## Number Theory Essay Topics

A wonderful book is *Proofs from THE BOOK* by Ziegler and Aigner (the capitals are in the title - I am not actually shouting). This book contains lots of wonderfully elegant proofs from number theory, geometry, analysis and so on. The proofs are almost always at a low-enough level for a good undergraduate student to understand (the idea is that they are simple and elegant, and therefore are understandable!). Clicking on the following link will download a table of contents for the 4th edition: link.

The name *Proofs from THE BOOK* is in honour of Paul Erdős. From wikipedia: "(The book is) a book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, *You don't have to believe in God, but you should believe in The Book*...He accused (God) of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, *This one's from The Book!*."

answered Feb 27 '14 at 12:00

Some of our faculty have listed ideas for undergraduate research work. These ideas could also be used as the basis of a senior thesis to earn Latin Honors. These suggestions may spark some other idea that interests you. They also give you an idea about what areas interest some of our faculty.

Among the ideas posted here, some are harder and some easier. Some involve working on actual problems while others involve learning about a problem and why it's important or interesting. In some cases, there may be an easier version (special cases) of a problem that is more accessible. If one of the areas sounds interesting to you, contact the faculty member to discuss the topic and your background in more depth.

You can also feel free to contact other faculty, or to propose other ideas with individual faculty members. You might suggest ideas for projects that are based on your own reading, coursework, or perhaps on earlier work you've done (for example, in a summer REU). This list is not meant to be limiting in relation to topics or faculty member.

You can also contact Dr. Blake Thornton for suggestions about faculty to talk with.

### Professor Renato Feres (Geometry)

Some of following problems are meant to introduce you to advanced but well-established topics in graduate level math, others are much more open ended and may lead to original results. Some are more "theoretical" while others invite you to do some computer exploration. A few are probably pie-in-the-sky problems that, to me at least, are amusing to contemplate. Whatever the case, I'd be happy to discuss any of them with you and suggest reading material for anything in this list that strikes your fancy.

1) **The kinematics of rolling**. (Riemannian geometry/Non-holonomic mechanical systems)

On a smooth stone, draw a curve beginning at a point *p*, and hold the stone over a flat table with *p* as the point of contact. Now roll the stone over the plane of the table so that at all times the point of contact lies on the curve, being careful not to allow the stone to slip or twist. We may equally well think that we are rolling the plane of the table over the surface of the stone along the given curve. Mechanical systems with this type of motion are said to have "non-holonomic" constraints, and are common fare in mechanics textbooks.

Now imagine a tangent vector to the plane at p. This rolling of the plane over the surface provides a way to transport v along the curve, keeping it tangent at all times. The resulting vector field over the curve is said to be a "parallel" vector field. Show that there is a unique way to carry out this parallel translation. (Find a differential equation that describes the parallel vector field and use some appropriate existence and uniqueness theorem.) Let c be a short path joining *p* and *q*, whose velocity vector field is parallel. Show that c is the shortest path contained in the surface that joins *p* and *q*.

Whether or not you fully succeed, this mechanical idea will give you a concrete way of thinking about ideas in differential geometry that might seem a bit abstract at first, such as Levi-Civita connection, parallel translation, geodesics, etc. Also look for an engineering text on Robotic manipulators and explain why such non-holonomic mechanical systems are important in that area of engineering.

I don't know of many places where these things are explained in a simple way. Perhaps *Geometric Control Theory* by Velimir Jurdjevic is a place to start. In the engineering literature, *A Mathematical Introduction to Robotic Manipulation* is a particularly good reference.

2) **Geometry in very high dimensions**. (Convex geometry)

Geometry in very high dimensions is full of surprises. Consider the following easy exercise as a warm-up. Let *B(n,r)* represent the ball of radius *r*, centered at the origin, in Euclidian *n*-space. Show that for arbitrarily small positive numbers *a* and *b*, there is a big enough *N *such that (100 - *a*)% of the volume of *B(n,r)* is contained in the shell *B(n,r)* - *B(n,r - b)* for all *n* > *N*.

Here is a much more surprising fact that you might like to think about. Let *S(n-1)* denote the sphere of radius 1 in dimension *n*. (It is the boundary of *B(n,1 )*.) Let f be a continuous function from *S(n-1)* into the real line that does not increase distances, that is, | *f(p) - f(q)* | is not bigger than | *p* - *q* | for any two points *p* and *q* on the sphere. ( *f* is said to be a "1-Lipschitz" function.) Then there exists a number *M* such that, for all positive a, no matter how small, the set of points *p* in *S(n-1)* such that | *f(p) - M* |>*a* has volume smaller than exp(-*na*^2 / 2 ). In words, this means that, taking away a set with very small volume (if the dimension is very large), f is very nearly a constant function, equal to *M*.

This is much more than a geometric curiosity. In fact, such concentration of volume phenomenon is at the heart of statistics, for example. To make the point, consider the following. Let *S*(*n*-1, *n*^0.5) be the sphere in *n*-space whose radius is the square root of *n*. Let *f* denote the orthogonal projection from the sphere to one of the *n* coordinate directions, which we agree to call the *x*-direction. Show that the part of the sphere that projects to an interval *a* < *x* < *b* has volume very nearly (when *n* is big) equal to the integral from *a* to *b* of the standard normal distribution. (This is easy to show if you use the central limit theorem).

For a nice introduction to this whole subject, see the article by Keith M. Ball in the volume *Flavors of Geometry*, Cambridge University Press, Ed.: S. Levy, 1997.

3) **Hodge theory and Electromagnetism**. (Algebraic topology/Physics)

Electromagnetic theory since the time of Maxwell has been an important source of new mathematics. This is particularly true for topology, specially for what is called "algebraic topology". One fundamental topic in algebraic topology with strong ties to electromagnetism is the so called "Hodge-de Rham theory". Although in its general form this is a difficult and technical topic, it is possible to go a long way into the subject with only Math 233. The article "Vector Calculus and the Topology of Domains in 3-Space", by Cantarella, DeTurck and Gluck (The American Mathematical Monthly, V. 109, N. 5, 409-442) is the ideal reference for a project in this area. (It has as well some inspiring pictures.)

Another direction to explore is the theory of direct current electric circuits (remember Kirkhoff's laws?). In fact, an electric circuit may be regarded as electric and magnetic field over a region in 3-space that is very nearly one dimensional, typically with very complicated topology (a graph). Solving circuit problems implicitly involve the kind of algebraic topology related to Hodge theory. (Hermann Weyl may have been the first to look into electric circuits from this point of view.) The simplification here is that the mathematics involved reduces to finite dimensional linear algebra. A nice reference for this is appendix B of *The Geometry of Physics* (T. Frankel), as well as *A Course in Mathematics for Studentsof Physics* vol. 2, by Bamberg and Sternberg.

4) **Symmetries of differential equations**. (Lie groups, Lie algebras/Differential equations)

Most of the time spent in courses on ODEs, like Math 217, is devoted to linear differential equations, although a few examples of non-linear equations are also mentioned, only to be quickly dismissed as odd cases that cannot be approached by any general method for finding solutions. (One good and important example is the Riccati equation.) It turns out that there is a powerful general method to analyze nonlinear equations that sometimes allows you to obtain explicit solutions. The method is based on looking first for all the (infinitesimal) symmetries of the differential equation. (A symmetry of a differential equation is a transformation that sends solutions to solutions. An infinitesimal symmetry is a vector field that generates a flow of symmetries.) The key point is that finding infinitesimal symmetries amounts to solving linear differential equations and may be a much easier problem than to solve the equation we started with.

Use this idea to solve the Riccati equation. Choose your favorite non-linear differential equation and study its algebra of infinitesimal symmetries (a Lie algebra). What kind of information do they provide about the solutions of the equation? Since my description here is hopelessly vague, you might like to browse *Symmetry Methods for Differential Equations - A Beginner's Guide* by Peter Hydon, Cambridge University Press. It will give you a good idea of what this is all about.

5) **Riemann surfaces and optical metric**. (Riemannian geometry/Optics)

Light propagates in a transparent medium with velocity c/n, where c is a constant and n is the so called "refractive index" -- a quantity that can vary from point to point depending on the electric and magnetic properties of the medium. For a given curve in space, the time an imaginary particle would take to traverse its length, having at each point the same speed light would have there, is called the "optical length" of the curve. Therefore, the optical length is the line integral of *n/c* along the curve with respect to the arc-length parameter. According to Fermat's principle, the actual path taken by a light ray in space locally minimizes the "optical length". It is possible to use the optical length (for some given function n) to defined a new geometry whose geodesic curves are the paths taken by light rays. This is a particular type of Riemannian geometry, called "conformally" Euclidian. All this also makes sense in dimension 2.

One of the most famous paintings of Escher show a disc filled with little angels and demons crowding towards the boundary circle. What refractive index would produce the metric distortions shown in that picture?

A fundamental result about the geometry of surfaces states that, no matter what shape they have, you can always find a coordinate system in a neighborhood of any point that makes the surface conformally Euclidian. Why is this so? (This will require that you learn something about so called "isothermal coordinates".)

6) **Random walks and diffusion limits (I)**. (Probability theory/Elementary Geometry)

Imagine a long and narrow cylinder of radius r and a point particle that moves in the region bounded by the cylinder. The motion is specified as follows: starting at a point on the inner wall of the cylinder, choose at random a direction and let the particle move with constant speeduntil it hits another point of the cylinder. Once there, choose a new direction at random and repeat the process. A natural scheme (for reasons I won't describe here) is to pick the random direction with probability proportional to the cosine of the angle it makes with the (inward pointing) normal vector. The problem is to determine the probability that the particle will be given distance away from the initial point at a given time in the future. It is actually hard to find such a probability explicitly, but if the cylinder is very narrow and the particle moves very fast (with speed proportional to the reciprocal of the radius) you can use the central limit theorem to obtain an explicit (Gaussian) approximation. What is the variance of the resulting normal law? How does the variance change if the cross section of the tube is, say a square, instead of a circle?

7) **Random walks and diffusion limits (II)**. (Probability theory)

We can, of course, consider a two dimensional variant of the previous problem, in which the cylinder consists of two infinite parallel lines and the particle velocity after collision is chosen according to the same cosine law. However, after some thought you will realize that the hypothesis of the central limit theorem fail (barely!) to hold. Nevertheless, we can still ask what kind of limit process this random walk leads to. (Some key words: stable distributions, Levy processes.)

8) **Random Billiards**. (Billiard systems/Probability theory)

You may have heard a lot about the mathematical theory of chaos. It is part of the general subject of Dynamical Systems. In the tool box of the practitioners of this subject is a kind of toy system that is used to explore and illustrate almost any conceivable dynamical behavior (including chaos), called "billiard systems". It is just what you might expect: a billiard table and a point mass that moves about and bounces off the sides according to the law of mirror reflection. But the table is allowed any shape you want.

The problem I would like to propose is actually related to 6) and 7). Take the setting of RW2, except that the two parallel lines, when examined with strong lenses, reveal a periodic structure. More precisely, replace those lines with the graphs of, say, *C* sin(*x* / *C*) and 1+ *C* sin(*x* / *C*), where *C* is very small. Intuitively, as *C* approaches 0, the (deterministic) billiard system should behave more and more like the probabilistic system of 7). How can this intuition be made precise? What kind of scattering probability results after passing to the limit? What does the cosine law of 6) and 7), in particular, have to do with all this?

9) **Existence of surfaces**. (Computer Science/Differential Geometry)

You've probably heard of cellular =automata. The most celebrated example among them is John Conway's "game of life". They are, in general, a sort of beads game played over an infinite lattice (grid), which in our case will have dimension 3. At each moment, a lattice point may be empty or occupied by a bead of one among a number of colors. At the next moment, the state of that lattice point is renewed according to some function of the state of the nearest neighbor points. This function specifies the rules of the game.

Our problem is to find rules that will cause the beads to organize themselves into "surfaces". (Suggestion: try to find rules that imitate the behavior of amphiphilic molecules, likethe lipid bilayers that make up biological membranes. These molecules have one end that "likes" water and another that "hates" it.) If such surfaces can be obtained, is it possible to control how "crumpled" or "smooth" they are? or to control their curvature? Is it possible to make sense of notions such as differentiability and curvature in this discrete setting? (This would require the passage to some appropriate scaling limit.)

10) **Chemical varieties**. (Algebraic geometry/Chemical kinetics)

Algebraic geometry studies the geometry of sets of solutions of systems of polynomial equations (typically over the field of complex numbers) and how that geometry relates to the algebra of all polynomials that vanish on the set.

It is not difficult to show that to every system of chemical reactions with specified reaction speeds is associated a system of nonlinear first order differential equations describing how reactant concentrations change in time. These differential equations are of a very special kind: on the left-hand side is the first derivative of each reactant concentration (in moles) and on the right a polynomial function of the concentrations, whose coefficients are the stoechiometric constants. (Incidentally, the whole business of stoechiometry and its linear algebra underpinnings is in itself a great subject for a project.)

The set of zeros of the polynomial equation are equilibrium concentrations for the chemical reactions. Call the set of complex solutions of the polynomial equations the "Chemical Variety" of the system of reactions. These should be very special algebraic varieties. (They are typically of degree 2, for example, for any reasonable reaction mechanism.) Choose your favorite reaction mechanism and describe, in as much detail as you can, the geometric properties of the associated chemical variety. Are there interesting special properties shared by all chemical varieties?

### Professor Matt Kerr (Algebraic Geometry)

1) **Webs and polylogarithms**

The theory of webs was initiated by Blaschke and Bol in the 1930s. In its most basic form, it looks at the local analytic behavior of n intersecting foliations of complex 2-space by families of curves. For instance, you might fix three points and draw all the lines through each of them, or fix an algebraic curve and draw all the tangents to it. Then you look at the resulting configuration far from the curve or triple of points. Web geometry turns out to have numerous applications to differential equations, algebraic geometry and even physics.

Around the turn of the millenium, a group of French mathematicians made the very exciting discovery that so-called exceptional webs were intimately related to functional equations of polylogarithms. These are the functions you get by replacing the "k" in the denominator of the power-series expansion for the logarithm, by some power of k. Recent developments in algebraic K-theory have turned them from a curiosity into a major industry. (Capitalizing the k made it look more important.) One thing that, with some help, an undergraduate student might be able to do, is come up with a more algebro-geometric description of the (exceptional) Bol 5-web than I have seen in the literature.

There are related functions called Grassmanian polylogarithms, invented by A. Goncharov, which enjoy relatively simple functional equations. To try to relate these to webs, or to find a new (more geometric) approach to their functional equation, would also be interesting and potentially do-able.

2) **Orbifold singularities**

One of the great collaborative success stories of the past two decades has been that between complex algebraic geometers and string theorists in the mirror symmetry program. The quest to produce Calabi-Yau 3-manifolds (three complex dimensions!) required mathematicians and physicists to confront the kinds of singularities -- local failure of manifold structure -- that arise from quotienting complex 3-space by a finite group action. Moreover, they had to figure out how to resolve them -- the higher-dimensional analogue of lifting an (actual) string off

itself.

While this story has only recently been thoroughly understood, it would be well within the powers of an interested undergraduate student to provide a down-to-earth account with basic examples. This is something I have not seen in the literature, and shouldn't be thought of as an expository project -- it would require some original thought. It would also acquaint you with toric geometry, an extremely useful tool which gives a dictionary between algebro-geometric concepts and the geometry of convex bodies (like polygons and polytopes) considered relative to a lattice. In working out examples, the latter boils down to some surprisingly entertaining 3-dimensional linear algebra which ultimately tells you how to draw a triangulation.

Mirror symmetry comes into this story in a number of ways. In one version, the resolutions of singularities you will construct are "mirror" to certain families of Riemann surfaces. An ambitious student might want to investigate this too.

3) **Abel's theorem for amoebas ** (I will also accept non-amoebas as students on this project.)

The classical theorems of Abel and Jacobi describe the divisors (configurations of zeroes and poles with multiplicity) of meromorphic functions on compact Riemann surfaces. Attempts to generalize these results to noncompact or singular settings, as well as to higher dimension, have motivated a lot of modern algebraic and differential geometry -- like the Bloch-Beilinson and Hodge conjectures and the theory of webs. I don't know of a good write-up of the one-dimensional generalizations, and you could already learn a lot by trying to trying to understand the situation for unions of lines, or for multiply connected regions.

In algebraic geometry, roughly speaking, we study solution sets of algebraic equations. Replace everywhere multiplication by addition and addition by "taking the maximum," and you have an exciting new theory called tropical geometry -- which even has its own version of Abel's theorem! Amoebas are objects which provide a connection (via a limiting process) between algebraic curves and tropical curves, and it would be extremely interesting (though not necessary for an interesting project) to devise a connection whereby one Abel's theorem becomes the limit of another.

For this project, all I really ask is that a student be familiar with basic complex analysis. It would also be useful to know what a Riemann surface is, but this could be dealt with in summer reading.

4) **Expository projects**

I would be happy to direct a reading course and subsequent write-up as well, on any of the following topics (or on appropriate student-proposed topics):

- modular forms and elliptic curves
- representation theory: finite groups, Young tableaux, and crystallography; or Lie groups and Lie algebras (for a more ambitious project)
- algebraic number theory: Galois groups of number fields, class fields, easy cases of Fermat's last theorem; irrationality and transcendence

### Professor Greg Knese (Analysis)

1) **Failure of von Neumann's inequality.**

Von Neumann proved that if A is a contractive matrix (has operator norm <= 1) and p(z) is a complex polynomial, then p(A) has operator norm bounded by the supremum of p on the unit circle. A two variable version of this result is true (Andô's inequality) but the three variable version is false. Counterexamples can be shown to exist either through probabilistic arguments (i.e. a random polynomial will fail the inequality) and there are also a few examples constructed through ad hoc methods. This project would involve trying to construct more interesting families of counterexamples to the three variable von Neumann inequality in order to understand "how badly" the inequality fails.

2) **Multilinear Bohnenblust-Hille inequality**

This is a different kind of inequality for polynomials. Multilinear polynomials satisfy an inequality bounding certain little l^p norms of their coefficients by the supremum norm of the polynomial. This project would also involve looking for interesting examples to test the sharpness of known versions of this inequality.

3) **Reading projects**

I would be happy to supervise reading projects on other topics: topics in stable polynomials such as the Branden-Borcea theorems, Gurvits proof of the van der Waerden conjecture, or the solution of the Kadison-Singer problem; Grothendieck's inequality, or Khintchine inequalities.

### Professor Steve Krantz (Analysis, Complex Variables, Geometry)

1) It is known that, given any closed planar curve, there are four points on that curve which are the corners of a rectangle. It is an open problem whether there will be four points that are the corners of a square.

2) Let *U* be a planar region, and let *G* be the group of rigid motions of the plane that map *U* to itself. We call *G* the "automorphism group" of *U* , and we denote it by Aut( *U *). Now suppose that *U' * is a small perturbation of *U* . How is Aut( *U*' ), as a group, related to Aut( *U* ) ? How does the answer change as *U*' deviates farther and farther from *U* ?

3) (*Refer to (2) for terminology*.) Let *G* be any finite group. Is there a planar domain *U *such that Aut( *U *) = *G* ? Can we relate the topology of *U* to the structure of the group? What if we allow *U* to live in a higher dimensional space? Does that allow more groups *G *to give an affirmative answer? Given a group *G*, can we estimate the dimension of the space in which a domain *U* will live that has the desired property?

4) (*Refer to (2) for terminology*.) It is an intuitively obvious assertion that, of all planar domains, the disc has the "largest" automorphism group. Formulate a precise version of this statement and prove it. Given any group *G* that is the automorphism group of some planar domain, can we find a particular planar domain *U *that is as close to the disc as we please and so that Aut( *U *) = *G* ?

5) Consider the space *C*^\infty of infinitely differentiable functions and the space *C*^\omega of real analytic functions (i.e., functions with convergent power series expansions). Of course *C*^\omega is a subset of *C*^\infty. Is there a range of function spaces, perhaps a range that is parametrized, that spans the gamut between *C*^\omega to *C*^\infty ? (*This problem is important for the theory of partial differential equations*.)

6) Let *U* be a convex planar domain. Call a point *p* in *U *an equichordal point if all chords of *U * that pass through *p* have the same length. It is known that a convex planar *U* can have at most one equichordal point. But the proof is very abstract and extremely difficult. Problem 1: find an elementary proof. Problem 2: What is true in dimension three? Problem 3: What is true for non-convex domains?

### Professor Todd Kuffner (Statistics)

1) **The Berry-Esseen Theorem**. It is still of current interest to determine the maximal error of the asymptotic normal approximation to the scaled sample mean as specified by a central limit theorem. Recent results related to the bound in the Berry-Esseen theorem, for summands of both i.i.d. and non-identically distributed observations, are to due Shevtsova (2010, 2011). This project would involve studying the methods used to obtain such bounds and investigating the accuracy using simulated data.

2) ** Post-Selection Inference for Linear Regression**. Inferential correctness for testing hypotheses about regression coefficients after a variable selection procedure has been utilized requires a careful evaluation of the effects of the selection procedure on the final inference. This project involves studying, in real-data examples, how classical inference procedures are invalidated by the use of selection procedures. The performance of inference procedures designed to control, respectively, selective type I error and familywise error rates (FWER) will be compared in theory and practice.

3) ** Inference in Curved Exponential Families**. Inference in curved exponential families, following a principled approach, requires construction of exact or approximate ancillary statistics. Examples include the gamma hyperbola model, the bivariate normal correlation model, and ARMA(p,q) models used in time series analysis. This project involves motivating a principled, accurate approach to inference in such models, and real-data comparisons with conventional inference procedures which do not respect such statistical principles.

4) **The double bootstrap and the warp-speed double bootstrap**. The double bootstrap was proposed as a technique to improve the coverage accuracy of confidence intervals constructed via conventional single-bootstrap methods. The warp-speed double-bootstrap attempts to achieve this using a single double-bootstrap sample for each bootstrap sample, drastically reducing the computational costs. Recent results of Chang and Hall (2015) show that, unfortunately, warp-speed double-bootstrap confidence intervals do not in general share the same coverage accuracy properties as intervals constructed by the conventional double bootstrap. This project investigates the source of this discrepancy using large-scale simulations in different model settings of practical interest.

### Professor Mohan Kumar (Algebra)

1) If a1,a2,...an are integers with gcd = 1, then the Eulidean algorithm implies that there exists a (n x n)-matrix A with integer entries, with first row = (a1,a2,...,an), and such that det(A) = 1. A similar question was raised by J.P. Serre for polynomial rings over a field, with the a's being polynomials in several variables. This fundamental question generated an enormous amount of mathematics (giving birth to some new fields) and was finally settled almost simultaneously by D. Quillen and A. A. Suslin, independently. Now, there are fairly elementary proofs of this which require only some knowledge of polynomials and a good background in linear algebra. This could be an excellent project for someone who wants to learn some important and interesting mathematics. (*These results seem to be of great interest to people working in control theory. Though I am not an expert, I'm willing to learn with a motivated student.)*

2) A basic question in number theory and theoretical computer science is to find a "nice" algorithm to decide whether a given number is prime or not. This has important applications in secure transmissions over the internet and techniques like RSA cryptosystems. Of course, the ancient method of Eratosthenes (sieve method) is one such algorithm, albeit a very inefficient one. All the methods availabe so far has been known to take exponential time. There are probabilistic methods to determine whether a number is prime, which take only polynomial time. The drawback is that there is a small chance of error in these methods. So, computer scientists have been trying for the last decade to find a deterministic algorithm which works in polynomial time. Recently, this has been achieved by three scientists from IIT, Kanpur, India. A copy of their article can be downloaded from *www.cse.iitk.ac.in* A nice project would be to understand their arguments (which are very elementary and uses only a little bit of algebra and number theory) and maybe to do a project on the history of the problem and its ramifications.

### Professor Nan Lin (Statistics)

1)** DNA methylation **DNA methylation in vertebrates typically occurs at CpG sites and results in the conversion of the cytosine to 5-methylcytosine. Human DNA has about 80%-90% of CpG sites methylated. Methylation is important to embryonic development and cancer. With the current next-generation sequencing (NGS) technology, people identify regions with different methylation levels under different disease status to understand the mechanism of cancer and other disorders. NGS data from methylation experiments process complicated strictures and impose challenges to statisticians. We are developing statistical tools for the

analysis of NGS data from such experiments.

2)** Statistical analysis for anesthesiology data** Anesthesiology is a very important part of surgery and many other medical practice. Anesthesiologists are still in debate about proper ways to monitor patients' anesthetic status. A recent publication on the New England Journal of Medicine (vol 365, pages 591-600) suggest that a device approved by the Food and Drug Administration (FDA) to reduce the risk that patients will recall their surgery does not lower the risk of the problem, known as intraoperative awareness, any more than a less expensive method. Statistical analysis will help to find better ways of anesthetic practice.

3)** Statistical analysis for spatial cognition** It is intriguing to understand how people construct the memorized maps after seeing a real map. Statistical tools are crucial to understanding the transformation from the reality to the map in people's memory. We are developing statistical models to tackle these issues.

### Professor John McCarthy (Analysis)

1) **Fluid Dynamics** Consider a cylindrical tube, open at one end. At the closed end, a small quantity of gas is injected. It diffuses out the other end at a predictable rate. Now, suppose the quantity of gas injected is increased. The flow will not scale linearly, as the effect of the pressure of the introduced gas must be considered. I have a project with Professor Gregory Yablonksky in the Chemical Engineering department to model this flow.

2) **Linear Matrix Inequalities** A computer vision problem posed by Professor Robert Pless in the Computer Science Department. Imagine a large number of cameras arranged around a central object. One wants to match up the pictures, but there is some error in the measurement. Mathematically, the problem becomes approximating a large symmetric matrix by a rank 3 matrix that has 1's on the diagonal. It ties in to an active research area in systems theory: solving a linear matrix inequality with a rank constraint. Nobody knows how to do this well.

3) **Applied Statistics/Public Health** The "French paradox" is the claim that, despite having a high fat diet, French people have a low rate of heart disease. I believe this is a statistical artifact, due principally to cultural differences in filling out death certificates. I would be willing to supervise an undergraduate who wished to hunt down the data and analyze it.

4) **Galapagos Tortoises** My friend Stephen Blake has collected movement data on Galapagos Tortoises for two years. The data is available at www.movebank.org. We would like to understand more about the tortoises's movement, for example what makes them migrate, how are they influenced by climate changes, why smaller tortoises don't migrate, how they choose routes, etc. For more information about the tortoises, see Anne the Tortoise

### Professor John Shareshian (Algebra, topological combinatorics)

1) **Computation in topological combinatorics** Topological combinatorics includes the study of simplicial complexes (that is, geometric objects built from possibly higher dimensional analogues of the unit interval, the equilateral triangle and the equilateral tetrahedron) whose faces are indexed by combinatorial objects such as graphs. The Homology program of J.-G. Dumas, F. Heckenbach, D. Saunders and V. Welker has been used to investigate the structure of such complexes. There are many adjustments and additions which could be made to improve the program, the most ambitious of which is to make it amenable to parallel processing.

2) **Order complexes of subgroup lattices** The set of subgroups of a group G is partially ordered by inclusion. There are interesting open questions and proven theorems about relating the algebraic structure of G to the combinatorial structure of this partially ordered set. For any partially ordered set P, the set of all totally ordered subsets of P determines a simplicial complex. The topological structure of this complex is related to the combinatorial structure of P. One can hope to use this relationship productively when P is the set of subgroups of G. This area is appropriate for both reseach and expository projects.

3) **Symmetric functions** A symmetric function is a power series of bounded degree in infinitely many variables which is not changed by any permutation of the variables. Symmetric functions appear in many areas of mathematics, including combinatorics and representation theory (which involves studying a group G by understanding homomorphisms from G to various matrix groups). There are lots of interesting open combinatorial problems involving symmetric functions (many appear in the exercises after Chapter 7 of R. P. Stanley's book, Enumerative Combinatorics, Volume 2). This area is also appropriate for expository projects.

### Professor Edward Spitznagel (Statistics)

1) Learn about **propensity analysis** as a means of establishing causality and write your thesis using real data (which I can supply).

2) Learn about **zero-inflated and double-hurdle models**, and apply them to real data. (Both deal with the idea that certain variables predict whether a response is necessarily zero, and if the response is not necessarily zero, then other variables might predict its value.) A student worked on this last year, but you can extend what he did.

3) Learn about **interval-censored models**. The classic example is electro- convulsive therapy for depression. The machine typically has about five fixed charge levels (in Coulombs), and the charge is stepped up until the patient has a seizure. We thus know coarse lower and upper bounds on the seizure threshold, and the task is to estimate the exact seizure threshold.

4) Learn about **GIS** and write an application that maps pollution, river flows, drought, etc. SAS has a GIS application, but we also have special-purpose software on the public machines in Eads.

5) Learn about **analysis of complex survey data** (stratification, clusters, and weight) and analyze a real data set. Most large public-access data sets have this complex structure. This was covered in a course - Math 438 I think - but that was so long ago it's not listed in the catalog. There are, in fact, two forms of complex data, the "classic" form in which each stratum has exactly two clusters, and the "certainty PSU" form.

6) Learn about **census data**, and how census info can be merged by tract with special-purpose data. I have a friend who has some pathological gambling data, who has extracted most of the obvious results from her data, but might be looking for help in digging out some remaining gems.

7) Learn about the **relationship between the jackknife and the bootstrap**, and be able to illustrate what Bradley Efron meant when he said: "What actually happened was that Rupert Miller in our department was working on the jackknife. He had written a paper called the "Trustworthy Jackknife" in which he tried to figure out when the jackknife method gave dependable variance estimates. The Jackknife was considered very mysterious. It worked, but nobody could figure out why it worked. And sometimes it didn't work. What I thought was that the jackknife must be a differential, local kind of approximation for something else. And so when I started looking for the something else I came up with the bootstrap."

### Professor Xiang Tang (Geometry)

1) **Geometry of singular spaces**. In calculus, we learn that there are curves that do not have tangent lines. Such curves are usually called singular curves. In this project, we will study geometry of singular spaces which are generalizations of singular curves. One class of such singular spaces that are interesting to us is called orbifold. It is an important object in both mathematics and physics. We will mainly focus on orbifolds and compute some useful invariants about them.

2) **Noncommutative torus and quantization**. In mathematics, the shape of a donut is called torus. Noncommutative torus is the quantization of the usual torus, and appears naturally in both mathematics and physics. We will study some interesting applications of noncommutative tori in physics.

3) **Can you comb the hair on a coconut?** The answer is ``no". There is an interesting mathematics theory related to this answer, which is called index theory. We will apply index theory to study some interesting properties on geometric spaces.

### Professor Victor Wickerhauser (Applied and Computational Mathematics, Wavelets)

1) Read Daubechies and Sweldens "Factoring Wavelet Transforms into Lifting Steps," (J. Fourier Anal. Appl. 4:3(1998),245-267). Then implement the Euclidean algorithm for Laurent polynomials described in the paper. (*Thus, you will use ideas in abstract algebra and Fourier analysis to write an efficient computer program that is part of the JPEG-2000 image compression algorithm.*)

2) Read chapter 3 (pp. 67-101) of my book "Adapted Wavelet Analysis," and also Strang, "The Discrete Cosine Transform" (SIAM Review 41:1(1999),135-147). Synthesize a proof that the discrete Hartley transform is orthogonal. (*Thus, you will see how the Sturm-Liouville theorem from differential equations can save many tedious computations in the verification that a basis, such as one used in the JPEG (1990) image compression algorithm, is orthonormal*.)

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