Mathematics Colloquium

Dr. Liz Vivas, Ohio State University

Abstract

Given a polynomial f and a number x, we are interested on understanding (in general) the orbit of x under the action of f: that is the sequence of numbers: x, f(x), f(f(x)), etc. We will survey relevant results when we take x to be a complex number. I will also explain the challenges when we take more general maps and spaces.

Bio Sketch

Dr. Vivas received her Ph.D. in mathematics from the University of Michigan under the direction of Prof. Berit Stensones. After graduating, she had postdoctoral positions at Purdue University at Lafayette, IN, Institute Henri Poincaré at Paris, and Universidad Federal Fluminense at Rio de Janeiro.

Hongyi Sheng, UC Irvine

Abstract

Recent laureates of the Nobel Prize in Physics have inspired people to notice the beauty of black holes and general relativity once again. On Apr. 10, 2019, astronomers captured the first image of a black hole. This was a huge progress, for once we know the area of the event horizon, we actually get an estimate of the mass of the black hole. In fact, earlier in 1973, R. Penrose conjectured that given the dominant energy condition, the total mass of a space-time which contains black holes with event horizons of total area A should be at least (A/(16π))^1/2. An important special case in Riemannian geometry is now known as the Riemannian Penrose inequality. This inequality was first established by G. Huisken and T. Ilmanen in 1997 using the inverse mean curvature flow for a single black hole and then by H. Bray in 1999 for any number of black holes, using the technique of a conformal flow. Later in 2009, H. Bray and D. Lee generalized Bray’s result to dimension up to 7. In this talk, we will mainly focus on the construction of Bray's conformal flow in dimension 3. We will see that the area of the horizon is a constant along the flow, and that the mass is decreasing, using the well-known positive mass theorem. The convergence to the Schwarzschild metric finally gives us the inequality.

Bio Sketch

Hongyi Sheng, Ph.D. student at UC Irvine, will earn his doctorate degree this May under the supervision of Richard Schoen. Then he will work as Stephen E. Warschawski Assistant Professor at UC San Diego. His main research interests are in general relativity and geometric analysis.

Kai-Wei Zhao, UC Irvine

Abstract

Jang's equation arose from an embedding problem when P.S. Jang attempted to use Geroch's approach to prove positive mass conjecture in general relativity. This approach was not developed due to lack of existence and regularity of solutions to Jang's equation. In 1981, Schoen and Yau introduced a regularization procedure to construct a smooth solution to Jang's equation, and first gave a complete proof of the positive mass theorem using a different approach. In this talk, we will briefly introduce the background knowledge of general relativity and related topics, including initial data sets, marginally outer trapped surfaces and mass.

Bio Sketch

Kai-Wei Zhao, Ph.D. student at UC Irvine, is going to get the doctor degree in math this May under the supervision of Richard Schoen. Then he will work at University of Notre Dame as Kenna Visiting Assistant Professor of Mathematics. His main research interests lie in geometric analysis, general relativity and geometric flow.

Dr. Vilma Mesa, University Of Michigan

Abstract

Recent laureates of the Nobel Prize in Physics have inspired people to notice the beauty of black holes and general relativity once again. On Apr. 10, 2019, astronomers captured the first image of a black hole. This was a huge progress, for once we know the area of the event horizon, we actually get an estimate of the mass of the black hole. In fact, earlier in 1973, R. Penrose conjectured that given the dominant energy condition, the total mass of a space-time which contains black holes with event horizons of total area A should be at least (A/(16\pi))^1/2. An important special case in Riemannian geometry is now known as the Riemannian Penrose inequality. This inequality was first established by G. Huisken and T. Ilmanen in 1997 using the inverse mean curvature flow for a single black hole and then by H. Bray in 1999 for any number of black holes, using the technique of a conformal flow. Later in 2009, H. Bray and D. Lee generalized Bray’s result to dimension up to 7. In this talk, we will mainly focus on the construction of Bray’s conformal flow in dimension 3. We will see that the area of the horizon is a constant along the flow, and that the mass is decreasing, using the well-known positive mass theorem. The convergence to the Schwarzschild metric finally gives us the inequality.

Bio Sketch

Dr. Vilma Mesa is professor of education and mathematics at the University of Michigan. She is also a Faculty Associate at the Center for the Study of Higher and Postsecondary Education. She investigates mathematics instruction and resource use in post-secondary settings, with an emphasis on community colleges. She has conducted several analyses of textbooks and evaluation projects on the impact of innovative mathematics teaching practices for students in science, technology, engineering, and mathematics. She collaborated in the development of an instrument to assess the quality of videos of math instruction in postsecondary settings. In her collaborative work, she is studying the use of free, open source, open access, and interactive textbooks for college mathematics courses and developing an instrument to assess the mathematical knowledge needed to teach college algebra at community colleges. She is Associate Editor of Educational Studies in Mathematics. She has published over 40 articles in mathematics education. Prior to her career in education, Mesa was a systems programmer for the ministry of finances in Colombia and for the district of Bogotá, and a computing systems advisor for a large construction and hospitality firm in Colombia. She has a B.S. in computer sciences and a B.S. in mathematics from the University of Los Andes in Bogotá, Colombia, and a master's and a Ph.D. in mathematics education from the University of Georgia.

Dr. Neha Murad, Calico Life Sciences

Abstract

There are a multitude of pathological conditions that affect human health, yet we currently lack a predictive model for most diseases, and underlying mechanisms that are shared by multiple diseases are poorly understood. We leveraged baseline clinical biomarker data and long-term disease outcomes in UK Biobank to build prognostic multivariate survival models for over 200 most common diseases. We construct a similarity map between biomarker-disease hazard ratios and demonstrate broad patterns of shared similarity in biomarker profiles across the entire disease space. Further aggregation of risk profiles through density based clustering showed that biomarker-risk profiles can be partitioned into few distinct clusters with characteristic patterns representative of broad disease categories. To confirm these risk patterns we built disease co-occurrence networks in the UK Biobank and US HCUP hospitalization databases, and compared similarity in biomarker risk profiles to disease co-occurrence. We show that proximity in the biomarker-disease space is strongly related to the occurrence of disease comorbidity, suggesting biomarker profile patterns can be used for both predicting future outcomes as well as a sensitive mechanism for detecting under-diagnosed disease states.

Bio Sketch

Dr. Neha Murad is an Applied Mathematician with a Ph.D. in Biomathematics and a minor in Statistics from North Carolina State University. She is a senior data scientist in the Translational Computing team at Calico Life Sciences, where she uses a combination of epidemiological, bioinformatics and QSP models and tools to understand the biology that controls aging and lifespan. Prior to Calico, Neha was a postdoc at GSK where she was part of the Pharmacokinetics (PK) team and used systems models in combination with machine learning approaches to build Physiologically Based Pharmacokinetic (PBPK) Models.

When not building math models, Neha loves to volunteer for events promoting mathematics in women and minorities. She is a AAAS IF/THEN Ambassador and is deeply involved in STEM outreach for middle and high school girls.

Dr. Kimberly Ayers, CSU San Marcos

Abstract

Consider a continuous function f that maps a compact set to itself. By the Brouwer fixed-point theorem, such a function must have a fixed point: a point x₀ such that f(x₀) = x₀. Since f has the same domain and range, we can consider compositions of f with itself: f²(x), f³(x), f⁴(x), and so on. In this talk, we'll consider what happens to these iterates, and more specifically, what happens to particular points as we apply f to them over and over again. Sometimes, we'll see that this sequence converges to the fixed point; other times, it may move away from a fixed point. It may be sucked into an oscillator, or it may bounce all over the place. We will also examine how we can change the stability of systems to manipulate them to behave how we want - to make unstable fixed points stable again.

Bio Sketch

Dr. Kimberly Ayers is an Assistant Professor in the Mathematics Department at California State University, San Marcos. She's previously held positions at Carroll College in Helena, Montana, and Pomona College, in Claremont, California. She received her Ph.D. from Iowa State University in 2015. Her research is in the areas of dynamical systems, chaos theory, and ergodic theory, with special interest in applications to biological systems. She is also passionate about promoting diversity and equity in the mathematical sciences. When she's not doing math, you can find her riding her bike or rock climbing.

Dr. Priya Prasad, University of Texas at San Antonio

Abstract

The COVID-19 pandemic has had an enormous impact on instruction at institutes of higher education, but amid the myriad challenges, it has also provided instructors with opportunities. In this context, we were awarded an IUSE grant to encourage the use of evidence-based instructional practices in an introductory mathematics course; the instructors of this course, having moved all their lectures to pre-recorded videos, were thus open to trying new instructional strategies in class. In this talk, I will describe the features of the project that leveraged the opportunities presented by the move to virtual instruction. Additionally, I will discuss the preliminary results from interviews with participants, including some surprising consequences of the disruption caused by the COVId-19 pandemic. This project is supported by the National Science Foundation (#2116187).

Bio Sketch

Dr. Priya V. Prasad is an Associate Professor of Mathematics at University of Texas at San Antonio. She got her Ph.D. in Mathematics (with an emphasis in Math Education) from the University of Arizona in 2014. She is primarily interested in investigating mathematical knowledge for teaching at multiple levels.

Dr. Lara Alcock, Loughborough University, UK

Abstract

This talk will cover a series of research studies on mathematical reading and how we might improve it in undergraduates. The first study used eye-tracking to investigate and compare reading behaviours in undergraduates and mathematicians - I will present evidence that mathematical reading differs from ordinary reading in specific ways, and that students engage in key comprehension behaviours less than mathematicians. The second study investigated the effects of self-explanation training on students' reading behaviours. Using a combination of eye-tracking and experimental studies, we established that this training encouraged students to read in a manner more like that of professional mathematicians, and that this had positive effects on their mathematical proof comprehension.

Bio Sketch

Lara Alcock is a Reader in Mathematics Education at Loughborough University. She collaborates with colleagues, Ph.D. students and project students to conduct research on mathematical thinking and learning, specializing in reasoning among undergraduate mathematics students and professional mathematicians. She has also written four research-informed books for undergraduates and popular mathematics readers: How to Study as a Mathematics Major, How to Think about Analysis, Mathematics Rebooted, and How to Think about Abstract Algebra.

Dr. Amelia Stone-Johnstone, CSU Fullerton

Abstract

Research and state legislation have been the impetus for academic reform at postsecondary institutions nationwide. The corequisite model of academic support has been championed as an effective institutional structure for providing greater access to gateway mathematics and overall college degree/certificate completion at these institutions (e.g., Kashyap & Mathew, 2017; Logue et al., 2016; Richardson 2021). During this talk I will discuss the departmental change initiative at one institution towards adopting the corequisite model. Leveraging the four-frame model of organizational change (Bolman & Deal, 2008), as adopted for higher education (Reinholz & Apkarian, 2018), I will illustrate how an instructional team transformed College Algebra at a public four-year institution. Through the establishment of the core goals of course coordination, increased collaborative learning, and entwining metacognitive activities within the corequisite curriculum, the team of instructors developed a College Algebra corequisite course to better support their student population.

Bio Sketch

Dr. Stone-Johnstone is an Assistant Professor in the Mathematics department at California State University, Fullerton. Her research focuses on developing and assessing academic support mechanisms that support student learning in gateway mathematics courses. In addition, her research area includes equity, teacher professional development, and institutional change.

Brittany L. Marshall, Rutgers University

Abstract

As part of the larger COURAGE study, this presentation will discuss some of our findings which characterize forms of benevolence in calculus and precalculus instructors' perceptions of instructional practices and how they contrast with Black and Latinx students' perceptions of the same practices as potentially racialized and gendered. Instructors perceived themselves as lacking both agency and responsibility to reform calculus instruction in ways that supported student retention in the calculus sequence, which they largely attributed to not having enough time to teach required topics and concerns about high student enrollment in these courses.

Bio Sketch

Brittany L. Marshall is a third-year Ph.D. in Education student in Mathematics Education. She is interested in middle/high school mathematics teaching and learning as well as math identity development, specifically among Black girls. Prior to Rutgers, Brittany taught middle school and high school math in Chicago for almost a decade and practiced architecture in both Chicago and DC. She holds a master's degree in architecture from North Carolina State University and a bachelor of architectural studies from University of Illinois at Urbana-Champaign.

Priyanga Ganesan, Texas A&M University

Abstract

Quantum graphs are an operator space generalization of classical graphs that have appeared in different branches of mathematics including operator systems theory, non-commutative topology and quantum information theory. In this talk, I will review the different perspectives to quantum graphs and introduce a chromatic number for quantum graphs using a non-local game with quantum inputs and classical outputs. I will then show that many spectral lower bounds for chromatic numbers in the classical case (such as Hoffman's bound) also hold in the setting of quantum graphs. This is achieved using an algebraic formulation of quantum graph coloring and tools from linear algebra.

Bio Sketch

Priyanga Ganesan is a fifth year doctoral student at Texas A&M University working with Dr. Michael Brannan. Her research explores the interaction between operator algebras and quantum information theory through the study of quantum graphs and non-local games. Before starting her Ph.D., she completed an integrated Bachelors-Masters degree in Mathematics at the National Institute of Science Education and Research in India. She is actively involved with several diversity programs in mathematics, and has also served as the President of the TAMU Association for Women in Mathematics graduate student chapter, board member of the worldwide Operator Algebras Mentor Network, and co-founded an I DP program for Math graduate students at Texas A&M.