Mathematics Colloquium

Juan Pablo Mejía-Ramos, Rutgers University

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.

Yvonne Lai, University of Nebraska, Lincoln

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.