Georgia Institute of Technology College of Sciences

We study a particular distinguished component (the 'Hitchin component') of the space of surface group representations to SL(3,\R).  In this setting, both Hitchin (via Higgs bundles) and the more ancient subject of affine spheres associate a bundle of holomorphic differentials over Teichmuller space to this component of the character variety.  We focus on a ray of holomorphic differentials and provide a formula, tropical in appearance, for the asymptotic holonomy of the representations in terms of the local geometry of the differential.  Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Andrea Tamburelli, and all the constructions and definitions will be (likely briskly) explained.

Many fluid flows display at a wide range of space and time scales. Turbulent and multiphase flows can include small eddies or particles, but likewise large advected features. This challenge makes some degree of multi-scale modeling or homogenization necessary. Such models are restricted, though: they should be numerically accurate, physically consistent, computationally expedient, and more. I present two tools crafted for this purpose. First, the fast macroscopic forcing method (Fast MFM), which is based on an elliptic pruning procedure that localizes solution operators and sparse matrix-vector sampling. We recover eddy-diffusivity operators with a convergence that beats the best spectral approximation (from the SVD), attenuating the cost of, for example, targeted RANS closures. I also present a moment-based method for closing multiphase flow equations. Buttressed by a recurrent neural network, it is numerically stable and achieves state-of-the-art accuracy. I close with a discussion of conducting these simulations near exascale. Our simulations scale ideally on the entirety of ORNL Summit's GPUs, though the HPC landscape continues to shift.

The burning number $b(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be covered by $k$ balls of radii respectively $0,\dots,k-1$, and was introduced independently by Brandenburg and Scott at Intel as a transmission problem on processors \cite{alon} and Bonato, Janssen and Roshanbin as a model for the spread of information in social networks.

The Burning Number Conjecture \cite{bonato} claims that $b(G)\leq \left\lceil\sqrt{n}\right\rceil$, where $n$ is the number of vertices of $G$. This bound tight for paths. The previous best bound for this problem, by Bastide et al. \cite{bastide}, was $b(G)\leq \sqrt{\frac{4n}{3}}+1$.

We prove that the Burning Number Conjecture holds asymptotically, that is $b(G)\leq (1+o(1))\sqrt{n}$.

Following a brief introduction to graph burning, this talk will focus on the general ideas behind the proof.

Random and irregular growth is all around us. We see it in the form of cancer growth, bacterial infection, fluid flow through porous rock, and propagating flame fronts. Simple models for these processes originated in the '50s with percolation theory and have since given rise to many new models and interesting mathematics. I will introduce a few models (percolation, invasion percolation, first-passage percolation, diffusion-limited aggregation, ...), along with some of their basic properties.

TBA

We estimate the  Riesz basis (RB) bounds obtained in Hruschev, Nikolskii and Pavlov' s classical characterization of exponential RB. As an application, we  improve previously known estimates of the RB bounds in some classical cases, such as RB obtained by an Avdonin type perturbation, or RB which are the zero-set of sine-type functions. This talk is based on joint work with S. Nitzan

The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.   

Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.

(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others. The talk is suitable for both professors and students.)

The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.  

Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.

(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others.

The talk is suitable for both professors and students.)

Elastic beam Hamiltonians on single-layer graphs are constructed out of Euler-Bernoulli beams, each governed by a scalar valued fourth-order Schrödinger operator equipped with a real symmetric potential. Unlike the second-order Schrödinger operator commonly applied in quantum graph literature, here the self-adjoint vertex conditions encode geometry of the graph by their dependence on angles at which edges are met. In this talk, I will first consider spectral properties of this Hamiltonian with periodic potentials on a special equal-angle lattice, known as graphene or honeycomb lattice. I will also discuss spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a joint work with Mahmood Ettehad (University of Minnesota),https://arxiv.org/pdf/2110.05466.pdf.

A Coxeter group is a (not necessarily finite) group given by certain types of generators and relations. Examples of finite Coxeter groups include dihedral groups, symmetric groups, and reflection groups. They play an important role in various areas. In this talk, I will discuss why I am interested in Coxeter groups from a combinatorial perspective - the geometric concepts associated with the finite Coxeter groups form the language of Coxeter matroids, which are generalizations of ordinary matroids. In particular, finite Coxeter groups are related to Coxeter matroids in the same way as symmetric groups are related to ordinary matroids. The main reference for this talk is Chapter 5 of Borovik-Gelfand-White's book Coxeter Matroids. I will only assume basic group theory, but not familiarity with matroids.

https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

We will begin with a brief overview of several parallel and hybrid computing approaches including CUDA, OpenAcc, OpenMP, and OpenMPI, followed by a demonstration of how we can leverage these technologies to study complex dynamics arising from diverse nonlinear systems. First, we discuss multistable rhythms in oscillatory 4-cell central pattern generators (CPGs) of inhibitory coupled  neurons. We show how network topology and intrinsic properties of the cells affect dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. We then discuss symbolic methods and parametric sweeps to analyze isolated neuron dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits and animal CPGs. We also demonstrate how such symbolic methods can help identify the universal principles governing both simple and complex dynamics, and chaotic structure in various Lorenz-like systems, their key self-similar organizing structures in 2D parameter space, as well as detailed computational reconstructions of 3D bifurcation surfaces.
 

In the problem of online portfolio selection as formulated by Cover (1991), the trader repeatedly distributes her capital over $ d $ assets in each of $ T > 1 $ rounds, with the goal of maximizing the total return. Cover proposed an algorithm called Universal Portfolios, that performs nearly as well as the best (in hindsight) static assignment of a portfolio, with 

an $ O(d\log(T)) $ regret in terms of the logarithmic return. Without imposing any restrictions on the market, this guarantee is known to be worst-case optimal, and no other algorithm attaining it has been discovered so far. Unfortunately, Cover's algorithm crucially relies on computing the expectation over certain log-concave density in R^d, so in a practical implementation this expectation has to be approximated via sampling, which is computationally challenging. In particular, the fastest known implementation, proposed by Kalai and Vempala in 2002, runs in $ O( d^4 (T+d)^{14} ) $ per round, which rules out any practical application scenario. Proposing a practical algorithm with a near-optimal regret is a long-standing open problem. We propose an algorithm for online portfolio selection with a near-optimal regret guarantee of $ O( d \log(T+d) ) $ and the runtime of only $ O( d^2 (T+d) ) $ per round. In a nutshell, our algorithm is a variant of the follow-the-regularized-leader scheme, with a time-dependent regularizer given by the volumetric barrier for the sum of observed losses. Thus, our result gives a fresh perspective on the concept of volumetric barrier, initially proposed in the context of cutting-plane methods and interior-point methods, correspondingly by Vaidya (1989) and Nesterov and Nemirovski (1994). Our side contribution, of independent interest, is deriving the volumetrically regularized portfolio as a variational approximation of the universal portfolio: namely, we show that it minimizes Gibbs's free energy functional, with accuracy of order $ O( d \log(T+d) ) $. This is a joint work with Remi Jezequel and Pierre Gaillard. 

Dirac proved that every $n$-vertex graph with minimum degree at least $n/2$ contains a hamiltonian cycle. Moreover, every graph with minimum degree $k \geq 2$ contains a cycle of length at least $k+1$, and this can be further improved if the graph is 2-connected. In this talk, we prove analogs of these theorems for hypergraphs. That is, we give sharp minimum degree conditions that imply the existence of long Berge cycles in uniform hypergraphs. This is joint work with Alexandr Kostochka and Grace McCourt.

In this talk I will generalize a simple trick to produce splitting for the separatrices of (the time-one map of) a simple pendulum, to hyperbolic tori of any dimension $m\geq 2$. The examples will be constructed in the Gevrey class, and the splitting is bounded from below by a term of the form $\exp (-c(1/\eps)^a)$, where $a=\frac{1}{2(\alpha-1)(m-2)}$. This will be compared to usual upper bounds in the same setting.