Talk given at SEPC 2021 in honor of Elizabeth Meckes

SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">at this link (here).</a>

SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">here</a>

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Description of some work with Elizabeth Meckes at SEPC 2021

Talk given at SEPC 2021 in honor of Elizabeth Meckes

CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartan’s generalization of Klein’s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.

The problem of recovering a sparse Fourier signal from samples comes up in signal processing, imaging, NMR spectroscopy, and machine learning. Two major challenges involve dealing with off-grid frequencies, and dealing with signals lacking separation between frequencies. Without a minimum separation condition, the problem of frequency recovery is exponentially ill-conditioned, but the signal can still be efficiently recovered in an "improper" manner using an appropriate filter. I will explain such an algorithm for sparse Fourier recovery, and the theory behind why it works - involving some clever analytic inequalities for Fourier-sparse signals. Finally, I will discuss recent work with Xue Chen on applying these ideas to system identification. Identification of a linear dynamical system from partial observations is a fundamental problem in control theory. A natural question is how to do so with statistical rates depending on the inherent dimensionality (or order) of the system, akin to the sparsity of a signal. We solve this question by casting system identification as a "multi-scale" sparse Fourier recovery problem.

The cellular cytoskeleton is made up of protein polymers (filaments) that are essential in proper cell and neuronal function as well as in development. These filaments represent the roads along which most protein transport occurs inside cells. I will discuss several examples where questions about filament-cargo interactions require the development of novel mathematical modeling, analysis, and simulation. Protein cargoes such as neurofilaments and RNA molecules bind to and unbind from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since these transport models can be analytically intractable, we have proposed asymptotic methods in the framework of partial differential equations and stochastic processes which are useful in understanding large-time transport properties. I will discuss a recent project where we use stochastic modeling to understand how filament orientations may influence sorting of cargo in dendrites during neural development and axonal injury.

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al there were two nontrivial stationary distributions.

This talk consists of two aspects about solving the radiative transport through the integral formulation. The radiative transport equation has been numerically studied for many years, the equation is difficult to solve due to its high dimensionality and its hyperbolic nature, in recent decades, the computers are equipped with larger memories so it is possible to deal with the full-discretization in phase space, however, the numerical efficiency is quite limited because of many issues, such as iterative scheme, preconditioning, discretization, etc. In this talk, we first discuss about the special case of isotropic scattering and its integral formulation, then walk through the corresponding fast algorithm for it. In the second part, we try to trivially extend the method to anisotropic case, and talk about the method’s limitation and some perspectives in both theory and numerics.

A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology, which says in its simplest form that two species competing for one resource cannot coexist, and show how the random switching can facilitate coexistence.

I will give an informal overview of the history and status of my field. Local invariants of Riemannian metrics are called curvature, the weakest of which is known as "scalar curvature." The study of metrics with positive scalar curvature is very rich with >100 year old connections to General Relativity and smooth topology. Does this geometric condition have topological implications? The answer turns out to be "yes," but mathematicians continue to search for the true heart of the positive scalar curvature conditions.

Many problems that arise in mathematics, science, and engineering can be formulated as solving a parameterized system of polynomial equations which must be solved for given instances of the parameters. One way to solve these systems is to use a common technique within numerical algebraic geometry called homotopy continuation. My talk will start with background on homotopy continuation and parametrized polynomial systems, followed by applications to problems in computer vision and kinematics. Of these, I will first present a new approach which uses locally adaptive methods and sparse matrix calculations to solve parameterized overdetermined systems in projective space. Examples will be provided in 2D image reconstruction to compare the new methods with traditional approaches in numerical algebraic geometry. Second, I will discuss a new definition of monodromy action over the real numbers which encodes tiered characteristics regarding real solutions. Examples will be given to show the benefits of this definition over a naive extension of the monodromy group (over the complex numbers). In addition, an application in kinematics will be discussed to highlight the computational method and impact on calibration.

Quasi-Stationary Distributions (QSDs) describe the long-time behaviour of killed Markov processes. The Fleming-Viot particle system provides a particle representation for the QSD of a Markov process killed upon contact with the boundary of its domain. Whereas previous work has dealt with killed Markov processes, we consider killed McKean-Vlasov processes. We show that the Fleming-Viot particle system with McKean-Vlasov dynamics provides a particle representation for the corresponding QSDs. Joint work with James Nolen.

The optimal transport (Monge-Kantorovich) problem is a variational problem involving transportation of mass subject to minimizing some kind of energy, and it arises in connection with many parts of math, both pure and applied. In this talk, I will discuss a numerical algorithm to approximate solutions in the semi-discrete case. We propose a damped Newton algorithm which exploits the structure of the associated dual problem, and using geometric implications of the regularity theory of Monge-Amp{\`e}re equations, we are able to rigorously prove global linear convergence and local superlinear convergence of the algorithm. This talk is based on joint work with Quentin M{\’e}rigot and Boris Thibert.

Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.