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Mathematics Colloquium

Upcoming Colloquium

Potential dependence of the density of states for Schrödinger operators
Dr. Chris Marx, Oberlin College

January 22, 2021
12:00pm-1:00pm via Zoom

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Meeting ID: 910 8240 4817
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Abstract

In this talk we will explore the dependence of the density of states for Schrödinger operators on the potential. The density of states characterizes the averaged spectral properties of a quantum system. Formally, it can be obtained as an infinite volume limit of the spectral density associated with finite-volume restrictions of a quantum system. Such limit is known to exist for certain quantum mechanical models, most importantly for Schrödinger operators with periodic and random potentials.

Following ideas by J. Bourgain and A. Klein, we will consider the density of states outer measure (DOSoM) which is well defined for all Schrödinger operators. We will explicitly quantify the parameter dependence of the DOSoM by proving a modulus of continuity with respect to the potential (in $L^\infinity$-norm and weak topology). This result is obtained for all discrete Schrödinger operators on infinite graphs and captures the geometry of the graph at infinity.

This talk is based on joint work with Peter Hislop (University of Kentucky).

About the Presenter

Chris Marx is an Associate Professor of Mathematics at Oberlin College. His research interests lie at the intersection of analysis and mathematical physics, more specifically in spectral theory of Schrödinger operators. He earned a masters degree in theoretical and physical chemistry at the University of Vienna, Austria in 2007, and a PhD in mathematics from UC Irvine in 2012 (with S. Jitomirskaya). From 2012 to 2014, he held a postdoctoral position at Caltech as Harry Batement Instructor with Barry Simon. Since 2014 he has taught at Oberlin College, with promotion to Associate Professor in June 2020. In his spare time, Chris is a passionate amateur musician (harpsichord and voice), with a keen interest in early music and historical performance technique.

The Mathematics Colloquium is a unique opportunity for students to learn about new developments in mathematics and what mathematics and statisticians do after they graduate. Hosted by the Department of Mathematics and Statistics at California State University, Long Beach, the weekly meetings invite guests from universities, research laboratories, and industry to present and discuss current topics in mathematics. All students are encouraged to attend.

Schedule for Spring 2021

Unless otherwise noted, the times of the colloquia are 12:00-13:00.

4/30/21 (9:00-10:00): Title TBA

Tao He (San Francisco State University)

4/23/21: Title TBA

Aditya Adiredja (ASU)

4/16/21: Title TBA

Robin Wilson (CPP)

4/9/21: Title TBA

Prof. Ovidiu Munteanu (University of Connecticut)

3/26/21: Title TBA

Ying-fen Lin (Queen's University Belfast)

3/19/21: Title TBA

Naneh Apkarian (ASU)

3/12/21: Title TBA

Ben Hayes (UVA)

3/5/21:Title TBA

Mario Banuelos (Fresno State)

2/26/21: Title TBA

Belin Tsinnajinnie (SFCC)

2/5/21: Title TBA

Prof. Luis Leyva (Vanderbilt)

1/29/21: Title TBA

Prof. Hsioa-Fan Liu (Tamkang University)

1/22/21: Potential dependence of the density of states for Schrödinger operators

Prof. Chris Marx, Oberlin College

Abstract

In this talk we will explore the dependence of the density of states for Schrödinger operators on the potential. The density of states characterizes the averaged spectral properties of a quantum system. Formally, it can be obtained as an infinite volume limit of the spectral density associated with finite-volume restrictions of a quantum system. Such limit is known to exist for certain quantum mechanical models, most importantly for Schrödinger operators with periodic and random potentials.

Following ideas by J. Bourgain and A. Klein, we will consider the density of states outer measure (DOSoM) which is well defined for all Schrödinger operators. We will explicitly quantify the parameter dependence of the DOSoM by proving a modulus of continuity with respect to the potential (in $L^\infinity$-norm and weak topology). This result is obtained for all discrete Schrödinger operators on infinite graphs and captures the geometry of the graph at infinity.

This talk is based on joint work with Peter Hislop (University of Kentucky).

About the Presenter

Chris Marx is an Associate Professor of Mathematics at Oberlin College. His research interests lie at the intersection of analysis and mathematical physics, more specifically in spectral theory of Schrödinger operators. He earned a masters degree in theoretical and physical chemistry at the University of Vienna, Austria in 2007, and a PhD in mathematics from UC Irvine in 2012 (with S. Jitomirskaya). From 2012 to 2014, he held a postdoctoral position at Caltech as Harry Batement Instructor with Barry Simon. Since 2014 he has taught at Oberlin College, with promotion to Associate Professor in June 2020. In his spare time, Chris is a passionate amateur musician (harpsichord and voice), with a keen interest in early music and historical performance technique.

Schedule for Fall 2020

12/4/20: Calling Bull - rethinking the Q course

Dr. Carrie Diaz Eaton, Bates College

Abstract

We are in a new age of digital information. Information is a form of power, and our students consume and produce more unfiltered information than ever. They need agency as individuals and tools as members of a future workforce to ethically and responsibly process this information. What is the role of mathematics instruction in helping students in their role as a digital citizen? I talk about my journey to developing an information literacy course using Open Educational Resources, including Calling Bull, Figure of the Day, and RStudio. This course serves as a forum to think meaningfully about probability, data analysis, and data visualization, a gentle introduction to programming, and a context to examine the interplay of information, power, and social justice. It also asks students to use these tools to explore and develop their own agency as a digital citizen.

About the Presenter

Dr. Carrie Diaz Eaton's research in undergraduate interdisciplinary STEM education is grounded in community network theory and analysis. As an Associate professor of Digital and Computational Studies at Bates College in Maine, Dr. Diaz Eaton co-leads a number of digital community projects such as QUBES Director of Partnerships (Quantitative Undergraduate Biology Education and Synthesis) and Math Mamas. Carrie Diaz Eaton currently serves as the Chair for the Committee for Minority Participation in Mathematics for the Mathematical Association of America [MAA], is a MAA blogger for MathValues, and serves on the Editorial board of PRIMUS and CourseSource. She has also served as the past Program Chair and Electronic Communications Chair of BIO SIGMAA, as Education Subgroup Chair for the Society of Mathematical Biology, and for the editorial board for Letters in Biomathematics.

In 2012, Dr. Diaz Eaton was an MAA Project NExT Fellow, and in 2018 was selected as a Linton-Poodry SACNAS Leadership Institute Fellow. In 2020, Dr. Diaz Eaton was awarded the Society for Mathematical Biology John Jungck Excellence in Education Prize to recognize her for her work in interdisciplinary computational and mathematical biology education and mentorship. Dr. Diaz Eaton is also a proud 1st generation Latinx - her father is from Peru. She is also a mother. Dr. Diaz Eaton values the complex interplay at the intersection of her identities, professional activism in STEM education, and her research.

11/20/20: Non-local games from an operator algebraic perspective

Kari Eifler, Texas A&M

Abstract

Non-local games lie in the intersection of operator algebras and quantum information theory. In this talk, I will summarize some results of the operator algebraic approach to non-local games. I will especially focus on two examples: the graph isomorphism game and the metric isometry game, emphasizing the role of C*-algebras play in the study of these games. Finally, I will discuss how non-local games are able to shed light on the quantum symmetries of classical objects.

About the Presenter

Kari Eifler is a PhD candidate at Texas A&M with plans to graduate in the summer of 2021. Her research lies within the areas of operator algebras and quantum information theory, with her dissertation focusing on quantum symmetries relating to non-local games. She previously completed her Masters of Mathematics at the University of Waterloo and her Bachelor of Science at the University of Alberta. During quarantine, she's enjoyed the opportunity to improve her cooking and baking skills.

11/13/20: The Chromatic Polynomial of a Graph

Dr. Josh Hallam, Loyola Marymount University

Abstract

The chromatic polynomial was originally defined by George Birkhoff as a tool to solve the famous four-color problem. Although it was not used in the eventual proof of the four-color problem, it has been shown to possess many interesting and unexpected properties. We will survey some of the properties and along the way see connections of the chromatic polynomial with other mathematical objects.

No background knowledge is required.

About the Presenter

Josh Hallam obtained his PhD from Michigan State University in 2015. After that, he was a teacher-scholar postdoctoral fellow at Wake Forest University. He is currently an assistant professor at Loyola Marymount University. His research interests include enumerative and algebraic combinatorics.

11/06/20: Inference for Time-Course Count Data with Application to RNA-Sequencing Analysis

Dr. F. Jay Breidt, Colorado State University

Abstract

In this talk, I illustrate how many "small" but powerful statistical ideas (regression, interactions, likelihood ratios, multiple comparisons, mixtures, Monte Carlo, and more) can be combined to address a complex scientific problem. Experiments that longitudinally collect RNA sequencing (RNA-seq) data can reveal dynamic patterns of differential gene expression. Most existing tests are designed to distinguish among conditions based on overall differential patterns across time, but a variety of composite hypotheses may be of more scientific interest. Further, existing methods may lack power and some fail to control the false discovery rate (FDR). We propose a new model and testing procedure to address these issues simultaneously. Conditional on a latent Gaussian mixture with evolving means, we model the data by negative binomial regression, introduce a general testing framework based on the proposed model and show that the proposed test enjoys the optimality property of maximum average power. The test allows not only identification of traditional differentially-expressed genes but also testing of a variety of composite hypotheses of biological interest. We establish the identifiability of the proposed model, implement the proposed method via efficient algorithms, and demonstrate its good performance via simulation studies. The procedure reveals interesting biological insights when applied to data from an experiment that examines the effect of varying light environments on the fundamental physiology of a marine diatom.

This is joint work with Meng Cao, Novartis; Wen Zhou, Department of Statistics, Colorado State University; and Graham Peers, Department of Biology, Colorado State University

About the Presenter

Jay Breidt, Professor of Statistics at Colorado State University, has research interests in survey sampling, time series, nonparametric regression, and uncertainty quantification for complex scientific models. He received his PhD at Colorado State University in 1991 and spent nine years at Iowa State University before returning to Colorado State in 2000. Breidt has been an associate editor for eight journals and reviews editor for Journal of the American Statistical Association and The American Statistician. He has served on six review committees for the National Academy of Sciences. He is past Chair of the American Statistical Association National Committee on Energy Statistics, has served two terms on the Federal Economic Statistics Advisory Committee, and is currently a member of the Census Scientific Advisory Committee. Breidt is an elected Fellow of the American Statistical Association and an elected Fellow of the Institute of Mathematical Statistics.

10/30/20: Geometry of Turbulent Flows and the 3D Navier-Stokes regularity problem

Dr. Aseel Farhat, Florida State University

Abstract

We describe several aspects of an analytic/geometric framework for the three-dimensional Navier-Stokes regularity problem, which is directly inspired by the morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of three-dimensional turbulence. Among these, we present our proof that the scaling gap in the 3D Navier-Stokes regularity problem can be reduced by an algebraic factor within an appropriate functional setting incorporating the intermittency of the spatial regions of high vorticity.

About the Presenter

Aseel Farhat finished her PhD work in the Department of Mathematics at University of California Irvine in 2012. She was a Zorn Postdoctoral Fellow in the Department of Mathematics at Indiana University Bloomington between 08/2012 and 08/2015, and later joined the Department of Mathematics at University of Virginia as a Whyburn Instructor (Postdoc) till 08/2018. She is currently an Assistant Professor in the Mathematics Department at the Florida State University.

Aseel's main research area is fluid dynamics and analysis of non-linear PDEs. Her interests include well-posedness of geophysical models of ocean and atmosphere, continuous data assimilation (downscaling algorithms), feedback control, regularity criteria for the 3D Navier-Stokes equations, and geometry of turbulent flows.

10/23/20: From Classical Mechanics to Symplectic Geometry

Dr. Rui Wang, UC Berkeley

Abstract

Symplectic geometry originated from classical mechanics. In this talk we first give a mathematical introduction to classical mechanics and quantum mechanics. Then we explain how the formalism of Symplectic Field Theory introduced by Eliashberg-Givental-Hofer, which is the most fruitful theory in symplectic geometry in the past 20 years, beautifully matches such theoretical mechanics framework. This talk is accessible to a general audience.

About the Presenter

Rui Wang is working at UC, Berkeley. She got her Ph.D. degree in mathematics from University of Wisconsin-Madison. Her mathematical interests lie in symplectic geometry, contact geometry and mathematical physics. She enjoys research and teaching.

10/16/20: A primer on survey statistics or "why do I need to use those weights?"

Dr. Jean Opsomer, Westat

Abstract

Many important observational datasets in social sciences and public health are obtained through statistical surveys. A key strength of surveys is that the results of statistical analyses of survey data, properly performed, are statistically valid for inference about the overall population, without the need of additional assumptions. This is in contrast with non-survey observational data, whose representativeness needs to be justified on non-statistical grounds. However, analysis of survey data requires the use of specialized (weighted) approaches, which are not always available in standard statistical software. We will present an overview of the principles of survey design and estimation, and describe how survey-weighted analyses are performed and interpreted.

About the Presenter

Jean Opsomer is Vice President at Westat, where he directs several large-scale survey and modeling projects for federal agencies and other clients. Previously, he spent 23 years as a faculty member in statistics, the majority of which at Iowa State University. His recent research has focused on the introduction of shape-constrained and nonparametric methods in survey estimation and on several interdisciplinary projects with survey components on a range of topics (higher education, public health, nutrition, employment, fisheries management, methane emissions, forest health, and agricultural erosion). The author or coauthor of 65 peer-reviewed articles, he has introduced a number of influential novel statistical methodologies into survey estimation. Dr. Opsomer is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics, and an Elected Member of the International Statistical Institute.

10/9/20: Changing Healthcare

Dr. Ian Duncan, UC Santa Barbara

Abstract

Healthcare budgets in most countries seem to be out of control, with the U.S. heading to 20% of GDP and other countries not far behind in terms of rapidity of increases. Recently, predictive analytics, big data and artificial intelligence have been proposed as a solution that will enable practitioners to identify high risk populations and conditions earlier and intervene more effectively with patients. Is this hope justified or is it another example of mis-placed optimism? What will it take for predictive analytics to make a significant impact on the cost and value of healthcare? We propose that three factors are required to work together to effect transformation: Payment Reform; Predictive Analytics and Behavioral Economics. In the future, more outcomes risk will be transferred to providers and consumers of healthcare services. As risk professionals, actuaries will be a significant contributor to this transformation.

About the Presenter

Ian Duncan is Adjunct Professor of Actuarial Statistics at the University of California Santa Barbara and president of Santa Barbara Actuaries Inc. a healthcare analytics firm. Dr. Duncan holds a graduate degree in Economics from Balliol College, Oxford and a PhD in actuarial statistics from Heriot-Watt University, Edinburgh. He is a fellow of a number of actuarial organizations. He is active in public policy and healthcare reform, and served on the board of directors of the Commonwealth of Massachusetts Health Insurance Connector Authority from 2007-2014 and the Society of Actuaries, 2012-5. He also serves on the advisory boards of several start-up healthcare companies. He is the author of more than 80 peer-reviewed papers, and several books and book chapters. His latest book, a second edition of "Healthcare Risk Adjustment and Predictive Modeling" (Actex Publications) was published in May 2018.

9/18/20: Resolution analysis in some scattering problems and super-resolution in certain scenarios

Dr. Yat Tin Chow, UC Riverside

Abstract

In this talk, we explore image resolution and ill-posed-ness of inverse scattering problems. In particular, we would like to discuss how certain properties of the inclusion might induce high-resolution imaging. We first explore the super-resolution phenomenon with certain particular high contrast inclusion. We then discuss how local sensitivity (and resolution) around a point is related to the extrinsic curvature of the surface of inclusion around the point. Along the line, we also discuss concentration of plasmon resonance (in a certain manner) at boundary points of high curvature leveraging the Heisenberg picture of quantization and quantum ergodicity first derived by Shnirelman, Zelditch, Colin de Verdiere and Hellfer-Martinez-Robert. This is a joint work with Habib Ammari (ETH Zurich), Hungyu Liu (CityU of HK), Keji Liu (Shanghai Key Lab), Jun Zou (CUHK).

About the Presenter

Yat Tin Chow is currently an Assistant Professor in the Department of Mathematics. He received his Ph.D. in Mathematics from the Chinese University of Hong Kong. He joined the faculty in UC Riverside after being a CAM assistant adjunct professor in Department of Mathematics in UCLA. His major research direction is applied mathematics. Dr. Chow's current research interests includes resolution analysis and enhancement of imaging from boundary measurements of various physical quantities, e.g. electric current, acoustic wave, light intensity, etc. He is also interested in computational methods of medical imaging and tomography, e.g. Electrical Impedance Tomography. Dr. Chow's other fields of interest include both theoretical and numerical aspects of large scale optimization method, computations of control methods and conflict modeling in high dimensional systems, as well as transportation plans and games between large populations in the mean field, and different phenomena that arise from this setting. If you are interested in his search areas, kindly visit Dr. Chow's personal website.

9/11/20: Fast Graph-based Algorithms for Analyzing Protein-Protein Interaction Networks

Dr. Junyuan Joanne Lin, Loyola Marymount University

Abstract

This research aims to predict proteins' functions from protein-protein interaction (PPI) networks. The PPI networks we study include physical and genetic interactions between labeled and unlabeled proteins. This allows us to predict proteins' unknown functions based on the function labels of closely interacted "neighbors". In this presentation, I will present our award-winning graph-based algorithms that achieve the best prediction accuracy worldwide in the 2016 Disease Module Identification DREAM Data Mining Challenge. We define the diffusion state distance (DSD) metric, which sets appropriate distances to measure proteins' proximity on PPI networks as well as many other close-knit networks including social and energy networks. Fast algorithms, such as the unsmoothed aggregation algebraic multigrid method with random projections, are adopted to compute the DSD efficiently. Based on random walks combining with random projections, we propose graph-based methods to construct k-nearest-neighbor (kNN) graphs under the DSD metric for function prediction. We test our proposed algorithms on different networks to demonstrate that the computational cost of the algorithms is nearly optimal.

9/4/20: Property/Casualty Insurance Ratemaking

Prof. Janet Duncan, FCAS, FSA, MAAA

Abstract

Insurance is a contractual promise to reimburse policyholders for future losses. Consumers often comment that they don't understand their insurance rates – it all seems very mysterious to them. But in reality, creating insurance rates is very logical when broken down into its component parts. The fundamentals of insurance ratemaking are very similar to the pricing of many other products, i.e., understanding cost and determining a target profit load. The major difference is that for many products, the cost is easily determined from the manufacturing process. However, for insurance, the cost involves significant uncertainty about the future. This presentation will introduce the audience to fundamental insurance principles and the mystery behind insurance ratemaking.

About the Presenter

Professor Janet Duncan has over 30 years of property/casualty financial analysis experience, including commercial and personal lines reserving and pricing, financial and capital modeling, planning, and management reporting. Janet's work experience includes six years as CNA's senior vice president and signing actuary, responsible for $17 billion of property/casualty reserves, including standard commercial lines, specialty lines, and discontinued operations. Prior to CNA, Janet worked at XL Capital, serving in roles of increasing responsibility including executive vice president and chief finance officer of XL Insurance Europe and Asia. She also worked with PricewaterhouseCoopers LLP (consulting and audit support), and served in various actuarial roles with Aetna Life & Casualty where she began her insurance career. Janet has a bachelor's degree in Math/Actuarial Science from the University of Connecticut. She has served on many actuarial committees including the CAS Committee on Professionalism Education, the CAS Committee on Reserves, the AAA IFRS Task Force, the AAA Opinion Seminar Committee, and the SOA Strategic Planning Task Force. She is now working as a lecturer at the Department of Applied Probability and Statistics at UC, Santa Barbara.

8/28/20: Creativity-in-Progress Reflections (CPR) on Proving and Problem Solving

Dr. Gülden Karakök, University of Northern Colorado

Abstract

Numerous reports, policy and standards documents, and research studies emphasize the importance of creativity. For example, the recent report from the World Economic Forum noted that creativity at work is one of the top-three demanded skills, and that it "has jumped from 10th place to third place in just five years" (Schöning & Witcomb, 2017, para. 12). Within the domain of mathematics, similar emphases are made by mathematicians, mathematics education researchers and policy/standards makers. For example, the Mathematical Association of America's (MAA) CUPM Curriculum Guidelines (Schumacher & Siegel, 2015) for majors in the mathematical sciences states that "a successful major offers a program of courses to gradually and intentionally lead students from basic to advanced levels of critical and analytical thinking, while encouraging creativity and excitement about mathematics" (p. 9). In this talk, I will briefly summarize some of the research on mathematical creativity at the K-16 levels and introduce the work of the Creativity Research Group focusing on undergraduate mathematics courses. Our research group aims to explore ways in which undergraduate students' mathematical creativity can be fostered and explicitly valued in mathematics courses that include proof-construction and/or problem solving activities. I will introduce the Creativity-in-Progress Reflections (CPR) on Proving and Problem Solving tools that we designed. These formative assessment tools were created to enhance mathematical creativity (of users) while facilitating proof-construction and problem-solving heuristics as well as fostering metacognition. With two categories, Making Connections and Taking Risks, these formative assessment tools aim to develop mathematical discourse centered around aspects of creativity related to fluency, elaboration, flexibility, and originality. I will provide some examples of how one might implement these tools in various mathematics courses as well as discuss some illustrative empirical examples from our research studies.

Previous Colloquia

The Mathematics Colloquium Archive has the Colloquia from previous semesters.