Mathematics Colloquium

Dr. Evan T. R. Rosenman, Claremont McKenna College

Abstract

Statistics and data science play a crucial role in modern presidential campaigns. Models are used to estimate the probability that each registered voter casts a ballot, and the probability that each voter supports a given candidate. Moreover, analysts' recommendations play a key role in the broader campaign strategy, including the decision to invest resources in particular states.

In this talk, I will discuss my experience leading the turnout modeling effort on the Biden for President 2020 campaign, including the unique challenges for campaign analytics in the midst of a global pandemic. I will also highlight some of the broader research questions that emerged out of the campaign, including questions about polling bias and forecasting error.

Bio Sketch

Evan T. R. Rosenman is an Assistant professor of Statistics at Claremont McKenna College. His research focuses primarily on problems in data science and causal inference, with applications to political science and public health. He is particularly intrigued by problems involving hybridizing observational and experimental data to better estimate causal effects, and by challenges in modern electioneering, such as ecological inference and prediction calibration. He earned his Ph.D. in Statistics from Stanford University, advised by Art Owen and Mike Baiocch. He also completed a postdoctoral fellowship at the Harvard Data Science Initiative, advised by Kosuke Imai and Luke Miratrix.

Dr. Juan Pablo Mejía-Ramos, Rutgers University

Abstract

Proof has several different roles in mathematical practice and in the mathematics classroom. One of those roles is to explain why a given proposition holds (as opposed to, for example, demonstrating that it does). Indeed, explanation is a particularly important role of proof in the mathematics classroom, but not one that it always fulfills. Philosophers of mathematics and mathematics educators have attempted to identify what makes a proof explanatory, or what makes a proof of a given proposition more explanatory than another one. While there have been promising, recent developments in this area, explicit criteria for what makes a proof (more or less) explanatory are still being debated. In this presentation, I illustrate a novel empirical approach to investigating the notion of explanation in proof-based mathematics. In particular, I report on a couple of studies aimed at investigating the extent to which mathematicians and undergraduate students share a notion of explanatoriness in proof-based mathematics (both among and between themselves). Results from these studies suggest this is a fruitful approach for studying what makes a proof (more or less) explanatory in mathematics.

Kyeong Hah Roh, Arizona State University

Abstract

Fostering logical thinking is a paramount objective, as it underpins the ability to convey ideas and construct rigorous arguments in mathematics. This presentation delves into the critical landscape of undergraduate students' engagement with logic within mathematical contexts. Drawing upon the outcomes of empirical studies conducted by my research teams, I will illuminate the intricate tapestry of undergraduate students’ reasoning about logic. This presentation encompasses three fundamental facets of logical thinking: (1) the idiosyncratic meanings students attribute to quantifiers, (2) students' logical consistency while evaluating mathematical statements and accompanying arguments, and (3) students' learning of logical principles for proof of conditional statements. By sharing key insights and research findings, I aim to offer valuable perspectives to enhance teaching and learning experiences in mathematical logic.

Bio Sketch

Kyeong Hah Roh is an associate professor of mathematics education in the School of Mathematical and Statistical Sciences at Arizona State University in Tempe, Arizona, USA. She earned her Ph.D. in mathematics (differential geometry) from Seoul National University in 2000 and her Ph.D. in mathematics education from the Ohio State University in 2005. She served as Program Chair for SIGMAA on RUME (2012-2013) and a member of the Analysis Course Study Group of the MAA Committee on the Undergraduate Program in Mathematics (CUPM) in 2012. Her research program focuses on undergraduate students' reasoning about logic and its role in learning mathematical concepts and proofs. National Science Foundation has funded her work on designing a research-based real analysis curriculum (DUE-0837443), modeling undergraduate students' reflection and abstraction of proof structures in transition to proofs courses (DUE-1954613), and generating a research-informed transition to a mathematical proof curriculum (DUE-2141925). Kyeong Hah enjoys listening to classical music and playing the piano in her spare time.

Dr. Yvonne Lai, University of Nebraska, Lincoln

Abstract

What are the best activities you have used as a teacher? What made them the best? In this colloquium, I will discuss enduring problems of secondary mathematics teacher education and design principles for activities that address these enduring problems.

Bio Sketch

Yvonne Lai is an expert in the field of mathematical knowledge for teaching. Her current research program seeks to improve the education of secondary mathematics teachers and early mathematics majors by bridging disciplinary perspectives from mathematics and education. Lai currently chairs the MAA’s Committee on the Mathematical Education of Teachers, is a current editor of PRIMUS, and is an incoming editor of the Notices of the American Mathematical Society.

Dr. Daniel Appelö, Virginia Tech

The 1958 Christmas issue of The New Scientist contained two pages with puzzles posed by Sir Roger Penrose (and his father S. L. Penrose). One of these puzzles asks the reader to design a smooth closed reflecting surface (a mirror) which contains two regions and has the property that a source of light placed in one region cannot be seen from the other region. This "room" has become known as the unilluminable room and there are now numerous fascinating solutions to the problem.

The original puzzle assumes that the light is described by rays (a so-called billiards problem) so that the light cannot "bend around corners." Here we model light by solving the Helmholtz equation with a point source in one of the regions of the room and study (among other things) how dark the other region of the room actually is as we change the frequency of the light-source.

To model the unilluminable room we introduce and discuss the WaveHoltz iteration for solving the Helmholtz equation. This method makes use of time domain methods for wave equations to design frequency domain Helmholtz solvers. We show that the WaveHoltz iteration results in a positive definite linear system whose solution gives the solution to the (indefinite) Helmholtz equation.

Bio Sketch

Before joining the CMDA program and the Department of Mathematics at Virginia Tech Daniel was a faculty at CMSE and the Department of Mathematics at MSU, the University of Colorado Boulder and at The University of New Mexico in Albuquerque. Before that Daniel was a postdoc in Mechanical Engineering at Caltech with Tim Colonius. Prior to Caltech Daniel worked at Lawrence Livermore National Laboratory in the Applied Math. group at the Center for Applied Scientific Computing. At LLNL he was a part of the Serpentine project where he and Anders Petersson, Bjorn Sjögreen developed massively parallel numerical methods for seismology. Together with Anders he also developed high order accurate embedded boundary methods for the wave equation. While at LLNL Daniel also worked with Bill Henshaw on simulations of converging shocks and on a parallel overset grid solver for solid mechanics.

Daniel was a Hans Werthen (the founder of Electrolux) Prize postdoc at the Department of Mathematics and Statistics at UNM where he worked with Tom Hagstrom on a general formulation of perfectly matched layer models for hyperbolic-parabolic systems and Hermite methods.

Daniel obtained a PhD in Numerical Analysis at NADA, KTH under the supervision of Gunilla Kreiss. His thesis considered different aspects of the perfectly matched layer method. It turned out that the well-posedness of general pml models could always be guaranteed by a parabolic complex frequency shift and that stability can be established for a certain class of hyperbolic systems.

Dr. Roummel Marcia, UC Merced

Abstract

In everyday life, we are constantly inundated with digital signals and, in particular, images, from web browsing and video meetings to movie streaming and video gaming. These signals can carry large volumes of data, but the information they contain are often redundant, meaning they have inherent structures that can be exploited to facilitate storage and transfer. In this talk, I will discuss some of the mathematics underlying image processing techniques (specifically linear algebra and optimization) and describe how they can be used in several important applications.