Seminars and Colloquia by Series

Improved Bounds for Szemerédi’s Theorem

Series
Additional Talks and Lectures
Time
Monday, April 29, 2024 - 17:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mehtaab SawhneyMIT

We discuss recent improved bounds for Szemerédi’s Theorem. The talk will seek to provide a gentle introduction to higher order Fourier analysis and recent quantitative developments. In particular, the talk will provide a high level sketch for how the inverse theorem for the Gowers norm enters the picture and the starting points for the proof of the inverse theorem. Additionally, the talk (time permitting) will discuss how recent work of Leng on equidistribution of nilsequences enters the picture and is used. No background regarding nilsequences will be assumed. 

Based on joint work with James Leng and Ashwin Sah.

Generative modeling through time reversal and reflection of diffusion processes

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 29, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Nicole YangEmory University

Please Note: Speaker will present in person.

In this talk, we discuss generative modeling algorithms motivated by the time reversal and reflection properties of diffusion processes. Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. We develop SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. Besides the quest for generating images at ever higher resolution, our primary motivation is to create a well-posed infinite-dimensional learning problem so that we can discretize it consistently at multiple resolution levels. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting by ensuring the well-posedness of forward and reverse processes, and derive the convergence of the approximation of multilevel training. We illustrate that approximating the score function with an operator network is beneficial for multilevel training.

In the second part of this talk, we propose the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and demonstrate its scalability in constrained generative modeling.

Constructive proofs of existence in differential equations on R^n

Series
CDSNS Colloquium
Time
Friday, April 26, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Matthieu CadiotMcGill University

Please Note: Zoom link to attend remotely: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

In this talk I will present a computer-assisted method to study solutions vanishing at infinity in differential equations on R^n. Such solutions arise naturally in various models, in the form of traveling waves or localized patterns for instance, and involve multiple challenges to address both on the numerical and on the analytical side. Using spectral techniques, I will explain how Fourier series can serve as an approximation of the solution as well as an efficient mean for the construction of a fixed-point operator for the proof. To illustrate the method, I will present applications to the constructive proof of localized patterns in the 2D Swift-Hohenberg equation and in the Gray-Scott model. The method extends to non-local equations and proofs of solitary travelling waves in the (capillary-gravity) Whitham equation will be exposed.

Computing High-Dimensional Optimal Transport by Flow Neural Networks

Series
GT-MAP Seminar
Time
Friday, April 26, 2024 - 15:00 for 2 hours
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Yao Xie H. Milton Stewart School of Industrial and Systems Engineering at Georgia Tech

Flow-based models are widely used in generative tasks, including normalizing flow, where a neural network transports from a data distribution P to a normal distribution. This work develops a flow-based model that transports from P to an arbitrary Q (which can be pre-determined or induced as the solution to an optimization problem), where both distributions are only accessible via finite samples. We propose to learn the dynamic optimal transport between P and Q by training a flow neural network. The model is trained to optimally find an invertible transport map between P and Q by minimizing the transport cost. The trained optimal transport flow subsequently allows for performing many downstream tasks, including infinitesimal density ratio estimation (DRE) and distribution interpolation in the latent space for generative models. The effectiveness of the proposed model on high-dimensional data is demonstrated by strong empirical performance on high-dimensional DRE, OT baselines, and image-to-image translation.

Max-sliced Wasserstein distances

Series
Stochastics Seminar
Time
Thursday, April 25, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
March BoedihardjoMichigan State University

I will give essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. A version of this result for Banach spaces will also be presented. From this, we will derive an upper bound for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure on a Euclidean space and its symmetrized empirical distribution.

The Giroux correspondence via convex surfaces

Series
Geometry Topology Seminar
Time
Wednesday, April 24, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vera VétesiUniversity of Vienna

The “hard direction” of the Giroux correspondence states that any two open books representing the same contact structure is related by a sequence of positive stabilisations and destabilisations. We give a proof of this statement using convex surface theory. This is a joint work with Joan Licata. 

Twist positivity, Lorenz knots, and concordance

Series
Geometry Topology Seminar
Time
Monday, April 22, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Siddhi KrishnaColumbia

There are lots of ways to measure the complexity of a knot. Some come from knot diagrams, and others come from topological or geometric quantities extracted from some auxiliary space. In this talk, I’ll describe a geometry property, which we call “twist positivity”, that often puts strong restrictions on how the braid and bridge index are related. I’ll describe some old and new results about twist positivity, as well as some new applications towards knot concordance. In particular, I’ll describe how using a suite of numerical knot invariants (including the braid index) in tandem allows one to prove that there are infinitely many positive braid knots which all represent distinct smooth concordance classes. This confirms a prediction of the slice-ribbon conjecture. Everything I’ll discuss is joint work with Hugh Morton. I will assume very little background about knot invariants for this talk – all are welcome!

Computing linear sections of varieties: quantum entanglement, tensor decompositions and beyond

Series
Algebra Seminar
Time
Monday, April 22, 2024 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Benjamin LovitzNortheastern University

Please Note: There will be a pre-seminar at 11am in Skiles 005.

Given a basis for a linear subspace U of nxn matrices, we study the problem of either producing a rank-one matrix in U, or certifying that none exist. While this problem is NP-Hard in the worst case, we present a polynomial time algorithm to solve this problem in the generic setting under mild conditions on the dimension of U. Our algorithm is based on Hilbert’s Nullstellensatz and a “lifted” adaptation of the simultaneous diagonalization algorithm for tensor decompositions. We extend our results to the more general setting in which the set of rank-one matrices is replaced by an algebraic set. Time permitting, we will discuss applications to quantum separability testing and tensor decompositions. This talk is based on joint work with Harm Derksen, Nathaniel Johnston, and Aravindan Vijayaraghavan.

Differential Equations for Continuous-Time Deep Learning

Series
PDE Seminar
Time
Friday, April 19, 2024 - 15:00 for 1 hour (actually 50 minutes)
Location
CSIP Library (Room 5126), 5th floor, Centergy one
Speaker
Dr.Lars RuthottoResearch Associate Professor in the Department of Mathematics and the Department of Computer Science at Emory University

In this talk, we introduce and survey continuous-time deep learning approaches based on neural ordinary differential equations (neural ODEs) arising in supervised learning, generative modeling, and numerical solution of high-dimensional optimal control problems. We will highlight theoretical advantages and numerical benefits of neural ODEs in deep learning and their use to solve otherwise intractable PDE problems.

Branching Brownian motion and the road-field model

Series
Stochastics Seminar
Time
Thursday, April 18, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nick CookDuke University

The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained via a connection with branching Brownian motion, beginning with works of McKean and Bramson in the 70s. I will discuss an extension of this probabilistic approach to the Road-Field Model: a reaction-diffusion PDE system introduced by H. Berestycki et al. to describe enhancement of biological invasions by a line of fast diffusion, such as a river or a road. Based on joint work with Amir Dembo.

 

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