Department of Mathematics, Duke University

• Monday, August 20, 2018, 9:30am, Physics 047, Applied Math And Analysis Seminar
  Primal dual methods for Wasserstein gradient flows
  Li Wang (University of Minnesota)

   

  In this talk, I will introduce a variational method for nonlinear equations with a gradient flow structure, which arise widely in applications such as porous median flows, material science, animal swarms, and chemotaxis. Our method builds on the JKO framework and a reformulation of the Wasserstein distance into a convex optimization with a linear PDE constraint. As a result, we end up with one nested structure of optimization problem with two time scales, and we adopt a recent primal dual three operator splitting scheme. Thanks to the variational structure, our method has a built-in positivity preserving, entropy decreasing properties, and overcomes stability issue due to the strong nonlinearity and degeneracy. Upon discretization of the PDE constraint, we also show the Γ−convergence of the fully discrete optimization towards the semi-discrete JKO scheme. This is a joint work with Jose Carrillo, Katy Craig, and Chaozhen Wei.

• Monday, August 20, 2018, 11:00am, Physics 047, Applied Math And Analysis Seminar
  Scaling limit analysis of Stein variational gradient descent
  Yulong Lu (Duke University)

   

  The Stein variational gradient descent (SVGD) was proposed by Liu and Wang as a deterministic algorithm for sampling from a given probability density with unknown normalization. The key idea is to involve a system of interacting particles in an optimized way so that the empirical measure approximates a target distribution. In this talk, I will first introduce the algorithm and compare it with some stochastic-dynamics-based sampling methods. I will also present some recent rigorous analysis results on the mean field limit and long time behavior of the resulting mean field partial differential equation. This is a joint work with Jianfeng Lu and James Nolen.

• Monday, August 20, 2018, 2:00pm, Physics 047, Applied Math And Analysis Seminar
  Aggregation diffusion to constrained interaction: minimizers and gradient flows in the slow diffusion limit
  Kathy Craig (University of California, Santa Barbara)

   

  Nonlocal interactions arise throughout the natural world, from collective dynamics in biological swarms to vortex motion in superconductors. Over the past fifteen years, there has been significant interest in aggregation diffusion equations, which consider the competing effects of nonlocal interactions and local diffusion. More recently, interest has also emerged in constrained aggregation equations, which consider the competition between nonlocal interactions and a hard height constraint on the density.
  
  In joint work with Ihsan Topaloglu, we prove that aggregation diffusion equations converge to the constrained aggregation equation in the slow diffusion limit. As an application of this theoretical result, we adapt Carrillo, Craig, and Patacchini’s blob method for diffusion to develop a numerical method for constrained aggregation equations, which we use to explore open conjectures in geometric shape optimization.

• Monday, August 20, 2018, 3:30pm, Physics 047, Applied Math And Analysis Seminar
  Fourth order models for crystal surface fluctations
  Jeremy Marzuola (University of North Carolina Chapel Hill)

   

  We’ll discuss derivations, dynamics, numerical approximations, recent analytical advances and open questions for a family of 4th order nonlinear PDEs that arise when modeling the fluctuations of a crystal surface. The microscopic problem follows from a continuous time jump Markov process where the jumps occur randomly with rates set from a generalized broken-bond Kinetic Monte Carlo model. The PDEs have a similar look to those of the thin film equations that have been studied by a large number of authors. We will discuss work with on this problem with Jonathan Weare, as well as Jian-Guo Liu, Jianfeng Lu and Dio Margetis; and Anya Katsevich.

• Thursday, August 23, 2018, 3:15pm, 235 Physics, Probability Seminar
  Conjugate gradient-accelerated Gibbs sampler for "large n and large p" sparse Bayesian logistic regression
  Aki Nishimura

   

  In a modern observational study based on healthcare databases, the number of observations is typically in the order of 10^5 ~ 10^6 and that of the predictors in the order of 10^4 ~ 10^5. Despite the large sample size, the data rarely provide enough information to reliably estimate such a large number of parameters. Sparse regression provides a potential solution to this problem. There is a rich literature on desirable theoretical properties of the Bayesian approach based on shrinkage prior. On the other hand, the development of scalable methods for the required posterior computation has largely been limited to the p >> n case. While shrinkage priors are designed to make the posterior amenable to Gibbs sampling, a major computational bottleneck still arises from the need to sample from a high-dimensional Gaussian distribution at each iteration. The closed form expression for the precision matrix $\Phi$ is available, but computing and factorizing such a large matrix is computationally expensive nonetheless. In this article, we present a novel algorithm to speed up this bottleneck based on the following observation: we can cheaply generate a random vector $b$ such that the solution of a linear system $\Phi \beta = b$ has the desired Gaussian distribution. An accurate solution of the linear system can then be found by the conjugate gradient algorithm with only a small number of the matrix-vector multiplications by $\Phi$, without ever explicitly inverting $\Phi$. We apply our algorithm to a large-scale observational study with n = 72,489 and p = 22,175, designed to assess the relative risk of intracranial hemorrhage from two alternative blood anti-coagulants. Our algorithm demonstrates an order of magnitude speed-up in the posterior computation.

• Monday, September 3, 2018, 3:15pm, 119 Physics, Geometry/topology Seminar
  Floer homology and Dehn surgery
  Faramarz Vafaee (Duke University, Mathematics)

   

  The past thirty years have witnessed the birth of a beautiful array of approaches to the field of low dimensional topology, drawing on diverse tools from algebra, analysis, and combinatorics. One particular tool that has made a dramatic impact on the field is the Heegaard Floer theory of Ozsvath and Szabo. Defined 17 years ago, this theory has produced an encompassing package of invariants, which have significantly impacted the study of many areas of low dimensional topology, including Dehn surgery. In this talk, we will focus on two questions: a) which 3-manifolds do arise by Dehn surgery along a knot in the 3-sphere? b) what are all ways to obtain a fixed 3-manifold by Dehn surgery along a knot in the 3-sphere?

• Wednesday, September 5, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  Primal-Dual Pi Learning Using State and Action Features
  Mengdi Wang (Princeton University)

   

  We survey recent advances on the complexity and methods for solving Markov decision problems (MDP) and Reinforcement Learning (RL) with finitely many states and actions - a basic mathematical model for reinforcement learning.
  
  For model reduction of large scale MDP in reinforcement learning, we propose a bilinear primal-dual pi learning method that utilizes given state and action features. The method is motivated from a saddle point formulation of the Bellman equation. The sample complexity of bilinear pi learning depends only on the number of parameters and is variant with respect to the dimension of the problem.
  
  In the second part we study the statistical state compression of general Markov processes. We propose a spectral state compression method for learning the state features from data. The state compression method is able to “ sketch” a black-box Markov process from its empirical data and output state features, for which we provide both minimax statistical guarantees and scalable computational tools.

• Tuesday, September 11, 2018, 3:15pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Kui Ren (Columbia University)

• Wednesday, September 12, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Andrej Zlatos (University of California-San Diego)

• Thursday, September 13, 2018, 3:15pm, 119 Physics, Probability Seminar
  TBA
  Houman Owhadi

• Wednesday, September 19, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Alexander Cloninger (UC San Diego)

• Thursday, September 20, 2018, 3:15pm, 119 Physics, Probability Seminar
  Stable coexistence of savannah and forest in a spatial model
  Carla Staver

   

  The goal of this talk is to further a joint project involving Carla Staver, Simon Levin, Rick Durrett, and Ruibo Ma. The puzzle is: why can savannah and forest display stable coexistence when this is not possible in a spatially homogeneous system.

• Wednesday, September 26, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Roman Shvydkoy (University of Illinois-Chicago)

• Thursday, September 27, 2018, 3:15pm, 119 Physics, Probability Seminar
  TBA
  Yu-ting Chen (U of Tennessee, Knoxviller)

• Wednesday, October 3, 2018, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
  TBA
  Jun Kitagawa (Michigan State University)

• Thursday, October 4, 2018, 3:15pm, 119 Physics, Probability Seminar
  TBA
  Erik Slivken (Paris Diderot)

• Wednesday, October 10, 2018, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
  TBA
  Ilse Ipsen (NC State University)

• Wednesday, October 17, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Jiequn Han (Princeton University)

• Wednesday, October 24, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Almut Burchard (U of Toronto)

• Wednesday, October 31, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Guillaume Bal (University of Chicago)

• Wednesday, October 31, 2018, 3:15pm, 119 Physics, Number Theory Seminar
  TBD
  Jessica Fintzen (Michigan/Cambridge)

• Monday, November 5, 2018, 3:15pm, 119 Physics, Geometry/topology Seminar
  TBA
  Hannah Schwartz (Bryn Mawr College, Mathematics)

• Wednesday, November 7, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
  TBA
  Weijie Su (University of Pennsylvania)

• Thursday, November 8, 2018, 3:15pm, 119 Physics, Probability Seminar
  title
  Pascal Maillard (CRM (Montréal) and Université Paris-Sud)

   

  abstract

• Friday, November 16, 2018, 3:15pm, 119 Physics, Algebraic Geometry Seminar
  

  tentative: J. Rana

• Monday, November 26, 2018, 3:15pm, 119 Physics, Triangle Topology Seminar
  TBA
  Lisa Piccirillo (University of Texas, Austin, Mathematics)

• Friday, February 22, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
  

  tentative: Sebastian Casalaina-Martin

• Friday, March 22, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
  

  tentative: Sebastian Casalaina-Martin

• Friday, April 5, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
  

  tentative: Sebastian Casalaina-Martin