In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.

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{\â??e}rigot and Boris Thibert.

In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds.

It is well-known that the asymptotic complexity of matrix-matrix product and matrix inversion is given by the rank of a 3-tensor, recently shown to be at most O(n^2.3728639) by Le Gall. This approach is attractive as a rank decomposition of that 3-tensor gives an explicit algorithm that is guaranteed to be fastest possible and its tensor nuclear norm quantifies the optimal numerical stability. There is also an alternative approach due to Cohn and Umans that relies on embedding matrices into group algebras. We will see that the tensor decomposition and group algebra approaches, when combined, allow one to systematically discover fast(est) algorithms. We will determine the exact (as opposed to asymptotic) tensor ranks, and correspondingly the fastest algorithms, for products of Circulant, Toeplitz, Hankel, and other structured matrices. This is joint work with Ke Ye (Chicago).

Many problems in science and engineering involve parameters in their mathematical models. Depending on the values of the parameters, the equations can differ greatly in nature. Asymptotic preserving (AP) methods are one type of methods which are designed to work uniformly with respect to different scales or regimes of the equations when the parameters vary.

In this talk, I will present our work in developing high order AP methods for some kinetic models, including discrete-velocity models in a diffusive scaling and the BGK model in a hyperbolic scaling. When the Knudson number approaches zero, the limiting equations of the former model can be heat equation, viscous BurgersÂ? equation, or porous medium equation, while the limiting equations for the latter are the compressible Euler equations. When the Knudson number is very small, the BGK model also leads to compressible Navier-Stokes equations. The proposed methods are built upon a micro-macro decomposition of the equations, high order discontinuous Galerkin (DG) spatial discretizations, and the globally stiffly accurate implicit-explicit Runge-Kutta (IMEX-RK) temporal discretizations. Theoretical results are partially established for uniform stability, error estimates, and rigorous asymptotic analysis. Numerical experiments will further demonstrate the performance of the methods.

The heart is a coupled electro-fluid-mechanical system. The contractions of the cardiac muscle are stimulated and coordinated by the electrophysiology of the heart; these contractions in turn affect the electrical function of the heart by altering the macroscopic conductivity of the tissue and by influencing stretch-activated transmembrane ion channels. In this talk, I will present mathematical models and adaptive numerical methods for describing cardiac mechanics, fluid dynamics, and electrophysiology, as well as applications of these models and methods to cardiac fluid-structure and electro-mechanical interaction. I will also describe novel models of cardiac electrophysiology that go beyond traditional macroscopic (tissue-scale) descriptions of cardiac electrical impulse propagation by explicitly incorporating details of the cellular microstructure into the model equations. Standard models of cardiac electrophysiology, such as the monodomain or bidomain equations, account for this cellular microstructure in only a homogenized or averaged sense, and we have demonstrated that such homogenized models yield incorrect results in certain pathophysiological parameter regimes. To obtain accurate model predictions in these parameter regimes without resorting to a fully cellular model, we have developed an adaptive multiscale model of cardiac conduction that uses detailed cellular models only where needed, while resorting to the more efficient macroscale equations where those equations suffice. Applications of these methods will be presented to simulating cardiac and cardiovascular dynamics in whole heart models, as well as in detailed models of cardiac valves and novel models of aortic dissection. Necessary physiological details will be introduced as needed.

In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.

Hilbert space frames have traditionally been used in signal/image processing. Recently, there have arisen a variety of new applications to speeding up the internet, producing cell phones which won't fade, quantum information theory, distributed processing and more. We will review the fundamentals of frame theory and then look at the myriad of applications of frames.

The talk is concerned with a matrix-valued Boltzmann equation derived from the Fermi-Hubbard or Bose-Hubbard model for weak interactions. The quantum analogue of the classical distribution function is the Wigner function, which is matrix-valued to accommodate the spin degree of freedom. Conservation laws and the H-theorem can be proven analytically, and numerical simulations illustrate the time dynamics.

Recently there has been extensive development of numerical methods for fluid flow interacting with moving boundaries or interfaces, using regular finite difference grids which do not conform to the boundaries. Simulations at low Reynolds number have demonstrated that, with certain choices in the design of the method, the velocity can be accurate to about O(h^2) while discretizing near the interface with truncation error as large as O(h). We will describe error estimates which verify that such accuracy can be achieved in a simple prototype problem, even near the interface, using corrections to difference operators as in the immersed interface method. We neglect errors in the interface location and derive uniform estimates for the fluid velocity and pressure. We will first discuss maximum norm estimates for finite difference versions of the Poisson equation and diffusion equation with a gain of regularity. We will then describe the application to the Navier-Stokes equations.

The forces generated by moving interfaces usually include the parts due to tangential stretching. We derive the evolution equation for the tangential stretching, which then forms the basis for an Eulerian formulation based on level set functions. The jump conditions are then derived using the level set and stretch functions. The derived jump conditions under this Eulerian formulation are clean. This makes possible a local level set method for immersed interface method to simulate membranes or vesicles where the tangential forces are present. This is a continuation of one piece of my work during my Ph.D. study.

Regularity of solutions in the optimal transport problem requires very rigid hypotheses (e.g., convexity of certain sets). When such conditions are not available, one can consider the question of partial regularity, in other words, the in-depth analysis of the structure of singular sets. In this talk, I will discuss the regularity of the set of ``free singularities`` which arise in an optimal transport problem with inner product cost, from a connected set to a disconnected set, along with the stability of such sets under suitable perturbations of the data involved. Some of these results are proven via a non-smooth implicit function theorem for convex functions, which is of independent interest. This talk is based on joint work with Robert McCann.

A neuron receives thousands of synaptic inputs from other neurons and integrates them to process information. Many experimental results demonstrate this integration could be highly nonlinear, yet few theoretical analyses have been performed to obtain a precise quantitative characterization. Based on asymptotic analysis of an idealized cable model, we derive a bilinear spatiotemporal integration rule for a pair of time-dependent synaptic inputs. Note that the above rule is obtained from idealized models. However, we have confirmed this rule both in simulations of a realistic pyramidal neuron model and in electrophysiological experiments of rat hippocampal CA1 neurons. Our results demonstrate that the integration of multiple synaptic inputs can be decomposed into the sum of all possible pairwise integration with each paired integration obeying a bilinear rule.

Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency of data-based prediction with the accuracy and insights provided by the physical models. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, need extra care as the errors tend to accumulate, and the results may quickly become unreliable. We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). We consider the Poisson bracket structure primary and require it to be satisfied exactly, whereas the Hamiltonian, only known from physics, can be satisfied approximately. By design, the method preserves all special integrals of the bracket (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body or satellite motion (the case of SO(3) group), Kirchhoff's equations for an underwater vehicle (SE(3) group), and others. Joint work with Chris Eldred (Sandia National Lab), Francois Gay-Balmaz (CNRS and ENS, France), and Sophia Huraka (U Alberta). The work was partially supported by an NSERC Discovery grant.

This talk describes the the gradient flow nature of dissipative fluid interface problems. Intuitively, the gradient of a functional is given by the direction of ``steepest descent''. This notion, however, depends on the geometry assigned to the underlying function space. The task is therefore to find a metric appropriate for the given dynamics.

For the problem of surface tension driven Hele-shaw flow, the correct metric turns out to have a remarkable connection to an optimal transport problem. This connection points the way to a diffuse interface description of Hele-Shaw flow, given by a degenerate Cahn-Hilliard equation. Some computational examples of this model will be given. The problem of viscous sintering, the Stokes flow counterpart to the Hele-Shaw problem, will also be discussed.

Adaptive methods for solving systems of partial differential equations have become widespread. Robust adaptive software for solving parabolic systems in one and two space dimensions is now widely available. Three spatial adaptive strategies and combinations thereof are frequently employed: mesh refinement (h-refinement); mesh motion (r-refinement); and order variation (p-refinement). These adaptive strategies are driven by a priori and a posteriori error estimates. I will present an adaptive h-refinement finite element code in three dimensions on structured grids. These structured grids contain irregular nodes. Solution values at these nodes are determined by continuity requirements across element boundaries rather than by the differential equations. The differential-algebraic system resulting from the spatial discretization is integrated using Linda Petzold's multistep DAE code DASPK. The large linear systems resulting from Newton's method applied to nonlinear system of differential algebraic equations is solved using preconditioned GMRES. In DASPK the matrix-vector products needed by GMRES are approximated by a ``directional derivative''. Thus, the Jacobian matrix need not be assembled. However, this approach is inefficient. I have modified DASPK to compute the matrix-vector product using stored Jacobian matrix. As in the earlier version of DASPK, DASSL, this matrix is kept for several time steps before being updated. I will discuss appropriate preconditioning strategies, including fast-banded preconditioners. In three dimensions when using multistep methods for time integration it is crucial to use a ``warm restart'', that is, to restart the dae solver at the current time step and order. This requires interpolation of the history information. The interpolation must be done in such a way that mode irregularity is enforced on the new grid. A posteriori error estimates on uniform grids can easily be generalized from two-dimensional results (Babuska and Yu showed that in the case of odd order elements, jumps across elemental boundaries give accurate estimates, and in the case of even order elements, local parabolic systems must be solved to obtain accurate estimates). Babuska's work can even be generalized to meshes with irregular modes but now they no longer converge to the true error (in the case of odd order elements). I have developed a new set of estimates that extend the work of Babuska to irregular meshes and finite difference methods. These estimates provide a posteriori error indicators in the finite element context. Several examples that demonstrate the effectiveness of the code will be given.

We discuss a new class of cubic interpolating and approximating "L₁ splines" that preserve the shape both of smooth data and of data with abrupt changes in magnitude or spacing. The coefficients of these splines are calculated by minimizing the L₁ norm of the second derivatives. These splines do not require constraints, penalties, a posteriori filtering or interaction with the user. Univariate and multivariate cases are treated in one and the same framework. L₁ splines are implemented using efficient interior-point methods for linear programs.

In spite of the fact that symmetries play one of the major roles in physics, the ir usage in finance is relatively new and, to the best of our knowledge, can be traced to 1995 when Kholodnyi introduced the beliefs-preferences gauge symmetry. In this talk we present another symmetry, foreign exchange option symmetry, int roduced by Kholodnyi and Price in 1996. Foreign exchange option symmetry associa tes financially equivalent options on opposite sides of the foreign exchange mar ket. In a two-currency market, the foreign exchange option symmetry is formalized in terms of the one-dimensional Kelvin transform. In a multiple-currency market the foreign exchange option symmetry is formalized in terms of differential geometr y on graphs, that is, in terms of vector lattice bundles on graphs and connectio ns on these bundles. Foreign exchange option symmetry requires no assumptions on the nature of a prob ability distribution for exchange rates. In fact, it does not even require the a ssumptions of the existence of such a distribution. Furthermore, the symmetry is applicable not only to a foreign exchange market but to any financial market as well. The practical applications of the foreign exchange option symmetry range from th e detection of a new type of true arbitrage to the detection of inconsistent mod els of foreign exchange option markets and the development of algorithms and sof tware to value and analyze portfolios of foreign exchange options.

In joint work, Suncica Canic, Eun Heui Kim and I have recently proved the existence of a local solution to the regular reflection problem in the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. There are two kinds of regular reflection, weak and strong, which are distinguished by whether the state immediately behind the reflected shock is subsonic (strong) or supersonic and constant, becoming subsonic further downstream (weak). In the more complicated case of weak regular reflection, the equation, in self-similar coordinates, is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation.

We combine techniques which have been developed for solving degenerate elliptic equations arising in self-similar reductions of hyperbolic conservation laws with an approach to solving free boundary problems of the type that arise from Rankine-Hugoniot relations. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection.

Ensembles of redistricting plans can be challenging to analyze and visualize because every plan is an unordered set of shapes, and therefore non-Euclidean in at least two ways. I will describe two methods designed to address this challenge: barycenters for partitioned datasets, and a novel dimension reduction technique based on Gromov-Wasserstein distance. I will cover some of the theory behind these methods and show how they can help us untangle redistricting ensembles to find underlying trends. This is joint work with Ranthony A. Clark and Tom Needham.