**Title:** A Nordhaus-Gaddum type problem for the normalized Laplacian spectrum and graph Cheeger constant

**Abstract:**

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# News Archive

## Discrete Math Seminar: Adam Knudson (BYU)

## Number Theory Seminar: Eric Moss (Boston College)

## Dynamics Seminar: Homin Lee – Northwestern University

## Number Theory Seminar: Weichen Gu, University of New Hampshire

## Applied Analysis Seminar: Vincent Martinez from CUNY-Hunter College

## Colloquium: Shane McQuarrie (Sandia National Laboratories)

## Number Theory Seminar: Hung Bui (University of Manchester)

## Discrete Math Seminar: Thomas Cameron (Penn State Behrend)

## Stephen McKean (Harvard University)

## Seminar: Deep Ray, University of Maryland

**Title:** A Nordhaus-Gaddum type problem for the normalized Laplacian spectrum and graph Cheeger constant

**Abstract:**

**Title:** Computing Bianchi-Maass Forms

**Abstract:** A Maass form is an important generalization of a modular form. Maass cusp forms are “transcendental” functions, and the state of the art for computing them explicitly is a numerical method called Hejhal’s Algorithm which I will explain briefly. A Bianchi group is PSL_{2}(O_{K}) where O_{K} is the ring of integers of an imaginary quadratic field. I will show how I have generalized Hejhal’s Algorithm to work over certain Bianchi groups. There are notable computational difficulties of actually implementing this in code, which I did in C++ and then ran it on the supercomputer at Boston College. I hope this talk will have a healthy balance of big formulas and pretty pictures!

**Title:** Smooth actions on manifold by higher rank lattices

**Abstract:** We will discuss about smooth actions on manifolds by higher rank groups, such as lattices in SL(*n*, ℝ) with *n* ≥ 3 or ℤ^{k} with *k* ≥ 2.

The higher rank property of the acting group suggests that the actions are rigid, which means that the action should have an algebraic origin, such as the Zimmer program and the Katok-Spatzier conjecture. One of the main topics is about how we can give an algebraic structure on the acting space which is a smooth manifold.

We survey some of recent breakthroughs and then focus mainly on actions of higher rank lattices. In particular, we focus on actions on manifolds with “positive entropy” by lattices in SL(*n*, ℝ), *n* ≥ 3. When the manifold has dimension *n*, then we will see that the lattice is commensurable to SL(*n*, *Z*) from a certain “algebraic structure” on *M* coming from the dynamics.

Part of the talk is ongoing work with Aaron Brown.

**Title:** Möbius disjointness for a class of exponential functions

**Abstract:** The entropy of arithmetic functions is the complexity of their value distributions. This complexity can be rigorously defined by the topological entropy of the continuous map induced by the arithmetic function. Sarnak’s Möbius Disjointness Conjecture asserts that any arithmetic function with zero entropy is disjoint from the Möbius function. In collaboration with Fei Wei, we show that a large class of exponential functions have zero entropy, and many of them are disjoint from the Möbius function.

**Title:** On the long-time statistical behavior of a stochastic Coleman-Gurtin equation in the memoryless limit

**Abstract:** We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporate the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials and sufficiently regular noise the system always admits invariant probability measures, defined on the extended phase space, that possess higher regularity properties dictated by the structure of the nonlinearities in the equation. Furthermore, we investigate the singular limit as the memory kernel collapses to a Dirac function. Specifically, provided sufficiently many directions in the phase space are stochastically forced, we show that there is a unique stationary measure to which the system converges, in a suitable Wasserstein distance, at exponential rates independent of the decay of the memory kernel. We then prove the convergence of the statistically steady states to the unique invariant probability of the classical stochastic reaction-diffusion equation in the desired singular limit. As a consequence, we establish the validity of the small memory approximation for solutions on the infinite time horizon. This is joint work with Nathan Glatt-Holtz (Tulane University) and Hung D. Nguyen (University of Tennessee—Knoxville).

**Title**: Learning Reduced-order Models with Uncertainty and Structure

**Abstract:** Many modern science problems require solving computationally expensive numerical models of complex physical systems for a variety of conditions. Model-order reduction seeks to alleviate the computational burden of such simulations by constructing computationally efficient surrogates, called reduced-order models (ROMs), which can be solved quickly to obtain approximate solutions of the emulated system. This talk develops two data-driven model reduction approaches based on Operator Inference, a paradigm in which the problem of learning the reduced operators that define a ROM is posed as a regression of state space data and corresponding time derivatives.

When time derivative data are not natively available, as is often the case in applications, they must be estimated from the state data with, e.g., finite difference approximations. The accuracy of the estimation greatly affects the quality of the learned ROM, hence learning accurate ROMs in this manner is challenging when available state data are sparse and/or noisy. To address these challenges, we incorporate Gaussian process surrogate modeling into the Operator Inference framework to probabilistically describe uncertainties in the state data and procure analytical time derivative estimates equipped with corresponding uncertainty estimates. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order operators that are defined probabilistically. The resulting ROM propagates uncertainties from the observed state data to reduced-order predictions.

In addition, we describe a novel method for embedding time-periodic structure into Operator Inference ROMs. The form of the model is motivated by relating the classical projection-based ROM of a polynomial system to a linear time-periodic system whose solutions contain the harmonics of the input frequency. The data-driven nature of Operator Inference allows us to choose the ROM inputs so that solutions are composed of the same higher-frequency content as would be observed in the linear time-periodic approximation. We demonstrate this approach for a highly nonlinear plasma glow discharge process.

*Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly-owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.*

**Title:** Moments of the Riemann zeta-function and applications

**Abstract:** The study of moments of the Riemann zeta-function was initiated by Hardy and Littlewood in 1910s, and has been a major theme in analytic number theory since then. We shall discuss its development and some applications to the distribution of the Riemann zeros.

**NOTE THE DIFFERENT DAY.**

**Title:** On the fort number, (fractional) zero-forcing, and maximum nullity of a graph.

**Abstract:** In 2018, Brimkov, Fast, and Hicks defined the concept of a Fort, which is a non-empty subset of vertices such that no vertex outside of the fort has exactly one neighbor in the fort. Forts play a vital role in integer programming models for computing the zero-forcing number of a graph since a subset of vertices is a zero-forcing set if and only if that set intersects every fort of the graph.

In this talk, we discuss the concept of a minimal fort and introduce two new graph parameters: the fort number and the fractional zero-forcing number. In particular, we investigate how many minimal forts a graph can have. Moreover, we discuss the relationship between these new parameters and the well-known graph parameters of the zero-forcing number and maximum nullity.

**Title:** Counting with quadratic forms

**Abstract:** Many theorems in enumerative geometry are restricted to algebraically closed fields. For example, a conic and cubic in the plane intersect 6 times over the complex numbers, but some of these intersections go missing over the reals. In this talk, I will discuss how to use tools from motivic homotopy theory to do enumerative geometry over arbitrary fields. Instead of integer-valued counts, these tools produce equations of quadratic forms. Invariants of these quadratic forms recover classical theorems from complex and real enumerative geometry and imply new theorems over other fields.

**TITLE:** A deep learning strategy for solving physics-based Bayesian inference problems

**ABSTRACT:** Inverse problems arise in numerous science and engineering applications, such as medical imaging, weather forecasting and predicting the spread of wildfires. Bayesian inference provides a principled approach to solve inverse problems by considering a statistical framework, which is particularly useful when the measurement/output of the forward problem is corrupted by noise. However, Bayesian inference algorithms can be challenging to implement when the inferred field is high-dimensional, or when the known prior information is too complex. In this talk, we will see how conditional Wasserstein generative adversarial networks (cWGANs) can be designed to learn and sample from conditional distributions in Bayesian inference problems. The proposed approach modifies earlier variants of the architecture proposed by Adler et al. (2018) and Ray et al. (2022) in two fundamental ways: i) the gradient penalty term in the GAN loss makes use of gradients with respect to all input variables of the critic, and ii) once trained, samples are generated from the posterior by considering an open ball around the measurement. These two modifications are motivated by a convergence proof that ensures the learned conditional distribution weakly approximates the true conditional distribution governing the data. Through simple examples we show that this leads to a more robust training process. We also demonstrate that this approach can be used to solve complex real-world inverse problems.