Seminars and Colloquia by Series

Hamiltonian Cycles in Uniform Hypergraphs with Large Minimum Degree

Series
Combinatorics Seminar
Time
Friday, February 8, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andrzej Rucinski – Emory and AMU Poznań

Abstract: Reiher, Rödl, Ruciński, Schacht, and Szemerédi proved, via a modification of the absorbing method, that every 3-uniform $n$-vertex hypergraph, $n$ large, with minimum vertex degree at least $(5/9+\alpha)n^2/2$ contains a tight Hamiltonian cycle. Recently, owing to a further modification of the method, the same group of authors joined by Bjarne Schuelke, extended this result to 4-uniform hypergraphs with minimum pair degree at least, again, $(5/9+\alpha)n^2/2$. In my talk I will outline these proofs and point to the crucial ideas behind both modifications of the absorbing method.

Travel Behavior Modeling Using Machine Learning

Series
ACO Student Seminar
Time
Friday, February 8, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xilei Zhao – ISyE, Georgia Tech –

The popularity of machine learning is increasingly growing in transportation, with applications ranging from traffic engineering to travel demand forecasting and pavement material modeling, to name just a few. Researchers often find that machine learning achieves higher predictive accuracy compared to traditional methods. However, many machine-learning methods are often viewed as “black-box” models, lacking interpretability for decision making. As a result, increased attention is being devoted to the interpretability of machine-learning results.

In this talk, I introduce the application of machine learning to study travel behavior, covering both mode prediction and behavioral interpretation. I first discuss the key differences between machine learning and logit models in modeling travel mode choice, focusing on model development, evaluation, and interpretation. Next, I apply the existing machine-learning interpretation tools and also propose two new model-agnostic interpretation tools to examine behavioral heterogeneity. Lastly, I show the potential of using machine learning as an exploratory tool to tune the utility functions of logit models.

I illustrate these ideas by examining stated-preference travel survey data for a new mobility-on-demand transit system that integrates fixed-route buses and on-demand shuttles. The results show that the best-performing machine-learning classifier results in higher predictive accuracy than logit models as well as comparable behavioral outputs. In addition, results obtained from model-agnostic interpretation tools show that certain machine-learning models (e.g. boosting trees) can readily account for individual heterogeneity and generate valuable behavioral insights on different population segments. Moreover, I show that interpretable machine learning can be applied to tune the utility functions of logit models (e.g. specifying nonlinearities) and to enhance their model performance. In turn, these findings can be used to inform the design of new mobility services and transportation policies.

Convex Relaxation for Multimarginal Optimal Transport in Density Functional Theory

Series
Applied and Computational Mathematics Seminar
Time
Friday, February 8, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Lexing Ying – Stanford University

Please Note: We will go to lunch together after the talk with the graduate students.

We introduce methods from convex optimization to solve the multi-marginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy.

Compactifying parameter spaces

Series
Intersection Theory Seminar
Time
Thursday, February 7, 2019 - 15:18 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tianyi Zhang – Georgia Tech
We continue the discussion of Chapter 8 in 3264 and All That. We will discuss complete quadrics, Hilbert schemes and Kontsevich spaces.

Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

Series
Stochastics Seminar
Time
Thursday, February 7, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Atilla Yilmaz – Temple University –
I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.

Interpolative decomposition and its applications

Series
School of Mathematics Colloquium
Time
Thursday, February 7, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Lexing Ying – Stanford University
Interpolative decomposition is a simple and yet powerful tool for approximating low-rank matrices. After discussing the theory and algorithms, I will present a few new applications of interpolative decomposition in numerical partial differential equations, quantum chemistry, and machine learning.

Translation and Systems Biology: Mathematical and computational modeling at the frontier of biomedical research

Series
Job Candidate Talk
Time
Thursday, February 7, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gesham Magombedze – Baylor Institute for Immunology Research –

A major challenge in clinical and biomedical research is on translating in-vitro and in- vivo model findings to humans. Translation success rate of all new compounds going through different clinical trial phases is generally about 10%. (i) This field is challenged by a lack of robust methods that can be used to translate model findings to humans (or interpret preclinical finds to accurately design successful patient regimens), hence providing a platform to evaluate a plethora of agents before they are channeled in clinical trials. Using set theory principles of mapping morphisms, we recently developed a novel translational framework that can faithfully map experimental results to clinical patient results. This talk will demonstrate how this method was used to predict outcomes of anti-TB drug clinical trials. (ii) Translation failure is deeply rooted in the dissimilarities between humans and experimental models used; wide pathogen isolates variation, patient population genetic diversities and geographic heterogeneities. In TB, bacteria phenotypic heterogeneity shapes differential antibiotic susceptibility patterns in patients. This talk will also demonstrate the application of dynamical systems in Systems Biology to model (a) gene regulatory networks and how gene programs influence Mycobacterium tuberculosis bacteria metabolic/phenotypic plasticity. (b) And then illustrate how different bacteria phenotypic subpopulations influence treatment outcomes and the translation of preclinical TB therapeutic regimens. In general, this talk will strongly showcase how mathematical modeling can be used to critically analyze experimental and patient data.

Strong edge coloring of subcubic planar graphs

Series
Graph Theory Working Seminar
Time
Wednesday, February 6, 2019 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joshua Schroeder – Georgia Tech

Strong edge coloring of a graph $G$ is a coloring of the edges of the graph such that each color class is an induced subgraph. The strong chromatic index of $G$ is the smallest number $k$ such that $G$ has a $k$-strong edge coloring. Erdős and Nešetřil conjecture that the strong chromatic index of a graph of max degree $\Delta$ is at most $5\Delta^2/4$ if $\Delta$ is even and $(5\Delta^2-2\Delta + 1)/4$ if $\Delta$ is odd. It is known for $\Delta=3$ that the conjecture holds, and in this talk I will present part of Anderson's proof that the strong chromatic index of a subcubic planar graph is at most $10$

On delocalization of eigenvectors of random non-Hermitian matrices

Series
High Dimensional Seminar
Time
Wednesday, February 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anna Lytova – University of Opole –

We study delocalization properties of null vectors and eigenvectors of matrices with i.i.d. subgaussian entries. Such properties describe quantitatively how "flat" is a vector and confirm one of the universality conjectures stating that distributions of eigenvectors of many classes of random matrices are close to the uniform distribution on the unit sphere. In particular, we get lower bounds on the smallest coordinates of eigenvectors, which are optimal as the case of Gaussian matrices shows. The talk is based on the joint work with Konstantin Tikhomirov.

Global Convergence of Neuron Birth-Death Dynamics

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, February 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joan Bruna Estrach – New York University –
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).

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