Department of Mathematics - University Of Illinois · 2014 Department of Mathematics ... assign a...

2 0 1 4Depar tment of Mathematics

College of Liberal Arts and Sciences • University of Illinois at Urbana-Champaign

The pentagram map Faculty Mentor: Rinat KedemGraduate Student: Panupong Vichitkunakorn The pentagram map was introduced by Richard Schwartz in 1992 on the projective plane: It is illustrated in the first picture. It maps a polygon with n vertices to another one. If n=5 the map is the identity, and if n=6 the map is 2-periodic. In general, it is a discrete integrable system. Here, we illustrate that it is a special case of a very well-studied discrete integrable system, the octahedron recursion, whose cluster-algebra quiver appears in the second figure. The latter is closely related to classical integrable systems via the discrete Hirota equation and tau functions of the KP hierarchy. To get the pentagram map from this system, simply wrap it around a torus, under proper identification of variables. This toric version gives the Boussinesq equation in the continuum limit. Pictured from left: map on polygons; the quiver on the infinite plane; the quiver on the torus.

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Department of Mathematics • University of Illinois at Urbana-Champaign • 1409 W. Green, Urbana, IL 61801 • Tel: (217) 333–3350 • Email: [email protected] • www.math.illinois.edu

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Kinship Accuracy: Comparing Algorithms for Big Genetic Data Faculty Mentor: Kay KirkpatrickUndergraduate Student: Danni Sun In genetic epidemiology, disease association is a popular test of whether alleles are correlated with a disease. In this project, we analyze methods for disease association on very large pedigrees, focusing on the strength of the linkage analysis and genome-wide association studies. In the diploid Wright-Fisher Model, each diploid individual has a paternal and a maternal allele; identity states are 4-node graphs on the alleles of two individuals, whose edges represent the common ancestry of two alleles. Then through Monte Carlo sampling, we get the probability distribution of identity states respect to the size of the population and the number of generations. Comparing the kinship coefficient computed as the expectation of the identity states probabilities and computed through dynamic programing recursion, the small error illustrates that kinship coefficients would be an effective and efficient tool to analyze relatedness in disease association.

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F E B R U A R Y 2012F

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Splines on polyhedral complexes Faculty Mentor: Hal SchenckGraduate Student: Michael DiPasquale A spline is a piecewise polynomial function on a simplicial or polytopal complex: assign a polynomial to each polytope of the complex, in such a way that when two polytopes meet, the corresponding polynomials glue to give a function possessing a prescribed number of continuous derivatives. Splines arise in many areas of applied mathematics, including approximation theory, geometric modeling, and solving systems of PDEs. One of the complications that arises in the polytopal case that does not manifest in the simplicial case is the existence of linear splines (each polynomial is a linear form) which have a large support, but which cannot be obtained as linear combinations of other linear splines with ‘local’ support. The image shows the graph of such a function which is a piecewise linear function over the displayed polytopal complex.

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St. Patrick’s Day March Equinox

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Ash Wednesday

Purim

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Mock theta functions Faculty Mentor: Scott AhlgrenGraduate Student: Nickolas Andersen Mock theta functions were introduced by S. Ramanujan in his famous “last letter” to G. H. Hardy in 1920. For the next eighty years, these strange functions were shrouded in mystery, but today they are realized as the “holomorphic parts” of certain functions related to modular forms. Of particular interest is the arithmetic information that is encoded in the Fourier coefficients of mock theta functions and mock modular forms. These functions are important objects in number theory and combinatorics, but they also have applications in many other branches of mathematics and physics, including the study of quantum black holes! Pictured above are “random walks” generated by the coefficients of Ramanujan’s mock theta function f(q) modulo 3 (top left), 4 (top right), and 7 (bottom). These coefficients appear to be “random” modulo 3 and 4. Modulo 7, however, the coefficients are biased toward being 0 mod 7, which explains the tendency for the graph to move farther to the right than we would expect from a true random walk. In fact, it is known that the coefficients of f(q) are biased toward being 0 mod p for every prime p larger than 3. Nick Andersen is a third year PhD student. He has completed three REGS projects (receiving a Gold Award for his presentation at REGS Day 2012) and written four papers since he started here.

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Palm Sunday Passover

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Creative Blocking Faculty Mentor: Bruce ReznickUndergraduate Students: Jeremy DeJournett, Daniel Hirsbrunner; Graduate Student: Ilkyoo Choi We define “creative blocking” which is a process that creates one polygon from another in the following way: build a block (typically, a square) on the sides of the given polygon, and connect the adjacent vertices of these squares to generate the new polygon. The image above is the 6th iterate of an isosceles right triangle oriented counter-clockwise. We investigated the relationship among the perimeters of the iterates of a fixed polygon p. We showed that the perimeters satisfy a certain recurrence, and moreover, our result generalizes when blocks other than squares are used as well. In particular, given a polygon with three points, we determined what the growth rate of the perimeter is depending on the shape of the polygon.

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Mother’s Day

Memorial Day


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3-D Models Faculty Mentor: Steven BradlowUndergraduate Student: Hiroshi Fuii Using the Illinois Geometry Lab’s 3-D printer, Hiroshi Fuii designed and printed colorful plastic models to illustrate how circles, ellipses, hyperbolas, and parabolas arise as conic sections. When stacked together the pieces of the model form a cone. Each successive piece is obtained by slicing the cone with a plane at an ever-decreasing angle to the cone axis of symmetry, starting at 90 degrees and progressing to zero. The cross sections revealed by the slices range through the full set of ‘conic sections.’ Professor Steven Bradlow used the models in a course on ‘Curves’ that he taught at the African Institute of Mathematical Sciences in January 2013.

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Father’s Day June Solstice


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Lithium Batteries: Structure and Efficiency Faculty Mentors: Jayadev Athreya, Yuliy Baryshnikov, Anil HiraniUndergraduate Students: Justin Faber, Keshav Regmi, Anthony Yunker, Cheng Wang, Jialiang Wang Graduate Students: Anton Lukyanenko, Grace Work, Kaushik KalyanaramanA lithium-ion battery charges as lithium ions flowing from the anode to the cathode of the battery bind to buffers in the electrode. Once there is a high enough concentration of lithium ions bound to the tin cells, the battery is fully charged. We are interested in finding an optimal distribution of buffers to minimize the time it takes to charge the battery. To do this, we created a model of the flow of lithium-ions through a cross-section of the cathode of the battery. This cross-section is represented as a grid and it’s dual graph where each section of the grid is a node in the graph. We perform iterations of lithium flow (in which 1 unit of lithium moves one unit of the grid) through the battery until the cumulative charge of the buffers is 95% of the total capacity of buffers in the battery. In order to minimize the charge time of the battery, we need to minimize the number of iterations. Our research has focused on testing all possible configurations of buffers in the battery. This is a brute force strategy and we have utilized the Campus Cluster in order to cover all the configurations.

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Independence Day


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Random Points, Broken Sticks, and Triangles: Adventures in Geometry, Probability, and Calculus in n Dimensions Faculty Mentor: A.J. Hildebrand Undergraduate Students: Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Ananya Uppal If a stick is broken up at two points, chosen at random along its length, what is the probability that the pieces obtained form a triangle? This question, which first appeared in an 1854 examination at Cambridge University and has since become a classic probability puzzle, served as motivation and starting point for an IGL project in 2012-13. The project team investigated, both theoretically and experimentally, generalizations of this question to broken sticks with n pieces. In particular, using higher-dimensional integrals, they obtained an exact formula for the probability that one can form at least one triangle from the pieces of an n-piece broken stick. Members of the team have presented their work at undergraduate conferences at the University of Texas, the Rose-Hulman Institute of Technology (where Lkhamsuren won the Best Presentation Award), Ohio State University, and at the Undergraduate Student Session of the MAA MathFest. Team members also engaged in a variety of outreach activities, including visits to local middle and high schools, and participation in the UI Public Engagement Symposium.

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Illinois BioMathematics Program Faculty Mentor: Zoi RaptiThe Illinois BioMathematics Program trains undergraduate students majoring in mathematics and biology to investigate original research questions combining skills and concepts from both disciplines. Under joint mentorship from faculty members at the School of Integrative Biology and the Department of Mathematics, students take a module-organized course that is offered in the fall semesters, participate in seminars and listen to lectures by the instructors and to talks by guest speakers, and take a project-based course that is offered in the spring semesters. They spend ten weeks in the summer working on the research problems in a faculty member’s lab. The various research groups engage in a diverse set of activities, such as field trips and data collection, designing and running experiments in the lab, analyzing the data the collected, performing analytical and numerical calculations in the mathematical models they developed, and writing research papers and preparing presentations for the local and national meetings they will attend.

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Icy Detail of the Mandelbox Faculty Mentor: Bruce BerndtGraduate Student: Dan Schultz Dan Schultz’s work in number theory was recently accepted for publication in one of the leading math journals in the world. In this paper, he gives a complete description of Ramanujan’s cubic theory of theta functions begun by Ramanujan almost 100 years ago. Dan also has a complete description of the cubic analogue of the Jacobi theory derived by Jacobi about 180 years ago. The above image is related to an IGL project with Nishant Nangia and Jeremy Tyson from the Spring of 2012. The fractal itself is defined as the prisoner set of some three dimensional iteration, in a similar fashion to the ordinary Mandelbrot set. Trillions of calculations involving ray tracing and lighting bring out the striking details of this set.

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Induced Saturation Faculty Mentors: Mike Ferrara, Douglas B. WestGraduate Students: Elyse Yeager, Sarah Behrens, Catherine Erbes, Mike Santana, Derrek Yager; Doob Postdoc: Theodore MollaGraph saturation studies structures that almost, but don’t quite, contain a forbidden subgraph. We considered a variation introduced by Martin and Smith to investigate structures that almost, but don’t quite, contain a forbidden induced subgraph. Such a structure has forbidden induced subgraphs lurking just around every corner--adding or deleting any edge gives rise to one. Every piece carries the potential to complete the whole. Pictured in the center is a paw-induced-saturated graph. The icosahedron (left) and dodecahedron (right) are C4 induced saturated and C8 induced saturated, respectively.

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Veterans Day

Thanksgiving Day


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Do numbers play dice? Visualizing order and chaos in number theory through random walks Faculty Mentor: A.J. HildebrandUndergraduate Students: Yiwang Chen, Yusheng Feng, Wenmian Hua, Natawut Monaikul, Mateusz Wala, Dylan Yang, Tong Zhang; Graduate student: M. Tip Phaovibul Many properties of the natural numbers can be encoded as sequences of 1’s and -1’s. On the surface, such sequences often show no obvious pattern and indeed seem to behave much like randomly generated sequences. In this project we seek to gain a deeper understanding of the behavior of such sequences by constructing and investigating certain “random walks” formed with these sequences. These random walks provide a natural way to visualize the degree of randomness inherent in a sequence and to detect, and possibly explain, hidden patterns, but they can also open up new mysteries that defy explanation. In Fall 2012 we focused on random walks associated with quadratic residue sequences, a particularly intriguing and mysterious case that is closely related to a famous result of Gauss. In Spring 2013, we investigated random walks defined in terms of the digital expansion of an integer and uncovered some interesting fractal-like patterns. Members of the team have given talks on this work at undergraduate conferences at the Rose-Hulman Institute of Technology, Ohio State University, and at the Undergraduate Student Session of the MAA MathFest. They also presented a poster at the 2013 University of Illinois Undergraduate Research Symposium which received a Best Poster Honorable Mention award.

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November 2014

Hanukkah

Now more than ever, the Department of Mathematics relies on the generosity of its alumni and friends. Here are some ways you can invest in our future.

Give to our current use funds:

• Mathematics Partnership Fund Unrestricted gifts have the highest impact. They can be applied to our most critical needs, which change from year to year as new opportunities emerge.

• Mathematics Legacy Fund Support our bold plan to renovate Altgeld Hall and Illini Hall to create a new collaborative environment for mathematics learning and discovery.

• Illinois Mathematics Scholarship Fund Scholarships enable and encourage the most promising admitted undergraduate mathematics students to pursue a superb education at Illinois.

Create an endowment: • Faculty Endowments

Help recruit and develop excellent faculty by supporting their scholarly and educational work.

• Endowed Graduate Fellowships Provide doctoral students with time and resources to pursue innovative ideas.

• If you are interested in creating a new endowment, please contact one of the people listed at right.

Invest in the future by enhancing our existing endowments: • Gene H. Golub Endowment Fund

Support graduate student research experiences.

• Mathematics Research Experience Endowment Fund Support undergraduate student research experiences.

• Please consider giving to the Math Excellence Fund where contributions can be pooled to create endowments.

Join in our future by supporting the Department of Mathematics

in its educational and scholarly missions.

Make your gift today atmath.il l inois.edu/giving/

or contact

Matthew AndoChair, Department of Mathematics

(217) 244-2846 [email protected]

Sheldon KatzDevelopment Committee Chair

Department of Mathematics(217) 265-6258 [email protected]

this year’s calendar highlights research by our students. Undergraduates have numerous opportunities to engage in research including the Illinois geometry Lab and the BioMathematics program. the department’s innovative REgS program (Research Experiences for graduate Students) helps graduate students make the transition to phD-level research.

this calendar was designed by tori Corkery for the Department of Mathematics at the University of Illinois at Urbana-Champaign © 2014.

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Ahlgren, Scott Albin, pierre Ando, Matthew Athreya, Jayadev Balogh, Jozsef Baryshnikov, Yuliy Bauer, Robert Bergvelt, Maarten Berndt, Bruce Boca, Florin Bradlow, Steven Bronski, Jared D’Angelo, John Deville, Lee Dey, partha Di Francesco, philippe van den Dries, Lou Dutta, Sankar Dunfield, Nathan Duursma, Iwan Erdogan, Burak Feng, Runhuan Fernandes, Rui Loja

Ford, Kevin Francis, george gorvett, Rick haboush, William hieronymi, philipp hinkkanen, Aimo hirani, Anil hur, vera Mikyoung Ivanov, Sergei Johnson, paul Junge, Marius Kapovich, Ilya Katz, Sheldon Kedem, Rinat Kerman, Ely Kirkpatrick, Kay Kirr, Eduard-Wilhelm Kostochka, Alexandr Laugesen, Richard Leininger, Christopher Lerman, Eugene Li, Xiaochun McCarthy, Randy

Merenkov, Sergiy Mineyev, Igor Muncaster, Robert Nevins, thomas Nikolaev, Igor palmore, Julian Rapti, Zoi Rezk, Charles Reznick, Bruce Rosenblatt, Joseph Ruan, Zhong-Jin Schenck, hal Solecki, Slawomir Song, Renming Sowers, Richard tolman, Susan tumanov, Alexander tyson, Jeremy tzirakis, Nikolaos Wu, Jang-Mei Yong, Alexander Zaharescu, Alexandru Zharnitsky, vadim

Department of MathematicsUniversity of Illinois at Urbana-Champaign

1409 W. Green Street, Urbana, IL 61801

www.math.illinois.edu Tel. (217) 333-3350 Email [email protected]

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