News Archive

Colloquium: Rachel Webb (University of California, Berkeley)

Title: An approach to computing Gromov-Witten invariants

Abstract: In the 1990s, theoretical physics gave rise to a new mathematical challenge: computing certain “virtual counts” of curves on manifolds. These counts, called the Gromov-Witten invariants of the manifold, model particle interactions in string theory. In principle one can determine the small-scale geometry of the universe by matching some numbers obtained in a physics lab to Gromov-Witten invariants computed in a math department (in practice, we are still waiting on the numbers from the lab). The challenge of computing Gromov-Witten invariants has motivated over two decades of mathematics, and there are still important open questions. I will discuss some explicit formulas for Gromov-Witten invariants that are available when the manifold is described by sufficiently “linear” data (representations of reductive groups), and the implications of these formulas for said open questions. These formulas also have many applications to both classical geometry and geometry motivated by physics.

Colloquium: Kyle Pratt (University of Oxford, England)

Title: Quadratic magic: Diophantine equations with squares

Abstract: We survey some classical and recent results on Diophantine equations involving perfect squares. We also discuss some open problems. The talk will be accessible to a broad, non-specialist audience.

Colloquium: Davi Obata (University of Chicago)

Title: Uniqueness of the measure of maximal entropy for the standard map

Abstract: Understanding the dynamics of a nonlinear system can be a very hard task, even for systems having a simple expression. A good example of such a system is the (Taylor-Chirikov) standard map. Sinai conjectured that the standard map has positive metric entropy for large parameters (i.e., it has a set of positive Lebesgue measure having non-zero Lyapunov exponents). The dynamics of the standard map is far from being well understood. In this talk, I will discuss some progress in the understanding of the dynamics of the standard map.

Colloquium: Shane McQuarrie (The University of Texas at Austin)

Title: Learning physics-based reduced-order models from data: Operator inference for parametric partial differential equations

Abstract: Large-scale numerical simulations of complex physical systems form the backbone of many modern scientific applications. For decades, mathematicians and computational scientists have focused on solving the forward problem of mapping initial/boundary conditions, system parameters, and auxiliary inputs to the corresponding solution of a known dynamical system. Next-generation scientific tasks such as physics-constrained optimization, optimal experimental design, and uncertainty quantification require many forward simulations (sometimes thousands or millions), each with different scenario parameters. Unfortunately, forward problems are often computationally intensive due to spatial and temporal resolution demands. Model order reduction seeks to alleviate the computational burden of forward solves by replacing expensive numerical simulations of complex physical systems with inexpensive surrogate models, called reduced-order models.

Classical model order reduction techniques construct reduced-order models by directly compressing the discretized governing equations, but this approach is infeasible for production-level codes where the discretization details are highly complex, proprietary, or classified. This talk presents Operator Inference, a data-driven model order reduction framework for constructing reduced-order models using only (i) knowledge of the structure of the governing equations and (ii) available simulation data. We detail a method for ensuring stability in the reduced-order model through a regularization selection procedure and use Bayesian inference to quantify the uncertainties associated with the data-driven learning. We also show how, for a large class of parametric systems, parametric dependencies can be embedded directly into the reduced-order model. In this setting, well-posedness conditions for the learning problem lead to a parameter selection criteria. The methodology is demonstrated on a variety of applications, including a single-injector combustion process and the FitzHugh-Nagumo neuron model.

Colloquium: Rebeca Paulsen (BYU)

Title: Taking Chances

Abstract: Many of the day-to-day decisions we make require us to weigh risks. Navigating the uncertainty in our lives is fraught with difficulty, as our intuition and our experience will, more often than not, lead us astray. Should I check my bag or carry it on? Should I finish the last leg of my road trip or find a hotel and drive home in the morning? Should I pay for the extended warranty? We will explore these questions and discuss other everyday topics in probability.

Colloquium: Kevin Miller (University of Texas at Austin)

Title: Doing More with Less: Graph-based Methods for Learning from Limited Observations

Abstract: Modern research in machine learning has primarily focused on the supervised learning of functions from massive amounts of labeled data, where inputs with their observed outputs (labels) are available to the learning algorithm. However, in applications, it is often more realistic to have plenty of unlabeled data (i.e., inputs without known labels) while only few labeled data. With a common theme of leveraging the geometric structure of data through similarity graphs, I will present my recent work on the theoretical understanding and computational application of semi-supervised and active learning paradigms for learning when labeled data are scarce but unlabeled data are plentiful. I will discuss my work to prove Bayesian posterior contraction in graph-based semi-supervised regression and how it inspired the subsequent design of a computationally efficient graph-based active learning method. I will also present a novel uncertainty sampling criterion for active learning in a graph-based model that has a well-defined continuum limit partial differential equation formulation; this continuum limit model facilitates the establishment of rigorous mathematical guarantees about the sampling complexity of the proposed method. Experimental results will demonstrate the utility of the methods to various applications like pixel classification in hyperspectral imagery and automatic target recognition in synthetic aperture radar imagery.

Colloquium: Matthew Romney (Stony Brook University)

Title: The metric geometry of surfaces

Abstract: A well-known part of classical mathematics is the differential geometry of smooth surfaces, developed by historical mathematicians such as Euler and Gauss. A fundamental result on the topic is the uniformization theorem, proved by Poincaré and Koebe in 1907, which states that any smooth Riemannian surface can be mapped conformally onto a surface of constant curvature. Since then, the geometry of surfaces has been investigated in increasing generality by several research communities. Non-smooth surfaces were perhaps first systematically studied as part of the field of Alexandrov geometry beginning in the 1930s, leading to a well-developed theory of surfaces of bounded integral curvature. Surfaces also form part of the modern field of analysis on metric spaces, which has led to several striking uniformization theorems for general classes of metric surfaces. In recent work with P. Creutz and D. Ntalampekos, we show how a geometric theory of surfaces in the same spirit as Alexandrov geometry can be developed under the single minimal geometric assumption of locally finite area. In particular, we use no assumption on curvature, or any of the previous assumptions from analysis on metric spaces. As an application, we obtain a new uniformization theorem for surfaces that is essentially the strongest possible for the non-fractal setting.

Colloquium: Erin Martin (William Jewell College)

Title: Applications of Linear Algebra and UPC Matroids

Abstract: Linear Algebra is a subject rich in applications. In this talk, we will discuss some of those applications and particularly how we can use linear algebra to help us solve an unsolved problem in graph theory. In 1973, Entringer posed the question: what simple graphs on n vertices have exactly one cycle of each length from 3 to n? These graphs are called uniquely pancyclic graphs or UPC graphs, and there are only seven known UPC graphs. To further study this idea, we will use matroids, which arise from the shared behaviors of vector spaces and graphs. In this way, we will determine properties that a matroid would need in order to be a UPC matroid and perhaps give rise to a UPC graph.

Careers in Math: Aaron Luttman (Pacific Northwest National Laboratory)

Speaker: Dr. Aaron Luttman Senior Technical Advisor at Pacific Northwest National Laboratory (PNNL)

Title: National Security Research and Careers in the US National Laboratories

Abstract: The US Department of Energy maintains 17 national laboratories, located around the country, that employ over 40,000 scientific and technical staff across all STEM disciplines. Many of the laboratories, along with additional production plants and experimentation sites, focus on research and development tailored to solving the nation’s greatest national security challenges. Supporting national security missions is among the most fulfilling professional trajectories that a scientist or engineer can follow, and, in this presentation, we will discuss how to pursue technical careers in the national laboratories. In addition to being fulfilling, the actual science and technology being discovered, explored, and developed in the national laboratories is world-leading. For this discussion, we’ll highlight some of the mathematics, physics, chemistry, and engineering in nuclear fusion research, artificial intelligence and machine learning, and materials science to demonstrate how undergraduate and graduate students, as well as career staff scientists, in the national laboratories are driving scientific discovery to solve national security problems.

Biography: Dr. Aaron Luttman is a Senior Technical Advisor at Pacific Northwest National Laboratory (PNNL), where his work focuses on understanding emerging challenges in nuclear security and developing scientific approaches to addressing them, both in support of the US nuclear weapons stockpile and in support of nuclear nonproliferation research. He has degrees in mathematics from Purdue University, the University of Minnesota, and the University of Montana, and his research has focused on computer vision and image processing. Over the last 15+ years, Aaron was a university professor, a research scientist, a Senior Technical Advisor to the US National Nuclear Security Administration (NNSA) in Washington, D.C., a personnel manager of teams of scientists and engineers, and a program manager. He is active in the mathematics community, as an officer of the Mathematical Association of America’s (MAA) Special Interest Group on the mathematics of Business, Industry, and Government, as a member of the Society of Industrial and Applied Mathematics’ (SIAM) Committee on Applied Mathematics Education, and as a member of the Embry-Riddle Aeronautical University’s Industrial Advisory Board for Computational Mathematics. He is also an active industrial partner in undergraduate research, providing research projects to the MAA’s PIC Math program, to numerous REU programs, and to Montana State University’s Data Science program. Aaron was the 2011 Outstanding New Teacher at Clarkson University in Potsdam, NY, and he has received numerous awards for his scientific contributions to the NNSA. He was an MAA Distinguished Lecturer, a SIAM Visiting Lecturer, and was a featured mathematician in 101 Careers in Mathematics (3rd Ed.). He also hires a lot of interns and staff and can help you get a job!

Applied Analysis Seminar: Anna Little (University of Utah)

Speaker: Anna Little (University of Utah)

Title: Robust Statistical Procedures for Finding Structure in Noisy Data

Abstract: This talk addresses two topics related to robust statistical procedures for analyzing noisy, high-dimensional data: (I) path-based spectral clustering and (II) robust multi-reference alignment (MRA). Both methods must overcome a large ambient dimension and lots of noise to extract the relevant low-dimensional data structure in a computationally efficient way. In (I), the goal is to partition the data into meaningful groups, and this is achieved by a novel approach which combines a data driven path metric with graph-based clustering. Using a data driven metric allows for strong theoretical guarantees, fast algorithms, and a flexible framework for balancing density and geometry considerations. Regarding (II), recent advances in applications such as cryo-electron microscopy have sparked increased interest in the mathematical analysis of multi-reference alignment (MRA) problems, where the goal is to recover a hidden signal from many noisy observations. The simplest model considers observations of a 1-d hidden signal which have been randomly translated and corrupted by high additive noise. We generalize this classic problem by incorporating random dilations into the data model, and explore multiple approaches to its solution based on translation invariant representations and nonlinear, data-driven unbiasing procedures.