Date and time: Sep 6 at 1:00 pm (Parker 328)

Title: Kuramoto Oscillators: synchronizing fireflies to algebraic geometry

Abstract: When does a system of coupled oscillators synchronize? This central question in dynamical systems arises in applications ranging from power grids to neuroscience to biology: why do fireflies sometimes begin flashing in harmony? Perhaps the most studied model is due to Kuramoto (1975); we  analyze the Kuramoto model from the perspectives of algebra and topology. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks); our work also tackles more general situations.

We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. The talk will include a surprising (at least to us!) connection to Segre varieties, and close with examples of computations using the Macaulay2 software package "Oscillator".

Joint work with Heather Harrington (Oxford/Dresden) and Mike Stillman (Cornell).

Date and time: Sep 20 at 1:00 pm (Parker 328)

Title: Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations.

Abstract: In this work, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we reformulate the original equations into an equivalent system that incorporates the energy evolution process. First- and second-order dynamically regularized Lagrange multiplier (DRLM) schemes are derived based on the backward differentiation formulas and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting the accuracy and stability of the numerical solutions. Various numerical experiments including the Taylor-Green vortex problem, lid-driven cavity flow, and Kelvin-Helmholtz instability are carried out to demonstrate the performance of the DRLM schemes. Extension of the DRLM method to the Cahn-Hilliard-Navier-Stokes system will also be discussed.

Date and time: Sep 27 at 1:00 pm (Parker 328)

Title: Hamiltonian spectral theory via the Maslov index

Abstract: As Arnol’d pointed out, Sturm’s 19th century theorem regarding the oscillation of solutions to a second-order selfadjoint ODE has a topological nature: it describes the rotation of a straight line in the phase space of the equation. The topological ingredient here is the Maslov index, a homotopy invariant which counts the (signed) intersections of a path of Lagrangian planes with a codimension-one subset of the set of all Lagrangian planes. In this talk, I’ll begin with a discussion of Sturm’s theorem, the main idea being that one can glean spectral information from the geometric structure of an eigenfunction. I'll then show how these ideas translate to study the eigenvalues of a class of Hamiltonian differential operators that I studied in my PhD, which do not enjoy the selfadjointness property of the operators that Sturm studied. In particular, by viewing the problem symplectically, one can use the Maslov index to give a lower bound for the number of positive real eigenvalues in terms of the Morse indices of two related selfadjoint operators, as well as a mysterious correction term. The Hamiltonian operators in focus here arise, for example, when determining the (spectral) stability of standing waves in NLS type equations; if time permits, I’ll go through some applications to such problems on both bounded and unbounded domains. Part of this talk was joint work with Graham Cox, Yuri Latushkin and Robby Marangell.

Date and time: Oct 4 at 11:00 am (Parker 328)

Title: Level Set Learning with Pseudo-Reversible Neural Networks for Nonlinear Dimension Reduction in Function Approximation

Abstract: Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

Date and time: Oct 25 at 1:00 pm (Parker 328)

Title: Space-time nonlocal integrable systems

Abstract: In this talk I will review past and recent results pertaining to the emerging topic of integrable space-time nonlocal integrable nonlinear evolution equations. In particular, we will discuss blow-up in finite time for solitons and the physical derivations of many integrable nonlocal systems.

Date and time: Oct 29 at 4:00 pm (Parker 228)

Title: On the Mathematical Perspective of the Missile Defense System

Abstract: We first provide an overview of the systems, weapons, and technology needed for detection, tracking, interception, and destruction of attacking missiles.  Then, we will outline the current research areas sought by the MDA to advance and solve complex technological problems, ultimately contributing to a more robust Missile Defense System (MDS). 

Date and time: Nov 8 at 1:00 pm (Parker 328)

Title: Multiple-scales perspective on moiré materials 

Abstract: In recent years, experiments have shown that twisted bilayer graphene and other so-called ``moiré materials'' realize a variety of important strongly-correlated electronic phases, such as superconductivity and fractional quantum anomalous Hall states. I will present a rigorous multiple-scales analysis justifying the (single-particle) Bistritzer-MacDonald PDE model, which played a critical role in the prediction of these phases in twisted bilayer graphene. The significance of this model is that it has moiré-periodic coefficients even when the underlying material is aperiodic at the atomic scale. This allows moiré materials to be studied via Floquet-Bloch band theory, a variant of the Fourier transform. I will then discuss generalizations of this model and other mathematical questions related to moiré materials.

Date and time: Nov 15 at 1:00 pm (on Zoom)

Title: Comparing the spectrum of Schrodinger operators on metric graphs using heat kernels

Abstract: We study Schrodinger operators on compact finite metric graphs subject to $\delta$-coupling and standard boundary conditions often known as \emph{Kirchoff-Neumann vertex conditions}. We compare the $n$-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of the eigenvalue deviations which represents a generalization to a recent result by Rudnick, Wigman and Yesha obtained for domains in $\mathbb{R}^2$ to the setting of metric graphs. We start this talk by introducing the basic notion of a metric graph and discuss some basic properties of heat kernels on those graphs afterwards. In this way, we are able to discuss a so-called \emph{local Weyl law} which is relevant for the proof of the asymptotic main result. If time permits, we will also briefly discuss the case of $\delta'$-coupling conditions and some possible generalizations on infinite graphs having \emph{finite total length}.

Date and time: Nov 22 at 1:00 pm (Parker 328)

Title: Convergence Analysis of the ADAM Algorithm for Linear Inverse Problems

Abstract: The ADAM algorithm is one of the most popular stochastic optimization methods in machine learning. Its remarkable performance in training models with massive datasets suggests its potential efficiency in solving large-scale inverse problems. In this work, we apply the ADAM algorithm to solve linear inverse problems and establish the sub-exponential convergence rate for the algorithm when the noise is absent. Based on the convergence analysis, we present an a priori stopping criterion for the ADAM iteration when applied to solve inverse problems at the presence of noise. The convergence analysis is achieved via the construction of suitable Lyapunov functions for the algorithm when it is viewed as a dynamical system with respect to the iteration numbers. At each iteration, we establish the error estimates for the iterated solutions by analyzing the constructed Lyapunov functions via stochastic analysis. Various numerical examples are conducted to support the theoretical findings and to compare with the performance of the stochastic gradient descent (SGD) method.