Seminars and Colloquia by Series

Collaborations with the digital computer: Simulation, Conjecture, and Proof in Dynamics

Series
Job Candidate Talk
Time
Thursday, May 1, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 or 006
Speaker
Jason Mireles-JamesFlorida Atlantic University

Computational methods have long inspired conjecture and counterexamples in mathematics, and  in recent years they appear more frequently in proofs of interesting mathematical theorems like the four color problem, Kepler's optimal sphere packing problem, and the proof of the Feigenbaum conjectures in nonlinear dynamics.  In this talk I'll discuss recent work of R. Calleja, C. Garcia, O. Henot, J.P. Lessard and myself on choreographic solutions of the gravitational N-body problem.  After reviewing some history and motivation, I'll explain the role of the digital computer in the (quite constructive) proofs of the theorems.


https://gatech.zoom.us/my/rkuske7?pwd=aHlLUFBFc2JndXlTelV1d3NlOEJBdz09

Recent progress on completely integrable systems

Series
Job Candidate Talk
Time
Tuesday, February 18, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Monica VisanUCLA

 We will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE.  These include a priori bounds, orbital stability of multisolitons, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.

 

 

From Optics to the Deift Conjecture

Series
Job Candidate Talk
Time
Monday, February 17, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rowan KillipUCLA

After providing a mathematical background for some curious optical experiments in the 19th century, I will then describe how these ideas inform our understanding of the Deift conjecture for the Korteweg--de Vries equation. Specifically, in joint work with Chapouto and Visan, we showed that the evolution of almost-periodic initial data need not remain almost periodic.

 

Extreme value theory for random walks in space-time random media

Series
Job Candidate Talk
Time
Wednesday, January 29, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shalin ParekhUniversity of Maryland

 

The KPZ equation is a singular stochastic PDE arising as a scaling limit of various physically and probabilistically interesting models. Often, this equation describes the “crossover” between Gaussian and non-Gaussian fluctuation behavior in simple models of interacting particles, directed polymers, or interface growth. It is a difficult and elusive open problem to elucidate the nature of this crossover for general stochastic interface models. In this talk, I will discuss a series of recent works where we have made progress in understanding the KPZ crossover for models of random walks in dynamical random media. This was done through a tilting-based approach to study the extreme tails of the quenched probability distribution. This talk includes joint work with Sayan Das and Hindy Drillick.

Zoom link:

https://gatech.zoom.us/j/96535844666

Recent progress on the horocycle flow on strata of translation surfaces - NEW DATE

Series
Job Candidate Talk
Time
Tuesday, January 28, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jon ChaikaUniversity of Utah

For about 2 decades the horocycle flow on strata of translation surfaces was studied, very successfully, in analogy with unipotent flows on homogeneous spaces, which by work of Ratner, Margulis, Dani and many others, have striking rigidity properties. In the past decade Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi proved some analogous rigidity results for SL(2,R) and the full upper triangular subgroup on strata of translation surfaces. This talk will begin by introducing ergodic theory and translation surfaces. Then it will describe some of the previously mentioned rigidity theorems before moving on to its goal, that many such rigidity results fail for the horocycle flow on strata of translation surfaces. Time permitting we will also describe a rigidity result for special sub-objects in strata of translation surfaces. This will include joint work with Osama Khalil, John Smillie, Barak Weiss and Florent Ygouf. 

 

https://gatech.zoom.us/j/95951300274?pwd=dZE89RkP2k6Ri4xbgJP3cSucsi9xna.1

Meeting ID: 959 5130 0274
Passcode: 412458

Expansion and torsion homology of 3-manifolds

Series
Job Candidate Talk
Time
Thursday, January 23, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan ZungMIT

We say that a Riemannian manifold has good higher expansion if every rationally null-homologous i-cycle bounds an i+1 chain of comparatively small volume. The interactions between expansion, spectral geometry, and topology have long been studied in the settings of graphs and surfaces. In this talk, I will explain how to construct rational homology 3-spheres which are good higher expanders. On the other hand, I will show that such higher expanders must be rather topologically complicated; in particular, we will demonstrate a super-polynomial-in-volume lower bound on their torsion homology.

Leveraging algebraic structures for innovations in data science and complex systems

Series
Job Candidate Talk
Time
Thursday, January 16, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Julia LindbergUniversity of Texas, Austin

Applied algebraic geometry is a subfield of applied mathematics that utilizes concepts, tools, and techniques from algebraic geometry to solve problems in various applied sciences. It blends tools from algebraic geometry, optimization, and statistics to develop certifiable computational algebraic methods to address modern engineeering challenges.

In this talk, I will showcase the power of these methods in solving problems related to Gaussian mixture models (GMMs). In the first part of the talk I will discuss a statistical technique for parameter recovery called the method of moments. I will discuss how to leverage algebraic techniques to design scalable and certifiable moment-based methods for parameter recovery of GMMs. In the second part of this talk, I will discuss recent work relating to Gaussian Voronoi cells. This work introduces new geometric perspectives with implications for high-dimensional data analysis. I will also touch on how these methods complement my broader research in polynomial optimization and power systems engineering.

https://gatech.zoom.us/j/97398944571?pwd=s8S02kNZd5dyVvSY8mZzNOfbNZrqfg.1

Constructing finite time singularities: Non-radial implosion for compressible Euler, Navier-Stokes and defocusing NLS in T^d and R^d

Series
Job Candidate Talk
Time
Tuesday, December 3, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jia ShiMIT

The compressible Euler and Navier-Stokes equations describe the motion of compressible fluids. The defocusing nonlinear Schr\"odinger equation is a dispersive equation that has application in many physics areas. Through the Madelung transformation, the defocusing nonlinear Schr\"odinger equation is connected with the compressible Euler equation. In this colloquium I will start from the compressible Euler/Navier-Stokes equation and introduce the blow-up result called implosion. Then I will introduce the defocusing nonlinear Schr\"odinger equation and the longstanding open problem on the blow-up of its solutions in the energy supercritical regime. In the end I will talk about the Madelung transformation and its application to transfer the implosion from the compressible Euler to the defocusing nonlinear Schr\"odinger equation. During the talk I will mention our work with Gonzalo Cao-Labora, Javier Gómez-Serrano and Gigliola Staffilani on the first non-radial implosion result for those three equations.

Chaos in polygonal billiards

Series
Job Candidate Talk
Time
Monday, December 2, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Francisco Arana-HerreraUniversity of Maryland

We discuss how chaos, i.e., sensitivity to initial conditions, arises in the setting of polygonal billiards. In particular, we give a complete classification of the rational polygons whose billiard flow is weak mixing in almost every direction, proving a longstanding conjecture of Gutkin. This is joint work with Jon Chaika and Giovanni Forni. No previous knowledge on the subject will be assumed.

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