Seminars and Colloquia by Series

Pointwise ergodic theorems along fractional powers of primes. (Note the special location)

Series
Analysis Seminar
Time
Wednesday, January 15, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Van Leer C456
Speaker
Leonidas DaskalakisWroclaw University

 We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences $\lfloor h(p)\rfloor$, where $h$ belongs in a wide class of functions, the so-called $c$-regularly varying functions. A key ingredient of our approach are certain exponential sum estimates, which we also use for establishing a Waring-type result. Assuming that the Riemann zeta function has any zero-free strip upgrades our exponential sum estimates to polynomially saving ones and this makes a conditional result regarding the behavior of our ergodic averages on $L^1$ to not seem entirely out of reach. The talk is based on joint work with Erik Bahnson, Abbas Dohadwala and Ish Shah.
 

Strongly exceptional Legendrian connected sum of two Hopf links

Series
Geometry Topology Seminar
Time
Monday, January 13, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Youlin LiShanghai Jiao Tong University

In this talk, I will present a complete coarse classification of strongly exceptional Legendrian realizations of the connected sum of two Hopf links in contact 3-spheres. This is joint work with Sinem Onaran.

Lorentzian polynomials and the incidence geometry of tropical linear spaces

Series
Algebra Seminar
Time
Monday, January 13, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jidong WangUniversity of Texas at Austin

Please Note: There will be a pre-seminar at 10:55 am in Skiles 006 (not 005).

The theory of stable polynomials features a key notion called proper position, which generalizes interlacing of real-rooted polynomials to higher dimensions. In a recent paper, I introduced a Lorentzian analog of proper position and used it to give a new characterization of elementary quotients of valuated matroids. This connects the local structure of spaces of Lorentzian polynomials with the incidence geometry of tropical linear spaces. A central object in this connection is the moduli space of codimension-1 tropical linear subspaces of a given tropical linear space. In this talk, I will show some new structural results on this moduli space and their implications for Lorentzian polynomials.

POSTPONED - Learning seminar about Margulis' inequalities

Series
CDSNS Colloquium
Time
Friday, January 10, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Asaf KatzGeorgia Tech

Margulis inequalities and Margulis functions (a.k.a Foster-Lyapunov stability) have played a major role in modern dynamics, in particular in the fields of homogeneous dynamics and Teichmuller dynamics.
Moreover recent exciting developments in the field of random walks over manifolds give rise to related notions and questions in a much larger geometrical content, largely motivated by upcoming work of Brown-Eskin-Filip-Rodriguez Hertz.

I will explain what are Margulis functions and Margulis inequalities and describe the main lemma due to Eskin-Margulis (“uniform expansion”) that allows one to prove such an inequality. I will also try to sketch some interesting applications.

No prior knowledge is needed, the talk will be self-contained and accessible.

Dehn twist and smooth mapping class group of 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, December 9, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anubhav MukherjeePrinceton

In this talk, I will present recent advancements in the study of smooth mapping class groups of 4-manifolds. Our work focuses on diffeomorphisms arising from Dehn twists along embedded 3-manifolds and their interaction with Seiberg-Witten theory. These investigations have led to intriguing applications across several areas, including symplectic geometry (related to Torelli symplectomorphisms), algebraic geometry (concerning the monodromy of singularities), and low-dimensional topology (involving exotic diffeomorphisms). This is collaborative work with Hokuto Konno, Jianfeng Lin, and Juan Munoz-Echaniz.

Leveraging low-dimensional structures in structure-preserving machine learning for dynamical systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 9, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/94954654170
Speaker
Qi TangGeorgia Tech CSE

In this talk I will discuss our recent effort to develop structure-preserving machine learning (ML) for time series data, focusing on both dissipative PDEs and singularly perturbed ODEs. The first part presents a data-driven modeling method that accurately captures shocks and chaotic dynamics through a stabilized neural ODE framework. We learn the right-hand-side of an ODE by adding the outputs of two networks together, one learning a linear term and the other a nonlinear term. The architecture is inspired by the inertial manifold theorem. We apply this method to chaotic trajectories of the Kuramoto-Sivashinsky equation, where our model keeps long-term trajectories on the attractor and remains robust to noisy initial conditions. The second part explores structure-preserving ML for singularly perturbed dynamical systems. A powerful tool to address these systems is the Fenichel normal form, which significantly simplifies fast dynamics near slow manifolds. I will discuss a novel realization of this concept using ML. Specifically, a fast-slow neural network (FSNN) is proposed, enforcing the existence of a trainable, attractive invariant slow manifold as a hard constraint. To illustrate the power of FSNN, I will show a fusion-motivated example where traditional numerical integrators all fail.

Absolute continuity of stationary measures-UPDATED DATE

Series
CDSNS Colloquium
Time
Friday, December 6, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Davi ObataBrigham Young University

In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired on Tsujii’s “transversality” method, which he used to show Palis conjecture for partially hyperbolic endomorphisms. This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan.

The Gibbs state of the mean-field Bose gas and a new correlation inequality

Series
Math Physics Seminar
Time
Friday, December 6, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
L2 Classroom Howey Physics
Speaker
Andreas DeuchertVirginia Tech

We consider the mean field Bose gas on the unit torus at temperatures proportional to the critical temperature of the Bose—Einstein condensation phase transition. We discuss trace norm convergence of the Gibbs state to a state given by a convex combination of quasi-free states. Two consequences of this relation are precise asymptotic formulas for the two-point function and the distribution of the number of particles in the condensate. A crucial ingredient of the proof is a novel abstract correlation inequality. This is joint work with Nam Panh Tanh and Marcin Napiorkowski. 

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