Seminars and Colloquia by Series

On the relativistic Landau equation

Series
PDE Seminar
Time
Tuesday, September 24, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Maja Taskovic – Emory University –
In kinetic theory, a large system of particles is described by the particle density function. The Landau equation, derived by Landau in 1936, is one such example. It models a dilute hot plasma with fast moving particles that interact via Coulomb interactions. This model does not include the effects of Einstein’s theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. These effects are captured by the relativistic Landau equation, which was derived by Budker and Beliaev in 1956. 
 
We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel in the relativistic collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data. This is joint work with Robert M. Strain.

Online algorithms for knapsack and generalized assignment problem under random-order arrival

Series
ACO Seminar
Time
Tuesday, September 24, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Arindam Khan – Computer Science and Automation, Indian Institute of Science, Bangalore –

For online optimization, the input instance is revealed in a sequence of steps and, after each step, the algorithm has to take an immediate and irrevocable decision based on the previous inputs. Online algorithms produce a sequence of decisions for such problems without the complete information of the future. In the worst-case analysis of online optimization problems, sometimes, it is impossible to achieve any bounded competitive ratio. An interesting way to bypass these impossibility results is to incorporate a stochastic component into the input model. In the random-order arrival model, the adversary specifies an input instance in advance but the input appears according to a random permutation. The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. In this talk, we will present improved competitive algorithms under random-order arrival for these two problems. This is joint work with Susanne Albers and Leon Ladewig.

Rational Tangles

Series
Undergraduate Seminar
Time
Monday, September 23, 2019 - 15:00 for
Location
Skiles 171
Speaker
Jennifer Hom – Georgia Tech

A knot is a smooth embedding of a circle into R^3. Closely related are tangles, which are properly embedded arcs in a 3-ball. We will model certain tangles using jump ropes and relate this to Conway's classification of rational tangles.

Intrinsic Combinatorics for the Space of Generic Complex Polynomials

Series
Geometry Topology Seminar
Time
Monday, September 23, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Dougherty – Colby College –

The space of degree-n complex polynomials with distinct roots appears frequently and naturally throughout mathematics, but its shape and structure could be better understood. In recent and ongoing joint work with Jon McCammond, we present a deformation retraction of this space onto a simplicial complex with rich structure given by the combinatorics of noncrossing partitions. In this talk, I will describe the deformation retraction and the resulting combinatorial data associated to each generic complex polynomial. We will also discuss a helpful comment from Daan Krammer which connects our work with the ideas of Bill Thurston and his collaborators.

Applied differential geometry and harmonic analysis in deep learning regularization

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 23, 2019 - 13:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wei Zhu – Duke University –

Deep neural networks (DNNs) have revolutionized machine learning by gradually replacing the traditional model-based algorithms with data-driven methods. While DNNs have proved very successful when large training sets are available, they typically have two shortcomings: First, when the training data are scarce, DNNs tend to suffer from overfitting. Second, the generalization ability of overparameterized DNNs still remains a mystery. In this talk, I will discuss two recent works to “inject” the “modeling” flavor back into deep learning to improve the generalization performance and interpretability of the DNN model. This is accomplished by DNN regularization through applied differential geometry and harmonic analysis. In the first part of the talk, I will explain how to improve the regularity of the DNN representation by enforcing a low-dimensionality constraint on the data-feature concatenation manifold. In the second part, I will discuss how to impose scale-equivariance in network representation by conducting joint convolutions across the space and the scaling group. The stability of the equivariant representation to nuisance input deformation is also proved under mild assumptions on the Fourier-Bessel norm of filter expansion coefficients.

The Jacobian Conjecture

Series
Student Algebraic Geometry Seminar
Time
Monday, September 23, 2019 - 13:15 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Stephen McKean – Georgia Tech –

The Jacobian Conjecture is a famous open problem in commutative algebra and algebraic geometry. Suppose you have a polynomial function $f:\mathbb{C}^n\to\mathbb{C}^n$. The Jacobian Conjecture asserts that if the Jacobian of $f$ is a non-zero constant, then $f$ has a polynomial inverse. Because the conjecture is so easy to state, there have been many claimed proofs that turned out to be false. We will discuss some of these incorrect proofs, as well as several correct theorems relating to the Jacobian Conjecture.

An Introduction to Braids and Complex Polynomials

Series
Geometry Topology Seminar Pre-talk
Time
Monday, September 23, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Dougherty – Colby College –

In this informal chat, I will introduce the braid group and several equivalent topological perspectives from which to view it. In particular, we will discuss the role that complex polynomials play in this setting, along with a few classical results.

Bounds on Ramsey Games via Alterations

Series
ACO Student Seminar
Time
Friday, September 20, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
He Guo – Math, Georgia Tech –

In this talk we introduce a refined alteration approach for constructing $H$-free graphs: we show that removing all edges in $H$-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdös and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers (each time extending recent results of Fox–He–Wigderson and Conlon–Fox–Grinshpun–He).
Based on joint work with Lutz Warnke.

Deep Generative Models in the Diffusion Limit

Series
Stochastics Seminar
Time
Thursday, September 19, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Maxim Raginsky – ECE Department, University of Illinois at Urbana-Champaign

In deep generative models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. In this talk, based on joint work with Belinda Tzen, I will discuss the diffusion limit of such models, where we increase the number of layers while sending the step size and the noise variance to zero. The resulting object is described by a stochastic differential equation in the sense of Ito. I will first show that sampling in such generative models can be phrased as a stochastic control problem (revisiting the classic results of Föllmer and Dai Pra) and then build on this formulation to quantify the expressive power of these models. Specifically, I will prove that one can efficiently sample from a wide class of terminal target distributions by choosing the drift of the latent diffusion from the class of multilayer feedforward neural nets, with the accuracy of sampling measured by the Kullback-Leibler divergence to the target distribution.

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