NSF Research Experiences for Undergraduates Summer 2019
REU Site: Research Training for Undergraduates in Mathematical Analysis with Applications in Allied Fields
“My mind rebels at stagnation. Give me problems, give me work, give me the most abstruse cryptogram, or the most intricate analysis, and I am in my own proper atmosphere. But I abhor the dull routine of existence. I crave for mental exaltation.”
—Sherlock Holmes, in Arthur Conan Doyle’s “The Sign of the Four”
This REU Site, and all its work, is supported by the National Science Foundation under Grant Number NSF-DMS-1852288.
On May 16, 2020 our REU summer program received a special mention in this year’s “Virtual CAS Awards” ceremony, which can be viewed in the time range 18:14-18:53. Also, check out our department blog 2019 NSF Mathematics REU Summer Program Ends with Student Research Presentations and page 2 of the newsletter Math Moments.
Exciting News from our REU Students
Steven Jin presented a 15 minute talk based on his REU work at the Joint Mathematics Meetings in Denver, Colorado. Steven’s talk will be given on Saturday, January 18, 2020 in the AMS Contributed Paper Session on Number Theory, I (8:30 AM, Room 602, Meeting Room Level, Colorado Convention Center).
Steven Jin presented a 20 minute talk based on his REU work at the the PAlmetto Number Theory Series XXXII on Sunday, September 22, 2019. The conference was held at University of North Carolina at Charlotte.
This REU Site is supported by the National Science Foundation, and this support is gratefully acknowledged. Our 2019 Mathematics REU Summer Program provides an intensive eight-week research experience that prepares undergraduates for the rigors of graduate level research in mathematics. Advanced undergraduates are invited to Chattanooga to participate in collaborative research projects with experienced faculty mentors. Our Special Summer Colloquium Series, augmented by other group activities, presents opportunities for REU participants to interact with mathematicians and scientists who are actively pursuing theoretical, computational, and methodological research at the frontiers of mathematical analysis and its applications.
We offer training for undergraduates in a distinctive combination of research areas:
- Multiplicative number theory
- Probability and mathematical statistics
- Dynamical systems and numerical analysis
- Differential and difference equations
- Functional analysis and operator theory
It is the common thread of mathematical analysis that connects all these research areas. Possible topics include the zeta function, prime numbers, exponential sums, sequences, Gaussian analytic functions, random polynomials, resolvent identities, spectral shift functions, mathematical modeling using ordinary and partial differential equations, biological applications, and optimal control simulation.
We are excited to have eleven talented and energetic undergraduate researchers participate in our summer program. We have formed four research groups basing on the diversity of interest.
Dr. Andrew Ledoan’s research group
- Emily Eckels, Emory University
- Steven Jin, University of Maryland, College Park
- Brian Tobin, Harvard University
is currently studying the L1 means of exponential sums and applications of harmonic analysis to number theory. Collateral reading for this project includes
- A. Balog and I. Z. Ruzsa, A new lower bound for the L1 mean of the exponential sum with the Möbius function, Bull. London Math. Soc., 31 (1999), pp. 415–418.
- A. Balog and A. Perelli, On the L1 mean of the exponential sum formed with the Möbius function, J. London Math. Soc. (2), 57 (1998), pp. 275–288.
- R. C. Vaughan, The L1 mean of exponential sums over primes, Bull. London Math. Soc., 20 (1988), pp. 121–123.
Dr. Jin Wang is directing two research groups. The first research group
- Margaret Brown, University of Maryland, College Park
- Shan (Miko) Jiang, Mount Holyoke College
is investigating cholera transmission dynamics under the impact of disease control measures. Collateral reading for this project includes
- D. Posny, J. Wang, Z. Mukandavire, and C. Modnak, Analyzing transmission dynamics of cholera with public health interventions, Math. Biosci., 264 (2015), pp. 38–53.
- C. Yang and J. Wang, On the intrinsic dynamics of bacteria in waterborne infections, Math. Biosci., 296 (2018), pp. 71–81.
Dr. Wang’s second research group
- Nafisa Tabassum, York College, City University of New York
- Carolyn Valenti, Bucknell University
is developing and analyzing a new multi-scale model for cholera that links the between-host and within-host dynamics and their interaction with the environment. Collateral reading for this project includes
- C. Ratchford and J. Wang, Modeling cholera dynamics at multiple scales: environmental evolution, between-host transmission, and within-host interaction, Math. Biosci. Eng., 16 (2) (2019), 782–812.
- H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review, Vo. 42, No. 4 (2000), 599–653.
Dr. Roger Nichols’s research group
- Blake Allan, Baylor University
- Justin Kim, Vanderbilt University
- Gregory Michajlyszyn, University of Rochester
- Donald Rung, Sewanee: The University of the South
is studying Krein-type resolvent identities for singular Sturm–Liouville operators and their connection to the spectral shift function. Collateral reading for this work includes
- J. Eckhardt, F. Gesztesy, R. Nichols, and G. Teschl, Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional potentials, Opuscula Math., 33 (2013), pp. 467–563.
- W. N. Everitt and H. Kalf, The Bessel differential equation and the Hankel transform, J. Comp. App. Math., 208 (2007), pp. 3–19.
The following are new papers from our 2019 Mathematics REU Summer Program:
- E. Eckels, S. Jin, A. Ledoan, and B. Tobin, Linnik’s large sieve and the L1 norm of exponential sums, submitted for publication.
- M. Brown, M. Jiang, C. Yang, and J. Wang, Modeling cholera transmission under disease control measures, submitted for publication.
- N. Tabassum, C. Valenti, and J. Wang, A multi-scale model for cholera dynamics, preprint.
- S. B. Allan, J. Kim, G. Michajlyszyn, R. Nichols, and D. Rung, Explicit Krein resolvent identities for singular Sturm–Liouville operators with applications to Bessel operators, submitted for publication.
- Wednesday, June 12, 2019: Our first group dinner with REU mentors and students at Big River Grille & Brewing Works in Downtown Chattanooga. Our group photo at the dinner.
- Thursday, June 20, 2019: A little rest and relaxation. Undergraduate researchers from our Mathematics REU Summer Program will get together with others from the UTChattSat, ICompBIO, and URaCE URAP Programs in Benwood Auditorium (room EMCS 230, 6:30 PM) for a screening of The Most Unknown, an epic documentary film that sends nine scientists to extraordinary parts of the world to uncover the answers to some of humanity’s biggest questions. The central theme is communication. Many thanks to Dr. Daniel Loveless from the Department of Electrical Engineering for organizing this fun event.
- Wednesday, July 3, 2019: 2019 Pops on the River Independence Day celebration, Coolidge Park. Entertainment will start at 5:00 PM, Chattanooga Symphony and Opera at 8:00 PM, and fireworks at 9:45 PM.
- Sunday, July 7, 2019: Second group dinner at Champy's Famous Fried Chicken. Our group photo at the dinner.
- Thursday, July 18, 2019: Our research groups will square off in EMCS 422 at 3:00 PM in a variation of the game
made popular by the American Mathematical Society. Prizes include Dover books, of course!
REU students rival for first place in the friendly contest.
- Thursday, August 1, 2019: The REU students will be presenting their research at the 2019 REU Special Mathematics Summer Colloquium Series. Here is the talks schedule.
Dr. Joe Wilferth (Interim Dean of College of Arts and Sciences) meets with the REU participants.
Interim Dean Wilfreth encourages REU students to pursue graduate study in STEM fields.
- Photos of our research groups hard at work.
Emily and Brian contemplate the next step in their proof.
Margaret and Miko analyze nontrivial bacterial dynamics.
Blake, Greg, and Justin compute the spectral shift function for Bessel operators.
Steven explains his calculations for the L1 norm of an exponential sum.
Nafisa and Carolyn work to produce computer simulation for a multi-scale model for cholera.
Don reviews a special case.
This series is presented in conjunction with the NSF-funded 2019 Mathematics REU Summer Program. We are pleased to announce six confirmed colloquium talks by speakers who work in number theory, approximation theory, functional analysis, mathematical biology, dynamical systems, and computational mathematics. The talks are intended to be accessible to advanced undergraduate and beginning graduate mathematics majors. They will be announced to the Department of Mathematics and to the UTC campus. All talks will be given at 2:00 PM in EMCS 422 on their scheduled dates. Here is the final talks schedule:
*** Colloquium Series Talks Schedule ***
Title & Abstract
Abstract. The subject of unitary dilation addresses the following type of problem. Given operators on a Hilbert space H (complete inner-product space), can we find unitary operators (unitary = Hilbert space isomorphism) on a larger Hilbert space K containing H having algebraic properties similar to the original operators (e.g., commuting relations, moment conditions, etc.), such that their restrictions to H recover the original operators? Unitary dilation has many benefits in operator theory. For instance, the Sz.-Nagy Dilation Theorem provides a concise proof of von Neumann’s inequality. In this talk we will discuss many aspects of unitary dilation including classical and recent results, counterexamples, applications, and open questions. We will begin with a brief summary of Hilbert space operators. No prior knowledge of Hilbert spaces will be assumed.
Dr. Scott Alexander Atkinson, Vanderbilt University
The Hardy–Littlewood Circle Method and Applications
Abstract. The Hardy–Littlewood circle method, also known as the exponential sum method, is a central tool in analytic number theory. I will describe what this classical and important method is, and how it applies to various problems in additive number theory, including Vinogradov’s three primes theorem (the ternary version of the Goldbach conjecture) and Roth’s theorem in combinatorics. I will also discuss the limitation of the circle method, and the modern development in the past decade to overcome this.
Dr. Fernando Xuancheng Shao, University of Kentucky
Working @ the Interface: the Challenges and Opportunities of Mathematical Biology
Abstract. The questions that drive Mathematical Biology research and the quest for their answers make working at the interface of mathematics and biology interesting and exciting. However, it is also a place of great challenge and struggle where two very different fields are being melded together to synergistically create something greater than the sum of the parts. The amount of time that must be spent in the great “in-between” of the interface is often taken for granted; yet, it is here where skill, logical thinking, and creativity (not to mention patience!) are greatly required. Using examples from immunology, I will illustrate some of the challenges and opportunities that may be encountered when working at the interface.
Dr. Judy D. Day, The University of Tennessee, Knoxville
Introduction to Applied Knot Theory
Abstract. Mathematical knots are simple closed curves in space and can be classified using topological invariants. In the last decades, more and more knots are found in physical systems with important implications. For example, knots in proteins and DNA are related to disease and entanglement in polymer melts determines their viscoelastic properties with many industrial applications. Physical knots usually do not satisfy the strict mathematical definitions of knottedness, giving rise to a new area of study, Applied Knot Theory. In this talk we will have a short introduction to applied knot theory and see how it can be applied in practice to measure entanglement in polymers.
Dr. Eleni Panagiotou, The University of Tennessee at Chattanooga
Sparse Optimization for Tensors
Abstract. The popularity of sparse ell one norm optimization problem was due to Emmanuel Candes and Terrence Tao via compressed sensing. I will start by introducing the little ell one norm. Then, I will describe how and why these sparse ell one norm optimization problems are useful in solving today’s challenging problems in data science and machine learning. In particular, I will discuss their applications to tensor decomposition. I will also include numerical examples in surveillance video analysis and matrix and tensor completion. In this talk, one can observe the interplay of (multi)linear algebra, optimization and numerical analysis with applications in computer science.
Dr. Carmeliza Navasca, University of Alabama at Birmingham
Continuing the Fraction
Abstract. Continued fractions play a key role in number theory, especially in understanding how well we can approximate irrational numbers by rational numbers. They also play an important role in function theory, in understanding how well we can approximate analytic functions by irrational functions. We discuss a few of the main achievements of the theory.
Dr. Doron Lubinsky, Georgia Institute of Technology
- Undergraduate status with minimum grade point average of 3.0
- Rising senior—outstanding rising juniors or sophomores who meet the other selection criteria will be considered
- Majoring in mathematics, statistics, or related fields
- Have completed linear algebra, differential equations, and at least two upper level courses such as abstract algebra, matrix theory, analysis, complex variables, elementary number theory, probability, and numerical analysis
Women, minorities from underrepresented groups, first-generation college students, and students from predominantly undergraduate institutions with limited research programs are especially encouraged to apply.
- U.S. citizens and permanent resident students will receive airfare and transportation (up to $400) to Chattanooga, room and board, and a $4,000 stipend.
- Accepted students will be notified of awards by March 25, 2019.
- All application materials must be submitted by March 15, 2019.
- Late materials will not be accepted.
- Incomplete applications will not be accepted.
- Completed Application: Please visit MathPrograms.Org to apply.
- Curriculum Vitae: Please indicate clearly U.S. citizenship or permanent residency.
- Personal Statement: In approximately 500 words, please answer the following questions:
- Why are you applying to this UTC REU program?
- What are your research interests?
- What are your future career goals?
- What have you achieved toward these goals?
- College or University Transcript: A copy of your college or university transcript is required. Unofficial transcripts will be accepted.
- Faculty Letter of Recommendation: Two faculty letters of recommendation are required. Each letter should be written on university letterhead and submitted to MathPrograms.Org by the reference writers.
- Dr. Andrew Ledoan, Associate Professor of Mathematics
- Dr. Jin Wang, Professor and Unum Chair of Excellence in Applied Mathematics
- Dr. Roger Nichols, Associate Professor of Mathematics
Evaluation of this UTC REU program will be performed annually throughout the duration of the program by Dr. Sherry Marlow Ormsby, Executive Director, Office of Panning, Evaluation, and Institutional Research. Ms. Heather Shirley Heinlein will provide additional administrative support.
UTC is a national model for regional metropolitan universities and has a history of excellence in undergraduate education and research. The campus is at the heart of Downtown Chattanooga and within walking distance of such attractions as
Chattanooga is a gorgeous and thriving city located on the Southeastern corner of Tennessee. Nestled between the majestic mountainscapes of Missionary Ridge, Lookout Mountain, and Signal Mountain, the city is situated perfectly between Atlanta, Knoxville, Nashville, and Birmingham. It has the fastest Internet speed in the nation, with its community-wide fiber optic network that can turbocharge Internet speed up to 1,000 megabits per second. Chattanooga is an ideal location that offers many avenues for the development of collegial relationships and interactions between REU participants.
Please write to Andrew-Ledoan@utc.edu with any questions.