Georgia Institute of Technology College of Sciences

Foundation is a powerful tool to understand the representability of matroids. The foundation of a matroid is a pasture which is an algebraic structure genrealize the field. I will briefly introduce matroids, algebraic structures (especially pastures) and matroid representability. I will also give some examples on how foundation works in representation of matroids.

We present a multiscale modeling and computational scheme for optical-
mechanical responses of nanostructures. The multi-physical nature of
the problem is a result of the interaction between the electromagnetic
(EM) field, the molecular motion, and the electronic excitation. To
balance accuracy and complexity, we adopt the semi-classical approach
that the EM field is described classically by the Maxwell equations,
and the charged particles follow the Schr ̈oidnger equations quantum
mechanically. To overcome the numerical challenge of solving the high
dimensional multi-component many- body Schr ̈odinger equations, we
further simplify the model with the Ehrenfest molecular dynamics to
determine the motion of the nuclei, and use the Time- Dependent
Current Density Functional Theory (TD-CDFT) to calculate the
excitation of the electrons. This leads to a system of coupled
equations that computes the electromagnetic field, the nuclear
positions, and the electronic current and charge densities
simultaneously. In the regime of linear responses, the resonant
frequencies initiating the out-of-equilibrium optical-mechanical
responses can be formulated as an eigenvalue problem. A
self-consistent multiscale method is designed to deal with the well
separated space scales. The isomerization of Azobenzene is presented as a numerical example.

It is a remarkable fact that some compact topological 4-manifolds X admit infinitely many exotic smooth structures, a phenomenon unique to dimension four.  Indeed a fundamental open problem in the subject is to give a meaningful description of the set of all such structures on any given X.  This talk will describe one approach to this problem when X is simply-connected, via cork twisting.  First we'll sketch an argument to show that any finite list of smooth manifolds homeomorphic to X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism.  In fact, allowing the cork to be noncompact, the collection of all smooth manifolds homeomorphic to X can be obtained in this way.  If time permits, we will also indicate how to construct a single universal noncompact cork whose twists yield all smooth closed simply-connected 4-manifolds.  This is joint work with Hannah Schwartz.

This presentation reviews different concepts of solution of a differential equation, in particular stressing the need to modify the classical theory when we want to deal with discontinuous systems.  We will review the concept of classical solution, and then of Caratheodory solution and Filippov solution, motivating with simple examples the need for these extensions.

We will discuss a deterministic, polynomial (in the rank) time approximation algorithm for counting the bases of a given matroid and for counting common bases between two matroids of the same rank. This talk follows the paper (https://arxiv.org/abs/1807.00929) of Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzant.

Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of partially ordered Reeb graphs, every Reeb graph with n leaves and first Betti number s, is equal to a coproduct of at most 2s trees with (n + s) leaves. An implication of this result, is that Reeb graphs are fixed parameter tractable when the parameter is the first Betti number. We propose partially ordered Reeb graphs as a natural framework for modeling time consistent phylogenetic networks.  We define a notion of interleaving distance on partially ordered Reeb graphs which is analogous to the notion of interleaving distance for ordinary Reeb graphs. This suggests using the interleaving distance as a novel metric for time consistent phylogenetic networks.

Basin of attraction for a stable equilibrium point is an effective concept for stability in deterministic systems. However, it does not contain information on the external perturbations that may affect it. The concept of stochastic basin of attraction (SBA) is introduced by incorporating a suitable probabilistic notion of basin. The criteria for the size of the SBA is based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small intensity but also with whose amplitude is proportional or in general is a function of an order parameter. The efficiency of the concept is presented through two applications.

A sub-Riemannian manifold M is a connected smooth manifold such that the only smooth curves in M which are admissible are those whose tangent vectors at any point are restricted to a particular subset of all possible tangent vectors.  Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators.  We will  construct a random walk on M which converges to a process whose infinitesimal generator  is  one of the natural sub-elliptic  Laplacian  operators.  We will also describe these  Laplacians geometrically and discuss the difficulty of defining one which is canonical.   Examples will be provided.  This is a joint work with Tom Laetsch.

It is a classical theorem of Alexander that every closed oriented manifold is a piecewise linear branched covering of the sphere. In this talk, we will discuss some obstructions to realizing a manifold as a branched covering of the sphere if we require additional properties (like being a submanifold) on the branch set.

We will survey different methods of proving functional inequalities for hypoelliptic  diffusions and the corresponding heat kernels. Some of these methods rely on geometric methods such as curvature-dimension inequalities (due to Baudoin-Garofalo), and some are probabilistic  such as coupling, and finally some use structure  theory and a Fourier transform on Lie groups. This is based on joint work with M. Asaad, F. Baudoin, B. Driver, T. Melcher, Ph. Mariano et al.

Consider the random surface given by the interface separating the plus and minus phases in a low-temperature Ising model in dimensions $d\geq 3$. Dobrushin (1972) famously showed that in cubes of side-length $n$ the horizontal interface is rigid, exhibiting order one height fluctuations above a fixed point. 

We study the large deviations of this interface and obtain a shape theorem for its pillar, conditionally on it reaching an atypically large height. We use this to analyze the law of the maximum height $M_n$ of the interface: we prove that for every $\beta$ large, $M_n/\log n \to c_\beta$, and $(M_n - \mathbb E[M_n])_n$ forms a tight sequence. Moreover, even though this centered sequence does not converge, all its sub-sequential limits satisfy uniform Gumbel tail bounds. Based on joint work with Eyal Lubetzky. 

 Tangles capture a notion of high-connectivity in graphs which differs from $k$-connectivity. Instead of requiring that a small vertex set $X$ does not disconnect the graph $G$, a tangle “points” to the connected component of $G-X$ that contains most of the “highly connected part”. Developed initially by Robertson and Seymour in the graph minors project, tangles have proven to be a fundamental tool in studying the general structure of graphs and matroids. Tangles are also useful in proving that certain families of graphs satisfy an approximate packing-covering duality, also known as the Erd\H{o}s-P\'osa property. In this talk I will give a gentle introduction to tangles and describe some basic applications related to the Erd\H{o}s-P\'osa property.