Georgia Institute of Technology College of Sciences

Polynomial systems can be effectively solved by exploiting structure present in their Galois group. Esterov determined two conditions for which the Galois group of a sparse polynomial system is imprimitive, and showed that the Galois group is the symmetric group otherwise. A system with an imprimitive Galois group can be decomposed into simpler systems, which themselves may be further decomposed. Esterov's conditions give a stopping criterion for decomposing these systems and leads to a recursive algorithm for efficient solving.

Given an immersed, Maslov-0, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-0, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-0, exact Lagrangian filling with genus g ≥ 1 and p double points can be obtained from such a Lagrangian surgery on a filling of genus g − 1 with p+1 double points. To show this, we establish the connection between the existence of an immersed, Maslov-0, exact Lagrangian filling of a Legendrian Λ that has p double points with action 0 and the existence of an embedded, Maslov-0, exact Lagrangian cobordism from p copies of a Hopf link to Λ. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-0, exact Lagrangian cobordisms between Legendrian links. Joint work with Noemie Legout, Maylis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.

After an introduction to how to think about the mapping class groupand its cohomology, I will discuss a recent theorem of mine saying
that passing to the level-l subgroup does not change the rational cohomology in a stable range.

Sleep and wake states are driven by interactions of neuronal populations in many areas of the human brain, such as the brainstem, midbrain, hypothalamus, and basal forebrain. The timing of human sleep is strongly modulated by the 24 h circadian rhythm and the homeostatic sleep drive, the need for sleep that depends on the history of prior awakening. The parameters dictating the evolution of the homeostatic sleep drive may vary with development or interindividual characteristics and have been identified as important parameters for generating the transition from multiple sleeps to a single sleep episode per day. Features of the mean firing rate of the neurons in the suprachiasmatic nucleus (SCN), the central pacemaker in humans, may differ with seasonality. In this talk, I will present our analysis of changes in sleep patterning under variation of homeostatic and circadian parameters using a mathematical model for human sleep-wake regulation. I will also talk about the fundamental tools we employ to understand the dynamics of the model, such as the construction of a circle map that captures the timing of sleep onsets on successive days. Analysis of the structure and bifurcations in the map reveals changes in the average number of sleep episodes per circadian day in a period-adding-like structure caused by the separate or combined effects of circadian and homeostatic variation. Time permitting, I will talk about some of our current work on modeling sleep patterns in early childhood using experimental data.

I will consider the long-time influence of deterministic and stochastic perturbations of dynamical systems and diffusion processes with a first integral . A diffusion process on the Reeb graph of the first integral determines the long-time behavior of the perturbed system. In particular, I will consider stochasticity of long time behavior of deterministic systems close to a system with a conservation law. Which of the invariant  measures of the non-perturbed system will be limiting for a given class of perturbations also will be discussed.

I present some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. Critical quantities measuring this size are the 3/2 norm of the magnetic field B and the 3 norm of the vector potential A.  The point is that the spinor structure enters the analysis in a crucial way. This is joint work with Rupert Frank at LMU Munich.

I will review several parallel GPU-based approaches to better understand multistable dynamics of simple neural networks and global bifurcation unfolding of systems with deterministic chaos. 
 

Quadratic forms and their hidden convexity have been studied for decades, dating back to famous theorems by Toeplitz-Hausdorff, Dines and Brickman. It has very rich connection to optimization via Yakubovich's S-lemma. I will introduce these results, as well as an ongoing work of obtaining convex hull via aggregations, where we introduced the closely related notion of hidden hyperplane convexity.

Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. We will discuss a construction of small symplectic caps, using ideas first laid out by Gay in 2002, for rational homology balls bounded by lens spaces. This allows us to explicitly understand embeddings of these rational balls in CP2 that were earlier understood only through almost toric fibrations. This is joint work with John Etnyre, Hyunki Min, and Lisa Piccirillo.

This is an ongoing project. We make use of two dual Lawrence polytopes $P$ and $P*$ of a graph $G$, to study invariants of the graph. The $h$-vector of the graphic (resp. cographic) matroid complex associated to $G$ coincides with the $h^*$-vector of the Lawrence polytope $P$ (resp. $P^*$). In general, the $h$-vector is an invariant defined for an abstract simplicial complex, which encodes the number of faces of different dimensions. The $h^*$-vector, a.k.a. the $\delta$-polynomial, is an invariant defined for a rational polytope, which is obtained by dilating the polytope. By dissecting the Lawrence polytopes, we may study the $h$-vectors associated to the graph $G$ at a finer level. In particular, we understand activities and reduced divisors of the graph $G$ in a more geometric way. I will try to make the talk self-contained.

The study of Poisson approximations of the process of recurrences to small subsets in the phase spaces of chaotic dynamical systems, started in 1991, have developed by now into a large active area of the dynamical systems theory. In this talk, I will present some new results. This is a joint work with Prof. Leonid Bunimovich.

1. I will start with some examples of dissipative hyperbolic systems,
2. then formulate the question of functional Poisson approximations for these systems.
3. To study Poisson approximations, I will present two difficulties, called short returns and ring conditions.
4. These two difficulties can be partially solved under some conditions of, e.g. the dimension of the dynamics, the Hausdorff dimension of the SRB measure, etc. I will present a new method which does not depend on dimensions but can completely solve these two difficulties for dissipative systems.