Posted January 12, 2022

Last modified January 25, 2022

Colloquium Questions or comments?

11:30 am - 12:20 pm Zoom
Guangqu Zheng, University of Edinburgh

Wiener chaos, Gaussian analysis and Stochastic partial differential equations

Abstract: This talk goes around the concept of Wiener chaos, which was first introduced by N. Wiener (1938) and later modified by K. Ito (1951, 56). It has been recurrently brought up in recent years, as it arises naturally in the study of stochastic partial differential equations, parameter estimation, nodal statistics of a Gaussian random fields and stochastic geometry, to name a few. Notably, Nualart-Peccati’s fourth moment theorem (2004) and Nourdin-Peccati’s Malliavin-Stein approach (2008) further push Wiener chaos to the center of Gaussian analysis, and it has turned out to be very effective in obtaining quantitative limit theorems in practice. In this talk, we will focus on the central limit theorem for the stochastic wave equation driven by Gaussian noise. We will present how the Wiener chaos enters the picture, and then highlight the key ideas and sketch main steps for obtaining relevant limit theorems. If time permits, we will talk about how this line of research (ideas, techniques) may lead to some other interesting results, for example: (i) extending random field solution theory for nonlinear SPDEs driven by colored noise, (ii) obtaining Gaussian fluctuations for (renormalized) singular stochastic dispersive/parabolic PDEs.

Posted December 13, 2021

Last modified January 19, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Gage Martin, Boston College

Annular links, double branched covers, and annular Khovanov homology

Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched cover construction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.

Posted January 25, 2022

Colloquium Questions or comments?

3:30 pm - 4:20 pm Zoom
Ana Balibanu, Harvard University

Poisson transversals in representation theory

Abstract: Geometric representation theory studies groups and algebras by realizing their representations geometrically, through actions on associated algebraic varieties. Symplectic and Poisson structures appear naturally in this setting, and give key insights into the geometry of the spaces that carry them. In turn, these spaces provide foundational examples for new research directions in Poisson geometry. The purpose of this talk is to illustrate this interplay in the framework of transversal structures. We will begin by introducing the notion of Poisson transversality, and by giving examples of several well-known representation-theoretic algebraic varieties that arise as Poisson transversals. Motivated by multiplicative analogues of these varieties, we will then define a general class of transversal slices for quasi-Poisson structures. This construction is based on the algebraic data that comes from an associated complex semisimple group, and can be used to produce canonical compactifications of these spaces which have interpretations in the setting of mathematical physics.

Posted January 26, 2022

Colloquium Questions or comments?

3:30 pm - 4:20 pm Zoom
Michael Lindsey, Courant Institute, NYU

A sampling of Monte Carlo methods

Abstract: In this talk I discuss recent work in several different areas involving Monte Carlo sampling. In the first part of the talk, I consider generic sampling problems, especially in low to moderate dimension, for which I introduce an ensemble Markov chain Monte Carlo (MCMC) method that overcomes the difficulty of slow transitions between isolated modes. In the second part, I consider the problems of computing ground states and excited states of quantum many-body systems, which are eigenpairs of exponentially high-dimensional Hermitian operators. I present a new optimization method within the framework of variational Monte Carlo (VMC). The VMC framework approaches these problems by stochastic optimization over a parametric class of wavefunctions. Of particular interest are recently introduced neural-network-based parametrizations for which this approach yields state-of-the-art results. In the last part, I consider a lattice model of quantum critical metals that captures a plausible mechanism for high-temperature superconductivity. This model can be studied numerically by Monte Carlo methods, but previous approaches cannot reach the large-volume limit needed to reveal critical scaling properties due to cubic computational cost in the lattice volume. I present recent work toward a linear-scaling approach.

Posted September 29, 2021

Last modified January 25, 2022

Mathematical Physics and Representation Theory Seminar

3:30 pm - 4:20 pm Zoom: https://lsu.zoom.us/j/98489192227
Iva Halacheva, Northeastern University

Welded tangles and the Kashiwara-Vergne group

Welded or w-tangles are a higher dimensional analogue of classical tangles, which admit a yet further generalization to welded foams, or w-trivalent graphs, a class of knotted tubes in 4-dimensional space. Welded foams can be presented algebraically as a circuit algebra. Together with Dancso and Robertson we show that their automorphisms can be realized in Lie theory as the Kashiwara-Vergne group, which plays a key role in the setting of the Baker-Campbell-Hausdorff series. In the process, we use a result of Bar-Natan and Dancso which identifies homomorphic expansions for welded foams, a class of powerful knot invariants, with solutions to the Kashiwara-Vergne equations.

Posted January 25, 2022

Colloquium Questions or comments?

3:30 pm - 4:20 pm Zoom
Spencer Leslie, Duke University

Periods, L-values, and stabilization

Abstract: The study of period integrals of automorphic forms originates in deep questions about cohomology of locally symmetric spaces. A particularly powerful tool for studying periods is a relative trace formula, which often allows one to relate these integrals to other arithmetic objects like L-functions. In this talk, I review some of this story, discuss this modern approach to relating period integrals to L-functions, and introduce an important case of active research: unitary Friedberg-Jacquet periods. These periods are conjecturally related to central values of certain L-functions and are thus connected to deep conjectures on the cohomology of the associated locally symmetric spaces. To prove these conjectural relationships, a promising approach is to use a relative trace formula. However, new problems (known as instability) arise in this setting that must be overcome if one is to prove this relation. I will discuss my work on a theory of endoscopy and a stable relative trace formula to overcome these problems. This gives a refinement of the relative trace formula amenable to proving this conjecture.

Posted January 14, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Sudipta Ghosh, Louisiana State University

TBA

Posted January 6, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
Marta Lewicka, University of Pittsburgh

Geodesics and isometric immersions in kirigami

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. We are concerned with two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the full family of polygonal geodesics can be simultaneously rectified into a straight line via a piecewise affine planar isometric immersion.

Posted December 10, 2021

Last modified January 14, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Shelly Harvey, Rice University

TBA

Posted January 14, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm
Petronela Radu, University of Nebraska, Mathematics Department

TBA

Posted December 17, 2021

Last modified January 14, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Fraser Binns, Boston College

TBA

Posted January 13, 2022

Control and Optimization Seminar Questions or comments?

9:30 am - 10:20 am Zoom (Click "Questions or comments?" to request Zoom link)
Emmanuel Trelat, Sorbonne Universite, Paris, France

On the Turnpike Property

The turnpike property was discovered in the 50's by Nobel prize winner Samuelson in economics. It stipulates that the optimal trajectory of an optimal control problem in large time remains essentially close to a steady state, itself being the optimal solution of an associated static optimal control problem. We have established the turnpike property for general nonlinear finite and infinite dimensional optimal control problems, showing that the optimal trajectory is, except at the beginning and the end of the time interval, exponentially close to some (optimal) stationary state, and that this property also holds for the optimal control and for the adjoint vector coming from the Pontryagin maximum principle. We prove that the exponential turnpike property is due to a hyperbolicity phenomenon which is intrinsic to the symplectic feature of the extremal equations. We infer a simple and efficient numerical method to compute optimal trajectories in that framework, in particular an appropriate variant of the shooting method. The turnpike property turns out to be ubiquitous and the turnpike set may be more general than a single steady-state, like for instance a periodic trajectory. We also show the shape turnpike property for PDE models in which a subdomain evolves in time according to some optimization criterion. These works are in collaboration with Gontran Lance, Can Zhang, and Enrique Zuazua.

Posted January 17, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm
Jonas Lührmann, Texas A&M University

Asymptotic stability of the sine-Gordon kink under odd perturbations

The sine-Gordon model is a classical nonlinear scalar field theory that was discovered in the 1860s in the context of the study of surfaces with constant negative curvature. Its equation of motion features soliton solutions called kinks and breathers, which play an important role for the long-time dynamics. I will begin the talk with an introduction to classical 1D scalar field theories and the asymptotic stability problem for kinks. After surveying recent progress on the problem, I will present a joint work with W. Schlag on the asymptotic stability of the sine-Gordon kink under odd perturbations. Our proof is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key aspects are a super-symmetric factorization property of the linearized operator and a remarkable non-resonance property of a variable coefficient quadratic nonlinearity.

Posted January 16, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 am Zoom
Chenjie Fan, Academy of Mathematics and Systems Science of the Chinese Academy of Sciences

TBA

Posted January 10, 2022

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm - 3:00 pm Lockett 233
Sam Shepherd, Vanderbilt University

TBD

Posted December 10, 2021

Last modified January 14, 2022

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Sam Shepherd, Vanderbilt University

TBA

Posted January 25, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:20 pm tba
Kasso Okoudjou, Tufts University

tba

Posted January 17, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm
Stan Palasek, UCLA

TBA

Posted January 13, 2022

Control and Optimization Seminar Questions or comments?

9:30 am - 10:20 am Zoom (Click "Questions or comments?" to request Zoom link)
Pierdomenico Pepe, University of L'Aquila

TBA

Posted January 18, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm ZoomTBA

Posted January 18, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
Mariana Smit Vega Garcia, Western Washington University

TBA

Posted January 19, 2022

Geometry and Topology Seminar Seminar website

Lockett 233
Dan Rutherford, Ball State University

TBA

Posted January 20, 2022

Applied Analysis Seminar Questions or comments?

3:30 pm - 4:30 pm Zoom
Davit Harutyunyan, University of California Santa Barbara

TBA