Research Experience for Undergraduates - REU - 2011 Abstracts of Talks REU students will speak on their research on the following dates at 3:00 PM in 2866 East Hall.   July 13 Name: Justin Campbell     Faculty Mentor: Mitya Boyarchenko Title: Weil's "analytic" interpretation of the Hilbert symbol Abstract: The Weil index is an invariant which assigns a complex 8th root of unity to a quadratic form over a local field. We will discuss a proof of the existence of the Weil index which, at least in the non-Archimedean case, that uses only the basic theory of Fourier transforms on locally compact abelian groups. We will also state Weil's formula for the Hilbert symbol in terms of the Weil index. Name: David B. McMillon      Faculty Mentor: Mark Conger Title: An Analysis of the Position Probability Distributions of Trilateration and Triangulation for Extremely Deep Space Navigation Abstract: This study investigates the nature of space vehicle position probability distributions resulting from each general method specifically for the extremely deep-space application. Important assumptions are: 1) the positions of the emitters are known; 2) pseudorange measurements, as well as angle measurements, are made with some reliable detector and have a systematic error that is normally distributed with mean zero and variance a predetermined function of the relative distance to the emitter. _______________ July 20 Name: Alex Cope     Faculty Mentor: Mark Conger Title: How people shuffle playing cards Abstract: The dovetail shuffle is the most common method of randomizing a deck of cards in the United States. We look at how this shuffle permutes a deck of cards, and the process of creating an accurate model for how people shuffle.   Name(s): Joseph Billian, Sifat Rahman      Faculty Mentor: Bhargav Bhatt Title: F-pure thresholds F-pure thresholds are important invariants in algebraic geometry which measure the singularities of polynomial functions by studying their behavior over finite fields. We compute these numbers in two distinct families and see what observations can be made/proved about the data.   Name: Rebecca Gleit      Faculty Mentor: Victoria Booth Title: Modeling human sleep patterns Abstract: Human sleep is composed of three main sleep stages, which exhibit fairly regular cycling throughout the night. We examine these patterns found in human sleep, and study how best to mathematically model the reported characteristics and the trends found in the data. ________________ July 27 Name: Daniel Smolkin      Faculty Mentor: Ralf Spatzier Title: Geodesics of periodic metrics on the real plane Abstract: The theory of geodesics on the flat two-dimensional torus is well-known. For instance, all geodesics on the flat torus are unbounded when lifted to the real plane via a universal covering. For my project I am studying geodesics on the real plane equipped with various Riemannian metrics that are periodic under integer translations. Such metrics on the plane induce metrics on the two-dimensional torus in a natural way. I looked at several classes of periodic metrics on the plane and managed to prove that all of their geodesics are unbounded. I numerically computed some geodesics for other periodic metrics, and my results suggest that the measure of the set of initial conditions yielding bounded geodesics is nonzero. Currently, I am trying to construct a periodic metric close to the flat metric with a closed geodesic. I am also trying to use the KAM theorem to extend my results about periodic metrics on the real plane with unbounded geodesics.   Name: Michael Norman and Nathan Golovich      Faculty Mentor: Charlie Doering Title: How What Kills You Makes You Stronger Abstract: The obvious goal of any biological species is to increase its number as much as possible. There are many possible approaches and we are interested in determining what actions will be beneficial. In particular we examine a phenomenon known as Population Increase through Mutual Predation (PIMP), a process where two species eat each other depending on age. We will discuss general facts about dynamical systems, introduce several ecological models and discuss their potential effect on real world evolution. Name: Hilary Monaco      Faculty Mentor: Trachette Jackson Title: Modeling Population Dynamics of Intratumoral Angiogenesis Abstract: Modeling the proliferation and death of cells is of critical importance in many areas of mathematical biology. Often, modelers write down standard phenomenological representations of growth without much thought to their underlying assumptions and resulting consequences. As an example, the logistic equation is a popular choice for describing population growth dynamics; and this formulation has been applied in many models of tumor growth and angiogenesis. Upon careful inspection, however, the logistic equation makes significant assumptions about the cell populations' behavior that do not hold for tumors and their vasculature. We build a model of co-cultured populations of tumor cells and vascular endothelial cells from first principles and examine the dynamics of the system. Our model uses the law of mass action to describe these cells' interactions with vascular endothelial growth factor (VEGF) and incorporates the effects of nutrient concentration and receptor recycling on the population density. ________________ August 3 Name: Joe Hidakatsu      Faculty Mentor: Angela Kubena Title: Automorphisms of Right-Angled Coxeter Groups Abstract: A right-angled Coxeter group is a group generated by elements of order 2 such that its only other defining relations indicate which pairs of these generators commute. If G is a right-angled Coxeter group, then the automorphism group of G is generated by three types of automorphisms. The first is the symmetries of what is known as the defining graph of G. The other two are what we call transvections and partial conjugations. We will discuss how to identify these automorphisms given the group’s defining graph. We will then use this to show how to find the outer automorphism group of the free product of certain right-angled Coxeter groups. Name: John Holler      Faculty Mentor: Brett Hemenway Title: Optimizing the Number of Players in a t out of n Perfect Ideal Linear Secret Sharing Scheme. Abstract: Often we find it useful to break up information in order to protect it from either being destroyed and stolen. Secret sharing offers a method of breaking a secret into shares that can later be combined to construct the secret - this is the basis of a secret sharing scheme. A scheme is said to be perfect if for a fixed t, any t shares construct and any t-1 learn nothing. It is linear if the secret is a linear combination of the shares and is ideal if the size of the shares are the same as the secret. But, we find that after fixing our t and the size of our finite field F, we necessarily restrict the number of total shares n we can have in a Perfect Ideal Linear (PIL) scheme. Research has led to results in the t=3 case, but bounding n has proven difficult in general. Our ongoing research has been concerned with the t=4 case.   ________________ August 10 Names: Molly Logue and Dominic Spadacene Faculty Mentor: Mike Zieve Title: Slowly growing backward orbits of a point under a rational function Abstract: Let f(x) be a degree-d rational function with complex coefficients, viewed as a function from the Riemann sphere (S^1) to itself. We show that, if d>1 and r>3, then any point P in S^1 for which #f^{-r}(P)>1 must in fact satisfy #f^{-r}(P) >= ceiling((d^r)/(d^2+d+1)). Conversely, we describe all functions for which equality holds; such functions exist for each d and r. The proofs involve the Riemann--Hurwitz genus formula and a study of the ramification of f(x) and its iterates. This result improves previous results of Silverman and Faber--Granville. As a corollary, we show that if f(x) is a rational function with rational coefficients has degree d>2 and f(f(x)) is not a polynomial, then there are only finitely many rational numbers, c, for which f(f(f(c))) is an integer. ====================================================== Name: Mark Liu Faculty Mentor: Hala Al Hajj Shehadeh Title: Evolution of one-dimensional crystal surfaces Abstract: In this talk, we will introduce a problem from material science. We are interested in crystal relaxation below the roughening temperature. Our 1-dimensional model is based on the Burton-Cabrera-Frank model, where the surface of a crystal is made of steps and terraces. Atoms diffuse on the terraces, and attach to and detach from the step edges. We investigate two profiles: A monotone step train connecting two facets, and an infinite bunch. Self-similarity has been observed for these configurations (as well as others). Starting with the step equations of motion, we are interested in analyzing the continuum consequences of the model, as well as the asymptotic self-similar behavior. ====================================================== Name: Kirill Serkh Faculty Mentor: Danny Forger Title: Reducing Jet lag: Optimal Light Schedules for Resetting the Circadian Clock Abstract: Suppose you are an out of state student from Arizona attending the University of Michigan. If you fly home over winter break and land at, say, 7 pm local time, your internal clock will tell you that it's 10 pm. Now if you saw that your watch were giving you the time 3 hours late, you would want to reset it, probably using some external cue. But how can you reset a biological clock? Researchers have found that light can be used to shift the internal time, called phase, of the circadian clock. The effects of light on the human circadian clock have been studied extensively, mostly using graphs called phase response curves, which map the timing of a light stimulus to the resulting shift in phase. However, if we want to find the best light schedule to correct this 3 hour jet lag, phase response curves can give only a limited answer. A Van der Pol oscillator model for the circadian clock has existed since 1990, and in its most recent iteration provides an excellent representation of experimental results. In this talk we look at ways to optimize light schedules on the model of the circadian clock. We will review methods which have been already been used, and discuss recent attempts to solve the problem. We will talk about, among other things, direct and indirect optimization methods, Pontryagin's maximum principle, and numerical methods for solving optimization problems of this sort.   ________________ August 17 Name: Geoffrey Iyer Faculty Mentor: Steven Miller (Williams College) Title: Constructing Generalized Sum-Dominant Sets Many of the biggest problems in additive number theory (such as Goldbach's conjecture, the Twin Prime conjecture, and Fermat's last theorem) can be recast as understanding the behavior of sums of a set with itself. A sum-dominant set is a finite subset of the integers, A, such that |A+A|>|A-A|. We expect such sets to be rare, as addition is commutative, and subtraction is not. Though it was believed that the percentage of subsets of {0,...,n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved a positive percentage are sum-dominant. We generalize their result to deal with the many different ways of taking sums and differences of a set. We first prove that |A+...+A-A-...-A|>|A+...+A-A-...-A| a positive percent of the time (where the number of plus/minus signs is different for each side of the inequality). Previous approaches proved the existence of infinitely many such sets given the existence of one; however, no method existed to construct such a set. Using techniques from probability and additive number theory, we develop a new, explicit construction for one such set, and then extend to a positive percentage of sets. We extend these results further, finding sets that exhibit different behavior as more sums/differences are taken. We find the limiting behavior of kA=A+...+A for an arbitrary set A as k tends to infinity and an upper bound of k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if A, A+A, ..., kA are all sum-dominant. Numerical searches were unable to find even a 2-generational set (heuristics indicate the probability is at most 10^-9, and almost surely significantly less). We prove the surprising result that for any k a positive percentage of sets are k-generational, and no set can be k-generational for all k. ====================================================== Name: Xiaotong Suo Faculty Mentors: Joel Smoller and Paul Federbush Title: Numerical analysis------Calculation Process for Stars with Magnetic Current. Abstract: Main functions: G1hat G2hat, F, P, I, and our independent variables are r,z, r’,z’ ; Constants: E, and P0 Programming language: C++ Given five main functions, the goal is to use a program to run through these functions numerically, thereby finding the rules inside functions. How to calculate them correctly and efficiently is a strong concern, since G1hat and G2hat require much more time and space than the other functions. The first mission is to translate the two dimensional points into single index for the purpose of calculating G1hat and G2hat. The following step is to calculate F, P by loops starting from the default values of F and P, since they are both recursive functions. After calculating five main functions, the next step is to graph the characteristic function I in order to visualize the changes after each iteration of recursion. In addition, the 2D plots for P can be generated to provide another way to look at the changes after each iteration. At the end of this program, our goal is to find proper sets of E and P0 in order to make our graphs grow during the first few iterations, but eventually stabilize. ====================================================== Name: Brandon Cloutier Faculty Mentors: Benson Muite and Jared Whitehead Title: Pseudo-Spectral methods and Convection Not all ordinary differential equations (ODE) and partial differential equations (PDE) have known solutions. In this talk we discuss methods for approximating solutions to PDEs using pseudo-spectral methods with three time-stepping schemes, forward and backwards Euler (first order accuracy) and the second order implicit midpoint rule, and use these to solve the linear heat equation and the nonlinear Allen-Cahn equation. Finally, we will describe the Navier-Stokes equations with Boussinesq approximations that we use to numerically simulate Rayleigh-Benard convection and internally heated convection on supercomputers. ====================================================== Name: Paul Rigge Faculty Mentor: Benson Muite Title: Computational Techniques for Nonlinear PDEs Abstract: In this talk, we will discuss using computers to solve and analyze PDEs. Inparticular, we will discuss the sine-Gordon equation and the nonlinear Schrodinger equation as model nonlinear PDEs. We will describe spectral methods and implicit Runge-Kutta schemes and their efficient implementation in FORTRAN. Additionally, we will talk about parallel computation on a large cluster and the challenges of scaling programs to larger systems. Finally, we will discuss tools and techniques for the visualization of large data sets. ________________ August 24 Name: Kit Clement Faculty Mentor: Patrick Boland Title: Face Vectors of Sphere Products A triangulation of a manifold is a collection of simplices that is diffeomorphic to the manifold. They can tell us about the manifold’s topology. For instance, the face vector of a triangulation, whose entries count the number of simplices, determines the Euler characteristic. Little is known about face vectors in general. They have been classified for surfaces and a short list of higher dimensional manifolds. The main goal of this project is to examine face vectors of T3, S2 x S2, and S1 x S4. In this talk, we will discuss triangulations of surfaces, introduce the Dehn--Sommerville Equations, and define the product triangulation. The product triangulation has proven useful in determining a minimal vertex triangulation of S2 x S2 (Lutz 2005). As a highlight, we will give a combinatorial proof that the Euler characteristic is multiplicative for a product of surfaces. ====================================================== Name: Nick Wasylyshyn Faculty Mentor: Brett Hemenway Title: Authentication and Pseudo-Random Functions using Learning Parity with Noise When two parties have information and they want to verify that they are the same without disclosing their secrets, we use what is called authentication: one player will encode her secret using a One-Way Function (OWF), and send the function with the output to the other player. A OWF is a function that is easy to evaluate but hard to invert. One such OWF is called Learning Parity with Noise, or LPN, which efficiently utilizes random binary matrices for authentication protocols. Another faculty of a OWF is to construct a Pseudo-Random Generator (PRG), from which we can create a Pseudo-Random Function (PRF). We will discuss our efficient method of creating a weak-PRF, and possibilities for extending it to a strong-PRF. ====================================================== Names: Sudarshan Balakrishnan and Gong Chen Faculty Mentors: Benson Muite and Peter Olver Title: Dispersive Quantization Abstract: The study of linear dispersive partial differential equations with piecewise constant periodic initial data leads to quantized structures at times which are rational multiples of pi and fractal profiles at irrational multiples of pi. Graphs for its solution at different times provide interesting insights into the behavior of the Fourier Series at rational and irrational multiples of pi. We will present an overview of these results for the linear and non-linear Schrodinger wave equation and the KdV on the torus and the real line. ====================================================== ________________ August 31 Name: David Bruce Mentor: Roman Vershynin Title: A Multivariate Median in Banach Spaces and Applications to Robust PCA With the rise in prominence of high dimensional data, multivariate measures of center have become very important. In this paper we focus on one multivariate measure of center - the geometric median, which is defined as the minimizer of the sum of distances to the data points. We study the quantitative robustness of the geometric median. Showing that for a non-degenerate distribution of $N$ points, altering $k$ points can only change the median by at most $O(k/N)$. Taking advantage of this robustness we introduce a robust form of Principle Component Analysis (PCA), which is based on what we call the median covariance matrix. Since there are several natural matrix norms, we look at the notion of the geometric median in general Banach Spaces. We conclude by proving that the geometric median is robust in all uniformly convex Banach spaces. ====================================================== Name: Paul Lewis Mentor: Christopher Lyons Title: The Weil Conjectures and Geometry over Finite Fields Zero sets of polynomials over the real and complex numbers give objects which can be understood geometrically. The "geometric" characteristics of these zero sets may be extended to give a sort of geometry over arbitrary fields, including finite fields. In particular, the zero set of a collection of polynomials over a finite field and its finite extensions can be used to define what is known as a zeta function. The Weil Conjectures (proven in 1973) describe many remarkable properties of this function and provide connections with conventional geometric concepts over the complex numbers. We give an introduction to the content of the Weil Conjectures and discuss how they can be used to effectively compute zeta functions. ====================================================== Name: Alex Carney Mentor: Ken Ono (Emory) Title: Partition numbers and the Hecke action on powers of the Dedekind eta-function Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono, we obtain closed formulas for the Hecke actions on all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. As corollaries to this result, we derive concise, efficient formulas to calculate large classes of partition numbers and congruences for powers of Ramanujan’s Delta-function. ====================================================== Name: Jake Levinson (new grad student) Mentor: Steven Miller (Williams College) Title: The n-level density of zeros of quadratic Dirichlet L-functions The statistical distributions of zeros of L-functions have wide-ranging applications in number theory and geometry. L-functions have been studied in connection with random matrix theory, which provides easier methods of computing these distributions. One important statistic, the n-level density of low-lying zeros for a family of L-functions, measures the distribution of zeros near the central point s = 1/2. According to the Density Conjecture of Katz and Sarnak, this statistic depends on a classical compact group associated to the family. We extend previous work by Gao, who computed the n-level densities of quadratic Dirichlet L-functions, and showed equality with random matrix theory up to n = 3. We develop a method to streamline the comparison with the prediction from random matrix theory. The key step is to find a ‘canonical’ form for several Fourier Transform identities, which allows us to verify them via combinatorial arguments. Our main result is to confirm up to n = 6 that, for test functions of suitable support, the density is that of the Symplectic Gaussian Ensemble. ======================================================