Georgia Institute of Technology College of Sciences

In this talk, we will consider semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^n. This class of equations includes Bellman equations containing operators of Levy-Ito type. Holder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.

I will discuss inverse problems involving elliptic partial differential equations with highly oscillating coefficients. The multiscale nature of such problems poses a challenge in both the mathematical formulation and the numerical modeling, which is hard even for forward computations. I will discuss uniqueness of the inverse in certain problem classes and give numerical methods for inversion that can be applied to problems in medical imaging and exploration seismology.

In numerical algebraic geometry the key idea is to solve systems of polynomial equations via homotopy continuation. By this is meant, that the solutions of a system are tracked as the coefficients change continuously toward the system of interest. We study the tropicalisation of this process. Namely, we combinatorially keep track of the solutions of a tropical polynomial system as its coefficients change. Tropicalising the entire regeneration process of numerical algebraic geometry, we obtain a combinatorial algorithm for finding all tropical solutions. In particular, we obtain the mixed cells of the system in a mixed volume computation. Experiments suggest that the method is not only competitive but also asymptotically performs better than conventional methods for mixed cell enumeration. The method shares many of the properties of a recent tropical method proposed by Malajovich. However, using symbolic perturbations, reverse search and exact arithmetic our method becomes reliable, memory-less and well-suited for parallelisation.

We will review the notion of Whitney differentiability and the Whitney embedding theorem. Then, we will also review its applications in KAM theory (continuation of last week's talk).

Morphogenesis of curved bilayer membranes Buckling of curved membranes plays a prominent role in the morphogenesis of multilayered soft tissue, with examples ranging from tissue differentiation, the wrinkling of skin, or villi formation in the gut, to the development of brain convolutions. In addition to their biological relevance, buckling and wrinkling processes are attracting considerable interest as promising techniques for nanoscale surface patterning, microlens array fabrication, and adaptive aerodynamic drag control. Yet, owing to the nonlinearity of the underlying mechanical forces, current theoretical models cannot reliably predict the experimentally observed symmetry-breaking transitions in such systems. Here, we derive a generalized Swift-Hohenberg theory capable of describing the wrinkling morphology and pattern selection in curved elastic bilayer materials. Testing the theory against experiments on spherically shaped surfaces, we find quantitative agreement with analytical predictions separating distinct phases of labyrinthine and hexagonal wrinkling patterns. We highlight the applicability of the theory to arbitrarily shaped surfaces and discuss theoretical implications for the dynamics and evolution of wrinkling patterns.

We consider generalized Bochner-Riesz multipliers $(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. For integers $m\ge 2$, we find domains such that $(1-\rho(\xi))_+^{\lambda}\in M^p(\mathbb{R}^2)$ for all $\lambda>0$ in the range $\frac{m}{m-1}\le p\le 2$, but for which $\inf\{\lambda:\,(1-\rho)_+^{\lambda}\in M_p\}>0$ when $p<\frac{m}{m-1}$. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the ``additive energy" of the boundary of a convex domain. Second, we associate a set of directions to a convex domain and define a sequence of Nikodym-type maximal operators corresponding to this set of directions. We show that domains that have low higher order energy, as well as those which have asymptotically good $L^p$ bounds for the corresponding sequence of Nikodym-type maximal operators, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.

There has been extensive research on cycle lengths in graphs with large minimum degree. In this talk, we will present several new and tight results in this area. Let G be a graph with minimum degree at least k+1. We prove that if G is bipartite, then there are k cycles in G whose lengths form an arithmetic progression with common difference two. For general graph G, we show that G contains \lfloor k/2\rfloor cycles with consecutive even lengths, and in addition, if G is 2-connected and non-bipartite, then G contains \lfloor k/2\rfloor cycles with consecutive odd lengths. Thomassen (1983) made two conjectures on cycle lengths modulo a fixed integer k: (1) every graph with minimum degree at least k+1 contains cycles of all even lengths modulo k; (2) every 2-connected non-bipartite graph with minimum degree at least $k+1$ contains cycles of all lengths modulo k. These two conjectures, if true, are best possible. Our results confirm both conjectures! when k is even. And when k is odd, we show that minimum degree at least $+4 suffices. Moreover, our results derive new upper bounds of the chromatic number in terms of the longest sequence of cycles with consecutive (even or odd) lengths. This is a joint work with Chun-Hung Liu.

In the 1970s, Girko made the striking observation that, after centering, traces of functions of large random matrices have approximately Gaussian distribution. This convergence is true without any further normalization provided f is smooth enough, even though the trace involves a number of terms equal to the dimension of the matrix. This is particularly interesting, because for some rougher, but still natural observables, like the number of eigenvalues in an interval, the fluctuations diverge. I will explain how such results can be obtained, focusing in particular on controlling the fluctuations when the function is not very regular.

A morph between two drawings of the same graph can be thought of as a continuous deformation between the two given drawings. In this talk we consider the algorithmic problem of morphing between any two planar drawings of a planar triangulation while preserving planarity during the morph. We outline two different solutions to the morphing problem. The first solution gives a strengthening of the result of Alamdari et al. where each step is a unidirectional morph. The second morphing algorithm finds a planar morph consisting of O(n²) steps between any two Schnyder drawings while remaining in an O(n)×O(n) grid, here n is the number of vertices of the graph. However, there are drawings of planar triangulations which are not Schnyder drawings, and for these drawings we show that a unidirectional morph consisting of O(n) steps that ends at a Schnyder drawing can be found. (Joint work with Penny Haxell and Anna Lubiw)