(This abstract is in TeX source code. Sorry!) Fix a small positive number $h$. Let $$G=h\mathbb{Z}^2=\{(ih,jh):i,j\in\mathbb{Z}\};$$ thus $G$ is a rectangular grid of points in $\mathbb{R}^2$. Let $\Omega$ be an bounded open subset of $\mathbb{R}^2$ with $C^1$ boundary and let $E=\{x\in G:x\in\Omega\}$. {\bf Question One.} Given $E$ can one determine the length of $\partial\Omega$ to within $O(h)$? The answer to this question is ``yes'', provided $\Omega$ satisfies a certain natural ``thickness'' condition; without this additional assumption the answer may be ``no''. {\bf Question Two.} Is there a fast algorithm for determining the length of $\partial\Omega$. The answer to this question also ``yes''. In this talk I will describe the proof that the answer to Question One is ``yes'' and I will describe the fast algorithm whose existence is implied in the answer to Question Two. If time permits, I will describe some applications.

In this talk I will introduce two stochastic voting systems and results we proved. (i) Axelrod's model generalizes the voter model in which individuals have one of Q possible opinions about each of F issues and neighbors interact at a rate proportional to the fraction of opinions they share. We proved that on large two-dimensional torus if Q/F is small, then there is a giant component of individuals who share at least one opinion and consensus develops on this percolating cluster. (ii) The latent voter model allows a latent period after each site flips its opinion. We will present Shirshendu's result on a random r-regular graph with n vertices that as the rate of exponential latent period $\lambda \gg \log n$, dynamics converge to coexistence behavior with quasi-stationary density = 1/2 at $O(\lambda)$ times. Using different technologies one can generalize it to the varying degree case, a.k.a. the configuration models. Joint work with Rick Durrett and Shirshendu Chatterjee

Motivated by several biological questions, we consider ODEs and PDEs with stochastically switching right-hand sides and PDEs with stochastically switching boundary conditions. In a variety of situations, we prove that the system exhibits surprising behavior. In this talk we will outline this class of problems and highlight some of the most interesting and unexpected results. The talk will be accessible to all graduate students.

We will give an update on our last talk on a new connection between differential geometry and astrophysics which involves a model for dark matter and a possible explanation for barred spiral patterns in galaxies. We will also briefly discuss the Tully-Fisher relation, a mysterious experimental fact relating the visible mass of a galaxy to the speed of the stars in the galaxy, which to this point defies a convincing theoretical explanation.

Kick off this year's flu season with a better understanding of within-host influenza dynamics. Influenza A is a rapidly-evolving RNA virus that typically escapes herd immunity through the generation of new antigenic variants every 3 to 8 years. An important part of this antigenic evolution is believed to occur at the intrahost level. I will present two competing models of intrahost dynamics and compare their predictions to empirical observations.

Motivated by the dynamic feedback systems in the kidney, we consider time-delayed transport equations with stochastic left-hand boundary conditions. We first prove the existence and uniqueness of the steady-state solution for the deterministic case with sufficiently small delay (e.g., zero left-hand boundary). Likewise, we prove those for the stochastic case using the similar analytic techniques. In this talk we will show the process of model formulation with biological motivation and address the role of time-delay and stochastic boundary on the solution behaviors. The talk should be accessible to all graduate students who are familiar with basic ODE theory.

Many machine learning algorithms are based upon estimating eigenvalues and eigenfunctions of certain integral operators. In practice, we have only finitely many randomly drawn points. How close are the eigenvalues and eigenfunctions of the finite dimensional matrix we construct in comparison to the infinite dimensional integral operator? In what way can we say these two are close if they do not even operate on the same spaces? To answer these questions, I will be showing some results from a paper "On Learning with Integral Operators" by Rosasco, Belkin, and De Vito.

Many machine learning algorithms today rely on finding the nearest neighbors of all points in a data set. In data sets which are too large and too complex, we cannot ask for finding the true nearest neighbors without paying the price of a full pairwise search. Therefore, we look to using approximate nearest neighbor searches to quickly construct rough approximations of the nearest neighbor graph. I will talk about many of the methods used today in practice.

We introduce a new quasi-local mass functional for a surface in spacetime and use it to prove the null Penrose conjecture under fairly generic conditions.

This talk will include some PDE, some probability, and some asymptotic analysis. The FKPP equation is a nonlinear partial differential equation that admits traveling wave solutions. It has been used as a simple model for many phenomena involving a stable state invading an unstable state (for example, an advantageous gene spreading through a population). Experiments and numerical simulations show that the speed at which the wave moves is much slower than what is predicted by the deterministic, continuum equation. One way to resolve this discrepancy is to account for the role of noise in the model by adding a stochastic term in the equation (i.e. a stochastic partial differential equation). Analysis of the SPDE has shown that very small noise in the equation introduces a very large correction to the speed associated with the deterministic model. I will explain the basics of the deterministic and stochastic equations, and I will explain some ideas about the asymptotic analysis of the stochastic waves. I hope to have time to explain some open problems related to this topic.

Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For most convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on. Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.) The interesting question is whether there are other convex curves, besides ellipses, for which there are infinitely many closed billiard n-gons for some n. In this talk, I'll discuss the above-mentioned phenomenon and show how it is related to the geometry of non-holonomic plane fields (which will be defined and described). This leads to some surprisingly beautiful geometry, which will require nothing beyond multivariable calculus from the audience.

The Conley Index is a topological tool in dynamical systems which makes heavy use of basic homological techniques, particularly exact sequence. I will present a short primer on relative homology, describe the homological version of the Conley Index, and then present a diagram chase that contains information about the unstable manifolds of invariant sets in a dynamical system. This talk is aimed at students in the Algebraic Topology I course as well as anyone interested in dynamical systems.

One of the exciting results of applied algebraic topology for data analysis has been a formulation of the field of "Stochastic topology." This is an intersection of topology and probability/statistics. I will present some research directions in this field: 1) Euler integration for stochastic models of surfaces and shapes: how topological summaries such as persistence homology or Euler characteristics curves can be used to model surfaces and measure distances between bones. 2) Random simplicial complex models: Given m points drawn from a distribution on manifold construct the union of balls of radius r around these points. As m goes to infinity and r goes to zero what can we say about the limiting distribution of Betti numbers or critical points of this random process ? 3) Spectral theory of simplicial complexes: There is a well developed spectral theory for graphs that provides insights on random walks, spectral clustering of graphs, and near linear time algorithms for solving a system of linear equations. How do these ideas extend to simplicial complexes, in particular: a) is there a notion of a Cheeger inequality for clustering to preserve holes ? b) how does one define a random walk on simplicial complexes that have limiting distributions related to the Harmonics of the (higher order) Hodge Laplacian ? c) we conjecture that the question of near linear time algorithms for linear systems is related to a notion of discrete Ricci curvature for graphs. I just expect knowledge of basic math and will focus on motivating concepts rather than details.

Graphics processing units (GPUs) are powerful accelerators that can launch many processes in parallel. Over the past decade, they have been utilized for scientific computation, including molecular dynamics, fluid mechanics, machine learning, and stochastic differential equations. Although dependent on the algorithm, GPUs may execute code faster than CPUs by several orders of magnitude. The mathematics department at Duke hosts 4 older generation GPUs on two nodes that are available for department use. In this seminar I will briefly introduce how GPUs are different than CPUs; the bulk of my time will be a tutorial on how to code CUDA so that attendees may begin to take advantage of these departmental resources for their research. Depending on the attendance, it may be a hands-on tutorial so bring your laptop.

The Patlak-Keller-Segel equations (PKS) are widely applied to model the chemotaxis phenomena in biology. It is well-known that if the total mass of the initial cell density is large enough, the PKS equations exhibit finite time blow-up. In this talk, I will present some recent results on applying additional fluid flows to suppress chemotactic blow-up in the PKS equations.

I will discuss the semi-classical Schrodinger equation with vector potentials, and its challenges in analysis and in numerical simulations. The time splitting spectral method method will be introduced to solve the equation directly, which is believed to have the optimal mesh strategy. Afterwards. a series of wave packet based approximation approaches will be introduced, like the Gaussian beam method, Hagedorn wave packets method and the Gaussian wave packet transformation method.

I will first explain with a few simple examples a beautiful picture in geometric invariant theory which relates quotient constructions in symplectic and algebraic geometry. Then we will naively apply this picture in a suitable infinite dimensional setting, leading us to the notion of extremal Kahler metrics and the Yau-Tian-Donaldson conjecture, which is an important problem in Kahler geometry today.

Suppose you need to compute an integral over a general surface numerically. How would you do it? You could triangulate the surface, or you might use coordinate charts. Either way is a lot of work, maybe more than you want to do if you have a large number of surfaces. I will describe a fairly simple method, appropriate for smooth, closed surfaces, developed by a former grad student here, Jason Wilson, in his Ph.D. thesis, including proofs that his algorithm works. I will then discuss the extension to integrals for potentials defined by densities on surfaces, such as harmonic functions. In that case the integrand has a singularity; special treatment is needed, and some interesting math comes in. Another of our former Ph.D.'s, Wenjun Ying, has contributed to that work (among many projects of his). Such integrals occur in several scientific contexts; I will especially mention Stokes flow (fluid flow dominated by viscosity), appropriate for modeling some aspects of biology on small scales. For more information, see J. t. Beale, W. Ying, and J. R. Wilson, A simple method for computing singular or nearly singular integrals on closed surfaces at my web site or at http://arxiv.org/abs/1508.00265

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.

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.

Many classes of mathematical objects turn out to be classified in the same way --- two infinite series and 3 "exceptional" objects. These include symmetries of 3-dimensional solids, rigid singularities, certain types of Lie algebras, positive definite even intersection forms, etc. Discovering why such classes should have the same classification has led to many beautiful ideas and observations. I will give a review of some of the basic ideas (assuming very little in the way of prerequisites) and I may have time to say why string theory has been important in this context.

How does the structure of a network effect how randomness propagates through it? This is a fun mix of probability, differential equations, and graph theory.

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.

We introduce a novel kind of percolation on graphs called jigsaw percolation intended as a simple model for collaborative problem solving and merging of ideas. Each node in a network (regarded as a person in a social network) has a unique piece of a jigsaw puzzle. At each stage, components of connected puzzle pieces merge if at least two people are adjacent in the social network and at least two puzzle pieces can join. I will discuss our recent results on this model, outline the proofs, and discuss some open problems. This is joint work with Charles Brummitt, Shirshendu Chatterjee and Partha Dey.

I will give an overview of some of the classical results on the topology of Lie groups, including Hopf's theorem which fully determines the cohomology algebra over the real numbers of any Lie group. We will also discuss how the deRham cohomology of a compact Lie group can be represented by bi-invariant forms. In addition, we will discuss first and the second homotopy groups of Lie groups.