (the Boltzmann equation, the neutron transport equation etc) is known to converge to fluid (the Euler equation or the heat equation) in some certain regimes, but the coupling of the two systems when both regimes coexist is still open. The key is to understand the half-space problem that resembles the boundary layer connecting the two systems. In this talk, I will present a unified proof for the well-posedness of a class of linear half-space equations with general incoming data, and propose a Galerkin method to numerically solve it in a systematic way. The main strategy is to use damping-recovering process for coercivity of the collision term, and the even-odd decomposition for resolving the singularity. Numerical results will be shown to demonstrate the accuracy of the algorithm.

The minimum number of edges in a 4-critical graph that is bipartite plus 3 edges

Benjamin Reiniger   [email] (UIUC Math)

Abstract: Rödl and Tuza proved that sufficiently large $(k+1)$-critical graphs cannot be made bipartite by deleting fewer than $\binom{k}{2}$ edges, and that this is sharp. Chen, Erdős, Gyárfás, and Schelp constructed infinitely many $4$-critical graphs obtained from bipartite graphs by adding a matching of size $3$ (and called them $(B+3)$-graphs). They conjectured that every $n$-vertex $(B+3)$-graph has much more than $5n/3$ edges, presented $(B+3)$-graphs with $2n-3$ edges, and suggested that perhaps $2n$ is the asymptotically best lower bound. We prove that indeed every $(B+3)$-graph has at least $2n-3$ edges. Our proof uses a potential function and the connection between orientations and colorings of graphs. This is joint work with A.V. Kostochka.

Algebraic geometry and geometric modeling (arXiv:1306.1445)

Hal Schenck (UIUC Math)

Abstract: I'll describe a recent application of algebraic geometry to a problem in geometric modeling. Let $P_d$ be a convex polygon with d vertices. The associated Wachspress surface $W_d$ is a fundamental object in approximation theory, defined as the image of the rational map $w_d$ from $P^2$ to $P^{d-1}$, determined by the Wachspress barycentric coordinates for $P_d$. We show $w_d$ is a regular map on a blowup $X_d$ of $P^2$, and if d>4 is given by a very ample divisor on $X_d$, so has a smooth image $W_d$. We determine generators for the ideal of $W_d$, and prove that in graded lex order, the initial ideal of $I(W_d)$ is given by a Stanley-Reisner ideal. As a consequence, we show that the associated surface is arithmetically Cohen-Macaulay, of Castelnuovo-Mumford regularity two, and determine all the graded betti numbers of $I(W_d)$. Joint work with Corey Irving, Santa Clara University

Topological tools for the real world

Jean-Luc Thiffault   [email] (University of Wisconsin, Mathematics)

Abstract: Topology is emerging as an important new tool for understanding our world. Computational homology, for example, has become standard for analyzing the connectivity of large-dimensional data sets. Here I present another approach, which is more dynamical in nature. The trajectories of `particles,' whether oceanic floats or people, can be regarded as mathematical objects called braids. By using traditional concepts from topological dynamics, such as topological entropy, we gain insight into the inherent complexity of motion.