Upcoming Events
Fri, Apr 10
Applied Mathematics Seminar
Yulong Lu (U Minnesota)
In-Context Learning in Scientific Computing
4:00 PM, MATH 250
Transformer-based foundation models, pre-trained on large datasets spanning a wide range of tasks, have shown remarkable adaptability to diverse downstream applications—even in low-data regimes. A particularly striking capability is in-context learning (ICL) : when given a prompt containing a few examples from a new task alongside a query, these models can produce accurate predictions without any parameter updates. This emergent behavior is often viewed as a paradigm shift for transformers, yet its theoretical foundations remain only partially understood. In this talk, I will present recent theoretical progress toward understanding ICL in scientific computing . I will focus on understanding how transformer architectures can implicitly perform task adaptation in three representative problem classes: learning solution operators of PDEs, dynamical system prediction and generative modeling.
Fri, Apr 10
Topology and Geometry Seminar
Roberta Shapiro (University of Michigan)
Geometry, topology, and combinatorics of fine curve graphs
4:00 PM, 122 Mathematics Building
The fine curve graph of a surface is a graph that encodes information about curves on a surface and their interaction. This is similar to the more classical curve graph, which encodes information about the isotopy classes of curves on a surface. In this talk, we construct both graphs and compare and contrast some properties, such as the groups that act on them, their geometry, their topology, and their combinatorics. Some shared results will be work joint with Ryan Dickmann, Zachary Himes, and Alex Nolte.
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.
