Seminars and Colloquia by Series

Quantitative finiteness of hyperplanes in hybrid manifolds

Series
CDSNS Colloquium
Time
Friday, September 6, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Anthony Sanchez – University of California - San Diego –

The geometry of non-arithmetic hyperbolic manifolds is mysterious in spite of how plentiful they are. McMullen and Reid independently conjectured that such manifolds have only finitely many totally geodesic hyperplanes and their conjecture was recently settled by Bader-Fisher-Miller-Stover in dimensions larger than 3. Their works rely on superrigidity theorems and are not constructive. In this talk, we strengthen their result by proving a quantitative finiteness theorem for non-arithmetic hyperbolic manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro. Perhaps surprisingly, the proof relies on an effective density theorem for certain periodic orbits. The effective density theorem uses a number of ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures that are nearly full dimensional. This is joint work with K. W. Ohm.

Sparse equidistribution in unipotent flows

Series
CDSNS Colloquium
Time
Friday, August 30, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 314
Speaker
Asaf Katz – Georgia Tech –

Equidistribution problems, originating from the classical works of Kronecker, Hardy and Weyl about equidistribution of sequences mod 1, are of major interest in modern number theory. 

We will discuss how some of those problems relate to unipotent flows and present a conjecture by Margulis, Sarnak and Shah regarding an analogue of these results for the case of the horocyclic flow over a Riemann surface. Moreover, we provide evidence towards this conjecture by bounding from above the Hausdorff dimension of the set of points which do not equidistribute.

The talk will be accessible, no prior knowledge is assumed.

Dynamic Stability in Stochastic Gradient Descent

Series
CDSNS Colloquium
Time
Friday, May 24, 2024 - 15:30 for
Location
Skiles 254
Speaker
Dennis Chemnitz – FU Berlin –

Please Note: Streaming via Zoom: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Most modern machine learning applications are based on overparameterized neural networks trained by variants of stochastic gradient descent. To explain the performance of these networks from a theoretical perspective (in particular the so-called "implicit bias"), it is necessary to understand the random dynamics of the optimization algorithms. Mathematically this amounts to the study of random dynamical systems with manifolds of equilibria. In this talk, I will give a brief introduction to machine learning theory and explain how almost-sure Lyapunov exponents and moment Lyapunov exponents can be used to characterize the set of possible limit points for stochastic gradient descent.

Validated enclosures of Fourier coefficients in Banach spaces of analytic functions

Series
CDSNS Colloquium
Time
Friday, May 10, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Jean-Philippe Lessard – McGill University –

Please Note: Streaming available via Zoom: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

This presentation introduces a methodology for generating computer-assisted proofs (CAPs) aimed at establishing the existence of solutions for nonlinear differential equations featuring non-polynomial analytic nonlinearities. Our approach combines the Fast Fourier Transform (FFT) algorithm with interval arithmetic and a Newton-Kantorovich argument to effectively construct CAPs. A key highlight is the rigorous management of Fourier coefficients of the nonlinear term Fourier series, achieved through insights from complex analysis and the Discrete Poisson Summation Formula. We demonstrate the effectiveness of our method through two illustrative examples: firstly, proving the existence of periodic orbits in the Mackey-Glass (delay) equation, and secondly, establishing the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation.

This is joint work with Jan Bouwe van den Berg (VU Amsterdam, The Netherlands), Maxime Breden (École Polytechnique, France) and Jason D. Mireles James (Florida Atlantic University, USA)

Thermodynamic formalism for and hitting time statistics for random open dynamical systems

Series
CDSNS Colloquium
Time
Wednesday, May 8, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jason Atnip – Queensland University –

Please Note: In this talk we present some recent results on thermodynamic formalism for random open dynamical systems. In particular, we poke random holes in the phase space and prove the existence of unique equilibrium states on the set of surviving points as well as find the rate at which mass escapes through these holes. If we consider small holes, through a perturbative approach, we are able to make a connection to extreme value theory and hitting time statistics. Furthermore, we prove a Gumbel's law and show that the distribution of multiple returns to small holes is asymptotically compound Poisson distributed.

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