Unless stated otherwise, colloquia are scheduled for Thursdays 4:15-5:15pm in WH-100E with refreshments served from 4:00-4:15 pm in WH-102.
Thursday November 3, 4:15-5:15pm, WH-100E
Speaker: Avy Soffer (Rutgers University)
Topic: The Asymptotic States of Nonlinear Dispersive Equations with Large Initial Data and General Interactions
I will describe a new approach to scattering theory, which allows
the analysis of interaction terms which are linear and space-time dependent, and nonlinear terms as well.
This is based on deriving (exterior) propagation estimates for such equations, which micro-localize the asymptotic states
as time goes to infinity.
In particular, the free part of the solution concentrates on the propagation set (x=vt), and the localized leftover is characterized in the phase-space as well.
The NLS with radial data in three dimensions is considered, and it is shown that besides the free asymptotic wave, in general, the localized part is smooth, and is localized in the region where |x|^2 is less than t.
Furthermore, the localized part has a massive core and possibly a halo which may be a self-similar solution.
This work is joint with Baoping Liu.
This is then followed by new results on the non-radial case and Klein-Gordon equations (Joint works with Xiaoxu Wu).
Thursday November 17, 2:50-3:50pm, WH-100E
Speaker: Emmett Wyman (University of Rochester)
Topic: Improved Weyl law remainders for products of spheres
Abstract: The Laplacian is a fundamental operator in mathematics. It arises, for example, in the heat, wave, and Schrödinger equations. Its eigenfunctions can be viewed as natural vibrational modes and their eigenvalues their respective frequencies.
A fundamental problem in harmonic analysis is to estimate the number $N(\lambda)$ of Laplace eigenvalues less than or equal to $\lambda$, counting multiplicity. The Weyl law gives a main term + remainder term estimate for $N(\lambda)$. The remainder is sharp in general but generically can be improved. It is hard to obtain polynomially improved remainders–let alone optimal remainders–except for a handful of very nice examples.
In this talk, we will discuss the remainder of the Weyl law for two
classic cases: the sphere and the torus. We will then compare these
results through the lense of the dynamics of the geodesic flow via the
Duistermaat-Guillemin theorem. Finally, I will present recent joint work
with Iosevich which explores the prospect of obtaining polynomially
improved Weyl remainders for products of manifolds, which we illustrate by
obtaining polynomially improved remainders for products of spheres.
Tuesday November 22, 4:15–5:15pm, WH-100E
Speaker: Alexander Dunlap (NYU Courant)
Topic: Stochastic partial differential equations in supercritical, subcritical, and critical dimensions
A pervading question in the study of stochastic PDE is how
small-scale random forcing in an equation combines to create nontrivial
statistical behavior on large spatial and temporal scales. I will discuss
recent progress on this topic for several related stochastic PDEs -
stochastic heat, KPZ, and Burgers equations - and some of their
generalizations. These equations are (conjecturally) universal models of
physical processes such as a polymer in a random environment, the growth
of a random interface, branching Brownian motion, and the voter model. The
large-scale behavior of solutions on large scales is complex, and in
particular, depends qualitatively on the dimension of the space. I will
describe the phenomenology, and then describe several results and
challenging problems on invariant measures, growth exponents, and limiting
Wednesday November 30, 4:30–5:30pm, WH-100E
Speaker: Anibal Medina-Mardones (Universite Sorbonne Paris Nord)
Topic: Effective Algebro-Homotopical constructions and their applications
It is necessary in order to incorporate ideas from homotopy theory
into concrete contexts – such as topological data analysis and topological lattice
field theory – to have effective constructions of concepts defined only indirectly
or transcendentally. In groundbreaking work, Mandell showed that the entire
homotopy type of a space was encoded in the quasi-isomorphism type of its
cochains enhanced with an $E_\infty$-structure. In this talk, we will present a concrete
construction of such structure by explicitly restoring up to coherent homotopies
the broken symmetry of the diagonal of cellular spaces, and, on the way, we
will explore connections of these ideas to data science, theoretical physics, knot
theory, higher category theory, and convex and toric geometry.
Friday December 2, 4:30–5:30pm, WH-100E
Speaker: Beibei Liu (MIT)
Topic: The critical exponent: old and new
The critical exponent is an important numerical invariant of
discrete isometry groups acting on negatively curved Hadamard manifolds,
Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I
will focus on discrete isometry groups acting on hyperbolic spaces. In
particular, I will explain how the numerical invariant is closely related to
geometry, dynamics, and representation of the group action on hyperbolic
Monday December 5, 4:30–5:30pm, WH-100E
Speaker: Jia Zhao (Utah State University)
Topic: Physics-informed Computational Modeling of Multiphase Complex Fluids with Applications in Life Science
Complex fluids are ubiquitous in nature and in synthesized
materials, such as biofilms, cytoplasm, mucus, synthetic and biological
polymeric solutions. Modeling and simulation of complex fluids have been
listed as one of the 21st-century mathematical challenges by DARPA, which is
therefore of great mathematical and scientific significance. In this talk, I
will first explain our research motivations by introducing several complex
fluids examples and traditional modeling techniques. We propose physics-
informed PDE models for multiphase complex fluid flows by integrating the
generalized Onsager Principle and phase-field approaches. Then, I will
introduce a numerical analysis platform for developing accurate, efficient,
and structure-preserving numerical approximations for solving complex-fluid
PDE models. The computational modeling strategy is rather general in that it
can be applied to investigate a host of complex-fluid problems. Finally, I
will present several applications in life science with our modeling and
numerical analysis toolkits.
Wednesday December 7, 4:30–5:30pm, WH-100E
Speaker: Minghao W. Rostami (Syracuse University)
Topic: Learning from data: a beginner’s journey
The behavior of a complex physical system is often modeled by Partial
Differential Equations (PDEs) derived from fundamental physical laws. We
can apply various numerical methods to the PDEs to make predictions about
the physical system, uncover the causal factors behind observed behaviors
of the system, and optimize the design of the system for a task at hand.
When sufficient data and computing resources are available, we can also
make discoveries by processing data instead. The speaker, as a classically
trained applied mathematician who is “learning the ropes” of this
alternative approach, will review and discuss the basics of PDE models,
numerical methods, and machine learning models. A new data-driven model
for calculating particle trajectories in a fluid flow will also be
presented, which, unlike conventional methods, does not entail costly flow
computations. This is joint work with Jianchen Wei (PhD student), Lixin
Shen from Syracuse University and Melissa Green from University of
Minnesota, Twin Cities.