Upcoming Events
Wed, Apr 15
Analysis Seminar
Rizwanur Khan, University of Texas at Dallas
Gaussian Behavior of Eisenstein Series
3:30 PM, 122 Mathematics building
Understanding the behavior of Laplace eigenfunctions inthe high-energy limit is a central question in analysis, with specialsignificance for number theorists when the underlying manifold is arithmetic,such as the modular surface. I will discuss recent work (joint with GoranDjanković) on the distribution of the continuous spectrum—namely, theEisenstein series—on the modular surface. We provide some rigorous evidence(through calculation of the fourth moment) that these functions exhibitGaussian random behavior, consistent with Berry’s Random Wave Conjecture andthe numerical observations of Hejhal and Rackner.
Fri, Apr 17
Applied Mathematics Seminar
Lili Ju (U of South Carolina)
Transferable Neural Networks for Partial Differential Equations
4:00 PM, MATH250
Transfer learning for partial differential equations (PDEs) aims to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information about the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to design transferable neural feature spaces for the shallow neural networks from purely function approximation perspectives without using PDE information. The construction of the feature space involves the re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. We use the proposed feature space as the predetermined feature space of a random feature model and use existing least squares solvers to obtain the weights of the output layer. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state-of-the-art methods. Finally, we discuss ongoing and future research topics along this direction.
Mon, Apr 20
Algebra Seminar
Douglas Rizzolo, University of Delaware
Ordered Chinese Restaurant Process up-down chains
4:00 PM, 250 Mathematics Building
Up-Down chains on branching graphs provide an interesting link between the algebraic structure of branching graphs and stochastic processes on the boundaries of these graphs. Up-down chains on branching graphs whose vertices are given by partitions of an integer are well understood, but up-down chains on branching graphs whose vertices are compositions of an integer have only begun to be studied recently. In this talk we will discuss up-down chains on graphs of compositions whose up-steps are based on the Ordered Chinese Restaurant Process. We will show how these can be used to construct diffusions on the boundary of this graph whose generators have simple expressions in terms of quasi-symmetric functions. We will give examples showing what sorts of new statistics can be understood based on the order structure, with an emphasis on a connection to phylogenetics where the order structure can be interpreted as the relative ages of alleles in a population.
Thu, Apr 23
Colloquium
Brandis Whitfield (University of Wisconsin-Madison/SLMath)
The geometries of subgroups of the mapping class group
4:00 PM, 250 Mathematics Building
Mapping class groups of surfaces play a prominent role in understanding the geometry and topology of three-manifolds. These connections have inspired decades-long research programs toward a notion of geometric finiteness for mapping class subgroups. In this talk, we’ll introduce the mapping class group, discuss methods of embedding different groups into it and how topological properties of a mapping class determine geometric data of its associated 3-manifold.
Fri, Apr 24
Topology and Geometry Seminar
Brandis Whitfield (University of Wisconsin-Madison/SLMath)
Constructing reducibly geometrically finite subgroups of the mapping class group
4:00 PM, 122 Mathematics Building
A longstanding goal within low-dimensional geometry and topology is to establish parallels between the theory of Kleinian groups and subgroups of the mapping class group. Convex-cocompact subgroups of the mapping class group have been well-studied in the past few decades; current research aims to extend this analogy to the more general class of geometric finiteness. Dowdall–Durham–Leininger–Sisto introduce parabolically geometrically finite (PGF) subgroups of the mapping class group whose coned-off Cayley graphs, with respect to a collection of twist subgroups, quasi-isometrically embed into the curve complex. In joint work with Aougab, Bray, Dowdall, Hoganson and Maloni, we extend this definition to allow the coned-off subgroups to be more generally reducible, and produce plentiful examples of this behavior. Our examples include free products of reducible subgroups and the beloved right-angled Artin constructions of Clay–Leininger–Mangahas.
