Monday, Apr 22 1:15 pm - 2:15 pm
Title: Convolutional neural networks and inverse problems for nonlinear wave equations.
Abstract: We consider inverse problems of determiningspace-time structures using the responses to probing waves (e.g. gravitationalwaves, electromagnetic waves etc) from the machine learning point of view.Based on the understanding of propagation of waves and their nonlinearinteractions, we construct a deep convolutional neural network in which theparameters are used to reconstruct the coefficients of nonlinear waveequations. We also discuss the depth and number of units of the network andtheir quantitative dependence on the complexity of the medium structure. Thisis a joint work with G. Uhlmann.
Monday, Apr 22 1:30 pm - 2:30 pm
Title: Analytic spread and symbolicanalytic spread
Abstract: The analytic spread of amodule M is the minimal number of generators of a submodule that has the same integral closure as M. In this talk, we will present a result that expresses the analytic spread of a decomposable module in terms of the analytic spread of its component ideals. In the second part of the talk, we will show an upper bound for the symbolic analytic spread of ideals of small dimension. The latter notion is the analogue of analytic spread for symbolic powers. These results are joint work with Carles Bivià-Ausina and Hailong Dao, respectively.
Monday, Apr 22 3:30 pm - 4:30 pm
Title: Exceptional Directions for the Teichmuller Geodesic Flow
Abstract: A translation surface is a pair (S,w) of a compact Riemann surface S equipped with a holomorphic 1-form w. These objects fit together in a moduli space which admits a natural action of SL(2,R). The work of Masur, Veech and many others has demonstrated that the dynamical behavior of the orbit of a translation surface (S,w) under the diagonal group in SL(2,R) is intimately tied to the ergodic properties of the translation flow induced by the vertical foliation of w on S. However, the hyperbolic nature of this diagonal flow often allows only for an understanding of the behavior of generic points. On the other hand, the remarkable results of Eskin-Mirzakhani-Mohammadi made it possible to understand the dynamical behavior of the SL(2,R) orbit of every (not just almost every!) such (S,w). In this talk, we show how one can leverage these powerful results to show that the subset of the circle of directions around any fixed (S,w) for which the time-averages along the diagonal flow deviate from the space-average by a definite amount have Hausdorff dimension strictly less than one. This is joint work with Al-Saqban, Apisa, Erchenko, Mirzadeh, and Uyanik.
Title: Adaptive Finite Element Approximation for the Fractional Laplacian
Abstract: We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations on non trivial domains in two dimensions. Our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and, where necessary, modifying them so that they can be applied to the fractional PDE case. Improved a priori error estimates are derived for the case of quasi-uniform meshes which are seen to deliver sub-optimal rates of convergence owing to the presence of singularities. Attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure. We assume that the resulting refined meshes are locally quasi-uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods, including the multigrid method. The storage of the dense matrices along with efficient techniques for computing the dense matrix vector products needed for the iterative solution is also considered. The performance and efficiency of the resulting algorithm is illustrated for a variety of examples.
Tuesday, Apr 23 3:30 pm - 4:30 pm
Title: Heat semigroup and BV functions
Abstract: In abstract Dirichlet spaces, we present a theory of Besov spaces which is based on the heat semigroup. This approach offers a new perspective on the class of bounded variation functions in settings including Riemannian manifolds, sub-Riemannian manifolds. In rough spaces like fractals it offers totally new research directions. The key assumption on the underlying space is a weak Bakry-\'Emery type curvature assumption.
The talk will based on joint works with Patricia Alonso-Ruiz, Li Chen, Luke Rogers, Nageswari Shanmugalingam and Alexander Teplyaev.
Title: Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential.
Abstract: I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.
Title: Resolutions of the Steinberg module for GL(n)
Abstract: The Steinberg module St(n,K) for the general linear group G of a number field K is the dualizing module for arithmetic subgroups of G. So St(n,K) can be used in studying the cohomology of arithmetic groups, and is especially suitable for computing Hecke operators. I will present several resolutions of St(n,Q) that are helpful in this regard. I will discuss applications to computations for congruence subgroups of SL(4,Z) (with Paul Gunnells and Mark McConnell) and some work related to Serre-type conjectures for GL(n,Z) (with Darrin Doud.)
Robust & Explicit A Posteriori Error Estimation Techniques in Adaptive Finite Element Method
Thursday, Apr 25 2:30 pm - 3:20 pm
Title: Finding and writing grants
Abstract: Seeking external funding typically starts in graduate school (e.g., requesting financial support to attend a conference), and it really never ends... In this talk I will share some tips on how to look for funding opportunities, and on how to write a compelling proposal. Bring your questions!
Friday, Apr 26 11:30 am - 12:30 pm
Title: Stochastic Gradient Descent
Abstract: Stochastic Gradient Descent (SGD), also known as stochastic approximation, refers to certain simple iterative structures used for solving stochastic optimization and root finding problems. The identifying feature of SGD is that, much like in gradient descent for deterministic optimization, each successive iterate in the recursion is determined by adding an appropriately scaled gradient estimate to the prior iterate. Owing to several factors, SGD has become the leading method to solve optimization problems arising within large-scale machine learning and ``big data" contexts such as classification and regression. This talk covers the basics of SGD with an emphasis on modern developments. The talk starts with examples where SGD is applicable, and then details important flavors of SGD and reported convergence rate calculations. I will present some numerical examples to aid intuition.
Title: Teaching Discussion -- Abstract: This weekly teaching discussion is a place to discuss our practice in the classroom. All are welcome and encouraged to participate, from experienced instructors who have seen it all to those who are just starting.
Title: Stability for Functional and Geometric Inequalities and a Stochastic Representation of Fractional Integrals and Nonlocal Operators
Title: Quasi-toroidal Varieties and Rational Log Structures in Characteristic 0
Spectral Methods for Boundary Value Problems in Complex Domains