Seminars and Colloquia Schedule

Degeneracy of eigenvalues and singular values of parameter dependent matrices

Series
Applied and Computational Mathematics Seminar
Time
Monday, May 6, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/93530218689?pwd=SFkzMXZyZXhZOTdRazhyL1BoVXprdz09
Speaker
Alessandro Pugliese – Università degli Studi di Bari Aldo Moro –

Speaker will present in person.

Hermitian matrices have real eigenvalues and an orthonormal set of eigenvectors. Do smooth Hermitian matrix valued functions have smooth eigenvalues and eigenvectors? Starting from such question, we will first review known results on the smooth eigenvalue and singular values decompositions of matrices that depend on one or several parameters, and then focus on our contribution, which has been that of devising topological tools to detect and approximate parameters' values where eigenvalues or singular values of a matrix valued function are degenerate (i.e. repeated or zero).

The talk will be based on joint work with Luca Dieci (Georgia Tech) and Alessandra Papini (Univ. of Florence).

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 –

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.

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 –

Streaming available via Zoom: <br />
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT0... />

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)