VIGRE Components: Invitation to Research

Invitation to Research Modules

Gregory Baker: Computational Dynamical Systems

Dynamical systems arise when we can describe the evolution of a system in terms of its current state. Drawing from several areas of application, we will describe how dynamical systems may be formulated. For example, Newton's Laws of motion give the acceleration in terms of forces; the laws of chemistry dedicate how chemical species react; the laws of electro-magnetism determine the behavior of electric circuits. Currently, there is growing interest in biological models, often based on concepts of probability. Some specific examples include the Lotka-Volterra equations that describe competing species and predator-prey systems, and the selection equation that describes the evolution of population genetics.

In all of these applications, the scientist is interested in the general properties of the solutions. One of the first questions concerns the likely long time behavior of the system. There are several candidates: the system may settled down to a stationary state (unchanging in time); it may maintain a persistent oscillatory pattern; or it may continue in a random, chaotic behavior. Since most of the dynamical systems of practical importance are described in terms of nonlinear equations, we use numerical methods to find the plausible long time behaviors.

We will describe the best numerical procedures in a series of four lectures. The first lecture will introduce the notion of a dynamical system and present several illustrative examples. Our first task will be to identify stationary solutions; those states that don't change in time. The next issue concerns the stability of these stationary states. If given a small perturbation (a small "kick"), will the system return to the stationary state (stable manifold), will it continue to jiggle around endlessly (center manifold) or will it depart substantial and evolve into some completely new behavior (unstable manifold). Which of these behaviors occurs depends on the eigenvalues of a linear system that describes the evolution near the stationary solution.

In the second lecture, we will use the information gained about the presence of stationary solutions and their stability. We will introduce the phase space to record the stationary solutions and how the system may move from unstable states to stable states. The curves in phase space along which the system moves are called trajectories, and once certain key trajectories are determined we know everything about the dynamical system.

Besides the unknown variables in the system, there are additional parameters that identify a specific case. For example, the strengths of the resistors in an electric circuit. Generally, scientists want to know how the system is affected by varying these parameters. Often the search is for a particular choice of the parameters that might optimize the system in some way. We will describe how changes in the phase space can be followed as parameters are varied. Some interesting things can happen: stable stationary solution may suddenly appear or disappear; stable stationary solution may suddenly become unstable and vice versa; and the system may suddenly start to oscillate. We will describe the numerical procedures needed to identify these situations.

In the final lecture, we will describe the more difficult task of locating persistent oscillatory solutions. We will also extend the numerical procedures to cover the cases where spatial variations in the system are important. Here, the mathematical formulation of the system involves partial differential equations. Some specific examples include the effects of diffusion in the Lotka-Volterra equations and the migration-selection equation in population genetics.

Vitaly Bergelson: Ergodic Theory: From Statistical Mechanics to Modern Number Theory

While the roots of ergodic theory are in the kinetic theory of gases and celestial mechanics, modern ergodic theory stands at the junction of many areas. The goal of this module is to acquaint the students with some of the promising directions of research by focusing the discussion on three central topics: recurrence, equidistribution, and independence.

In the first lecture we shall give an overview of the historical roots of ergodic theory, discuss the notion of a dynamical system and consider some instructive examples: billiards, unimodal maps, continued-fraction transformations, automorphisms of compact groups and others.

The main theme of the second lecture is recurrence. We shall discuss various examples from the theory of Diophantine approximations, celestial mechanics and Ramsey theory, and interpret them as recurrence theorems. We then try to formulate reasonable new conjectures.

The third lecture will deal with the concept of independence. We will discuss the connections of ergodic theory with probability theory and devote some time to chaotic dynamics.

The fourth lecture will be devoted to equidistribution in its broadest sense. The idea of invariant measures will be stressed, various convergence theorems will be discussed, and connections with and applications to statistical mechanics, number theory, and combinatorics will be pointed out.

Throughout these lectures we will constantly emphasize the ubiquity of dynamical ideas in diverse, seemingly static situations.

Dan Burghelea: Differential Topology

The first lecture of this module will be about differential manifolds, smooth maps and transversality. The key concept will be the ``degree of a map'' and some of its applications and generalizations (as described in [1]).

The second lecture will be about classification of surfaces. The student will be introduced to Morse Theory, as a natural tool to prove the classification theorem for surfaces. The lecture will end up by pointing out the difference between dimension two an higher, in particular dimension larger or equal to five, In this case one can have different differential manifolds which are homeomorphic as topological spaces.

The third lecture will begin with Milnor-Brieskorn examples of different differential structures on spheres of dimension larger than seven. Next, it will shown how S. Smale used Morse Theory to prove the h-cobordism theorem providing in this way one of the main tool topology has to count differential manifolds in dimension larger than five.

The fourth lecture will be about Lefschetz fixed point theorem and about Hopf theorem on zeros of vector fields, as well as its improvement in the case of a Hamiltonian vector fields (Morse inequalities). Again the role of Morse theory as a generalization of the min-max theorem from calculus will be emphasized. The lecture will provide the opportunity to say few words about symplectic topology, a very active new subfield of topology.

The fifth lecture will be about the great problems and the great people in Topology and how the field interacts with the rest of Mathematics and Physics, or about the dimensions three and four, the most challenging for present-day topology.

[1]	J. Milnor. Topology from the differential viewpoint. University Press of Virginia, Charlottesville, 1965.

Timothy Carlson: Mathematical Logic

Over the course of five lectures, we will give an overview of the subject and history of mathematical logic. We will describe the forces that gave rise to the four main areas within mathematical logic (model theory, proof theory, set theory and computability theory) and follow the evolution of these areas up to the problems that are driving them today.

In the first lecture we give a historical overview. We begin with the work of Boole and Frege who transformed the study of logic into a mathematical discipline. The need for the development of mathematical logic became crucial when Russell discovered his famous paradox in the naive set theory being used by Cantor as the foundational underpinning for work in analysis. In his address to the AMS in 1900, Hilbert proposed ten problems for the new century, one of which was to establish by ``finitistic'' means that basic mathematics was free of contradictions. Hilbert's program was derailed 30 years later by Gödel's celebrated incompleteness theorems.

We will describe how Hilbert's program, despite its failure, gave birth to proof theory, a profound study of the nature of mathematical proof, and how set theory emerged as the commonly accepted foundation for modern mathematics. Gödel's incompleteness theorems focussed attention on recursive functions, a class of computable functions on the natural numbers. We will see how the general study of algorithms, computability theory, naturally followed and lead to Church's Thesis: for functions on the natural numbers, the intuitive notion of ``computable by an algorithm'' is equivalent to the mathematically precise notion of recursive. The fourth area of mathematical logic, model theory, depends on semantics for formal logic. While the details of these semantics are too involved for this series of lectures, some examples will be given to provide the essential intuitions involved. We will consider ways model theory classifies a given theory in terms of its models i.e. the semantic interpretations in which the theory is true.

The remaining four lectures will be devoted to a closer look at contemporary model theory, proof theory, set theory and computability theory respectively.

Ruth Charney: Introduction to Geometric Group Theory

Topology, geometry, and group theory have long been intertwined with many groups arising naturally as symmetry groups of topological metric spaces and many topological spaces arising as orbit spaces of group actions. In this series of lectures we will discuss this interplay and introduce some of the modern techniques of geometric group theory.

In the first two lectures we will talk about groups acting on topological spaces, beginning with classical examples such as reflection groups and progressing to general constructions including Cayley graphs and Eilenberg-MacLane spaces.

In the third lecture we will talk about presentations of groups, amalgamated products, and Stallings theorem on ends of groups.

In the last lecture, we will introduce some of the modern concepts in geometric group theory including word metrics, hyperbolic groups and quasi-isometries.

Alexander Dynin: Mathematical Physics: Three Quantum Revolutions

The two way connection between pure (experimental) physics and pure (axiomatic) mathematics is long and winded via theoretical physics, mathematical physics, and applied mathematics. The hub is mathematical physics. It provides mathematical structures and intuition to physicists and physical structures and intuition to mathematicians. Modern mathematical physics has especially deep and rich interactions with pure geometry, topology, and analysis. In four lectures I present a perspective of the analysis venue following physicists quest for the fundamental reality.

First quantum leap. From classical particles to classical fields and quantum particles, from ordinary to partial differential equations, and from single-variable to multi-variable calculus.

Second quantum leap. From classical fields and quantum particles to quantum fields, from partial to variational differential equations, and from multi-variable to infinite-variable calculus.

Third quantum leap. From quantum fields to strings and M-theory, from variational differential equations to string and M(atrix) differential equations, and from infinite-variable calculus to ???.

A theory of variational differential equations has just started. It involves a justification of the celebrated Feynman integral. Even less is known about string and M(atrix) differential equations. Yet a certain mathematical permanence in all these transitions provides guidance.

Prerequisites:

Undergraduate linear algebra, multi-variable calculus, ordinary and partial differential equations.

Yuval Flicker: Modular Forms

The area of automorphic representations and their local components--called admissible representations--is a natural continuation of the main researches of the 19th and 20th centuries studies in number theory and modular forms. A guiding principle is a highly hypothetical reciprocity law that would relate these infinite dimensional representations of a reductive group, with finite dimensional representations of the Galois group of the number field over which we work. The initial case of one dimensional representations--namely that of GL(1)--is class field theory, a crowning achievement of early 20th century number theory. The case of two dimensional representations--namely that of GL(2)--is the theory of modular forms.

While this ``Principle of Functoriality'' is not expected to be accessible soon, it leads to expectations that phenomena on one side would be reflected on the other side. Simple relations on the Galois side would have deep counterpart relations on the automorphic side, named ``liftings''. These lifting relations are accessible purely on the automorphic side by several analytic techniques. The results are useful for the determination of zeta functions, which count rational points on related (Shimura) varieties of interest in arithmetic geometry.

The lectures would introduce some of these terms, illustrate them in simple cases, especially those of GL(1), GL(2), SL(2), GL(n), and give examples of work that can be attempted even without knowing all about algebraic groups and their representation theory, number theory and arithmetic geometry.

Avner Friedman: Partial Differential Equations

Partial differential equations (PDEs) arise naturally as models in chemistry and physics, in the engineering sciences as well as in the life sciences. This module gives an overview of the different types of PDEs, of ways to solve them and of typical applications where PDE models are important.

The first lecture of this module will introduce the three main types of partial differential equations (PDEs), namely elliptic, parabolic and hyperbolic equations represented by the Poisson equation, the heat equation, and the wave equation, respectively. It will be shown how these equations arise as models in various applications.

The second lecture will focus on Poisson equations. We seek, and find, explicit solutions for problems in electricity and in gravity. These solutions will be mathematically analyzed, and the result interpreted within the applied context to demonstrate how mathematics can be used to gain understanding of physical phenomena.

The third lecture will deal with the heat equation. We will again derive explicit solutions to solve problems such as those related to the cooling of a body or the spreading of population. In the first application, one could use the dependence of the solution on parameters to investigate, for instance, how the shape and size of a body affect its cooling rate.

In the last lecture, we will survey a number of methods that can be used when no explicit solution is available. One prominent example is the maximum principle for the Laplace and the heat equation. We will also talk about properties of solutions that one might be interested in such as the asymptotic behavior of solutions when time goes to infinity.

Harvey Friedman: Foundations of Mathematics

The formalization of mathematics (Lecture 1). Is mathematics a free form artistic enterprise, or is there a gold standard for correct mathematics? Interpreting mathematics in set theory. The axioms of set theory.

Rigorous logical reasoning (Lecture 2). Is completely rigorous proof a mirage, or is there a gold standard for rigor? Painless presentation of the axioms and rules for logic. The completeness theorem for logical reasoning.

Computability and incompleteness (Lecture 3). The mathematical theory of computation. Can we hope to answer all mathematical questions? The incompleteness theorems.

The tame and the wild (Lecture 4). Linear arithmetic is tame (the ordered group of integers). Elementary real algebra and Euclidean geometry are tame (the ordered field of real numbers). The ring of integers is wild.

Yuji Kodama: Integrable Systems

In this module, we will give an introduction to integrable systems which is one of the most important and active area of applied mathematics. It also has several deep connections to other fields of sciences including physics, biology and engineering.

The first lecture of this module introduces the basic concept of the integrable systems by taking several examples. We also introduce some nonlinear differential equations as prototype examples of integrable systems such as the Toda lattice equation (nonlinear mechanical chain), the Korteweg-de Vries equation (shallow water waves), and the nonlinear Schrödinger equation (optical pulse propagation).

The second lecture focuses on the Toda lattice equation. We introduce an analytic method, the so-called inverse scattering transform, that can be used to solve the initial value problem, and describe in some detail the topological structure of the solution manifold. Several extensions of the method based on Lie-algebra theory are also explained.

The remaining two lectures deal with the Korteweg-de Vries equation and the nonlinear Schrödinger equation. We describe how these equations arise in several physical phenomena by using a singular perturbation method. We derive explicit solutions, called solitons, and explain their significance in physics. We also introduce the tau-function, a universal function for integrable systems, to describe the analytic structure of the solutions.

Peter March: Probability and Stochastic Equations

This series of talks begins with a thorough discussion of the grand martingale; an apparently sure-fire way to make a dollar by betting on successive flips of a fair coin. This strategy consists of betting on heads and doubling the bet on the next flip if the previous flip was tails. In the course of the discussion martingales, stopping times and the discrete stochastic calculus make their appearance in a completely natural way.

By looking at the simple example of random walk on the integer lattice, we next make the connection between boundary value problems for difference equations and stochastic equations for random walk and its important modifications: random walk with reflection, absorption and killing. The famous Feynman-Kac formula comes up in the course of the discussion and its connection with discrete Schrödinger equations is made plain.

In the third talk we observe that stochastic equations for random walk are analogues of ordinary difference equations. Stochastic partial difference equations come up naturally when considering discrete analogues of diffusion, wave propagation and surface growth phenomena as we show by example.

In the last talk we present Einstein's derivation of brownian motion and we point out that brownian motion is the continuum limit of random walk. All the ideas that have been developed in the technically simple discrete context are shown (in a necessarily sketchy fashion) to have analogues in the continuum limit. We focus on just one example: the Kardar-Parisi-Zhang equation, a stochastic partial differential equation for the growth of a random surface. We conclude the talk and the series by pointing out a number of outstanding open problems related to the KPZ equation.

Jeffery McNeal: Several Complex Variables

The analysis of holomorphic (synonymously: analytic) functions of one complex variable is a beautiful and old subject. On the one hand, the assumption of holomorphicity implies very strong relationships between the various analytic, geometric, and algebraic properties such a function has, e.g. the size of its zero set, its average value on a given set, the kind of limits a sequence of such functions has. On the other hand, holomorphic functions in one variable occur naturally in topology, number theory, applied mathematics, etc., so results about the behavior of these functions is of wide interest.

The analysis of holomorphic functions of more than one variable is a much younger subject. There are many fundamental questions about these functions which remain unanswered and there are new phenomena (i.e., phenomena not present in one variable) for which the ``right'' questions have not been found. The field is in no sense self-contained and the analysis of functions of several complex variables uses a wide variety of methods from other fields, including partial differential equations, algebraic and differential geometry, and harmonic analysis.

The purpose of these six lectures is to present some of the major themes in complex analysis of more than one variable, illustrating some of the new phenomena which exist there and highlighting some of the analytic tools which dominate current work in the area. Since facts about holomorphic functions of one variable will be somewhat known to the audience, we will proceed, roughly, via the following plan: recall an aspect of function theory from one variable, examine what its higher dimensional analogue(s) might be, and then mention what is known/unknown about this problem. It is in the last step where we will indicate some of the tools and ideas which give the solution (or partial solution) to the problem at hand. The topics we will discuss might include: Reproducing kernels (Cauchy integrals), convergence of power series, the Cauchy-Riemann equations and approximation problems.

Henri Moscovici: Noncommutative Geometry

The study of certain ``spaces'' that arise in mathematics and physics forces us to adapt our geometric thinking to situations where the coordinates on the space do not commute. This has led to a novel geometric approach, based on a quantum perception of space, generalizing but distinct from the traditional description of a continuum filled with points (cf. Alain Connes's book Noncommutative Geometry). While the theory may seem strange and unfamiliar at first glance, its spirit and methods can be illustrated by relatively simple examples.

This series of lectures will begin with Connes's charming proof of one of the most surprising results in synthetic geometry: Morley's theorem asserting that the trisectors of the angles of any triangle meet at the vertices of an equilateral triangle. When cleverly restated, as a statement involving three elements of the group of affine motions of the line, it can be verified by a direct computation. More remarkably, it makes sense over an arbitrary field (of characteristic different from 3).

Rephrasing geometric properties in algebraic terms, without having to invoke commutativity, is only the beginning of the story. Deeper features of the theory, such as the natural occurrence of irrational dimensions, will be discussed next by means of a simple but highly nontrivial example: the noncommutative torus. This basic model of a noncommutative manifold can easily be introduced in terms of a simple differential equation describing the well-known Kronecker foliation. The fruitfulness of this study is exhibited even outside mathematics, by spectacular applications to the Quantum Hall Effect and more recently to string and M-theories.

Finally, the inherently quantum nature of the appropriate notion of symmetry in noncommutative geometry will be demonstrated by discussing the Connes-Kreimer Hopf algebra of rooted trees and by illustrating the way it encodes the combinatorics of Feynman graphs in perturbative renormalization--to date the most successful technique for computing physical quantities in quantum field theory.

András Némethi: Geometry of Polynomial Maps

The geometrical and topological properties of algebraic varieties provide many mysterious and beautiful formulas and unexpected connections between analytic and algebraic invariants, their topological counterparts and number theory. This module is an introduction to the wonderful world of these relationships. This will be exemplified via the quasi-homogeneous hypersurface singularities embedded in the three-dimensional complex affine space.

In the first lecture I plan to present the general background of singularities. I show how they appear in a natural way in different areas of mathematics, as in the study of function with more variables (analysis), polynomial equations (algebraic geometry), discriminants of polynomials in one variable (algebra), projection of smooth surfaces to the plane (topology).

In the second lecture I will consider the links of quasi-homogeneous polynomials in three variables. They are real 3-manifolds. This allows us to discuss classical Seifert manifolds and the arithmetical games between the weights of the original polynomial, the Seifert topological invariants of the link and continued fractions. The plumbing presentation of these 3-manifolds will add an additional graph-theoretical and combinatorial flavor to the subject.

The third lecture deals with the the nearby fiber, which in the classical theory of links corresponds to the Seifert surfaces. The main goal is to explain the role of the signature (of these real 4-dimensional manifold) and to present some signature formulas (of Hirzebruch) in terms of Dedekind sums.

Finally, in the last lecture we show that the signature can be determined by a lattice point computation in a tetrahedron (Brieskorn's result). Additionally, we will demonstrate Mordell's formula about the number of these lattice point in terms in terms of Dedekind sums.

Robin Pemantle: Map of Mathematics

Most students entering our mathematics graduate program have little knowledge of the active research areas of mathematics today. Moreover, their conceptual map of mathematics consists of only a few large regions (say: algebra, number theory, logic, analysis and topology, maybe differential equations and differential geometry) with no subdivisions drawn in or connections established between fields.

"Map of Mathematics" is a series of two lectures designed, along with the whole "Invitation to Research" lecture series, to remedy this. I have given similar lectures in 1993 and 1994 to new graduate students at Wisconsin, with a similar purpose in mind: that graduate students be able to make informed choices about their research directions, based on the background they will need and the scientific connections that will ensue.

The lectures begin with a map showing the main fields of mathematics listed above, together with some neighboring fields such as physics, computer science and economics. Proximity on the map relates roughly to proximity of the subject. After a quick overview and history of the heart of each area, the remainder of the lectures concentrate on the boundaries between the main fields. As in geological models, this is where most of the evolution is happening.

I will concentrate on syntheses with which I am most familiar. Some of these are as follows; of course many more could be mentioned, such as mathematical biology, complex dynamics, combinatorics and representation theory, etc.

1. At the juncture between algebra and computer science (and logic has something to say about theoretical decidability issues here) sits a wealth of recent work in computational group theory [1] and computational algebraic geometry [2].

2. The asymptotic evaluation of coefficients of multivariate generating functions is rooted in complex algebraic geometry, using methods as well from harmonic analysis and algebraic topology (specifically morse theory and stratified morse theory). The mathematics involved is roughly the same as in the study of singularities and bifurcations [3].

3. Percolation theory is a subfield of probability theory dealing with connectivity properties of random lattice graphs. Of longstanding interest to statistical physicists and chemists, this subject has taken on new importance via its conjectured relation to conformal field theories [4], which were just proved in 2001. Also related to this are stochastic partial differential equations known as Loewner evolutions.

4. Counting the longest increasing subsequence of a uniformly chosen permutation was a longstanding problem in probabilistic group theory. The recent solution [5] involved the distribution of eigenvalues of random matrices, and the solution of Riemann-Hilbert problems, which also arise in the analysis of equilibrium measures for certain random walks and queuing systems [6].

[1]	Kantor and Seress (eds.). Groups and computation. (vols I-III).
[2]	Cox, Little and O'Shea. Ideals, varieties and algorithms.
[3]	Arnold, Gusein-Zade and Varchencko. Singularities of differentiable maps. (vols I-II).
[4]	Langlands, Pouliot and Saint-Aubin. Conformal invariance in two-dimensional percolation.
[5]	Baik, Deift, and Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119-1178.
[6]	Fayolle, Iasnogorodsky, and Malyshev. Random walks in the quarter-plane.

Dijen Ray-Chaudhuri: Coding Theory

Coding Theory is the mathematical methodology to improve reliability of communication through a noisy channel in an efficient economically least expensive way. Mathematical coding theory has been applied in diverse areas including transmission of messages to earth from satellites in space missions, wireless telephones, improvement of quality of sounds in compact discs, cryptography for security of messages. These introductory lectures will give a brief description of the subject to attract students to this field. This field has expanded in the last decades at an exponential rate and is full of challenging mathematical problems whose solution will be of great benefit to our technological society.

Neil Robertson: Structural Graph Theory

The following is a series of lectures introductory to graph structure theory. A recreational problem and an unsolved problem will be given for each topic.

(1) Path connectivity in graphs. Examples of graphs, define graphs, subgraphs and isomorphism. Introduce paths, connected graphs, connected components, distance, circuits and trees. State Menger's theorem, what is vertex k-connectivity, and the structural theorems about 3-connected graphs. Introduce the deletion-contraction minors of graphs.

(2) The importance of trees. State some equivalent definitions of trees. Express the inductive construction of finite trees and spanning subtrees. Discuss properties of infinite trees such as Konig's infinity lemma and the equivalence of the axiom of choice and the existence of spanning subtrees in connected graphs. Define a "global" tree-structure of graphs that is preserved under taking minors.

(3) Graphs on surfaces. Elementary discussion of closed surfaces, examples of graph embeddings, their characteristic combinatorial properties and the Euler polyhedron formula. Discuss planar graphs, the Jordan curve theorem, and Kuratowski's theorem that there are only two topologically minimal nonplanar graphs.

(4) Finiteness aspects of tree and surface structure. State the Kruskal tree theorem that any infinite sequence of finite trees has some tree topologically contained in a tree further along in the sequence. Illustrate this by the special case of Higman's sequence theorem. Describe how the analogue of Kruskal's tree theorem, for minor inclusion over all finite graphs, generalizes Kuratowski's theorem and give the main "idea" behind its proof.

Warren Sinnott: Number Theory

In this series of lectures, we will survey the some of the problems that have directed the development of number theory throughout its long history, and illustrate some of the methods (some quite modern) that have been used to study and in many cases solve these problems.

We will begin with a discussion of some easily stated problems of number theory (many of them quite old): questions about prime numbers (especially questions about the regularities and irregularities of their distribution) and questions about the solutions to various Diophantine equations (such as Pell's equation, the Pythagorean equation, Fermat's equation, elliptic curves).

We will survey what is known and what is still unknown about such problems, and use them to illustrate the remarkable variety of methods that have been developed to treat them:

Algebraic methods: the study of rings of algebraic integers can be used give a complete description of the solutions to some simple Diophantine equations; and slightly more complicated equations lead to questions about algebraic numbers that are still unsolved today.
Geometric methods: some Diophantine equations can be studied from the perspective of algebraic geometry: the Pythagorean equation and elliptic curves are easily accessible examples.
Topological methods: the modern perspective in number theory is that one can effectively study number fields through their various completions. We will look at p-adic numbers and contrast them with the more familiar real completion of the rationals.
Analytic methods: zeta functions and L-functions were originally introduced to study the distribution of primes, and provide nice examples of applications of analysis to number theory. Furthermore, in the past century and a half, such functions have been found to contain an astonishing wealth of arithmetic information about number fields and algebraic varieties. We will discuss some of this information through examples. We may also discuss p-adic analysis and in particular look at the beginnings of the theory of p-adic L-functions.

We will conclude by describing some applications of number theory to fields outside number theory proper: coding theory and cryptology. Once the most prominent example of "pure mathematics", number theory today finds itself relevant for the practical problems of secure and reliable communication.

Some references for more information:

[1]	Kenneth Ireland and Michael Rosen. A classical introduction to modern number theory. Graduate Texts in Mathematics 84. Springer, New York, 1990.
[2]	Barry Mazur. Arithmetic on curves. Bull. Amer. Math. Soc. 14 (1986) 207-259.
[3]	Joseph H. Silverman and John Tate. Rational Points on Elliptic Curves. Springer, New York, 1992.
[4]	Andre Weil. Number Theory. An approach through history: From Hammurapi to Legendre. Birkhäuser, Boston, 1984.
[5]	Don Zagier. The first 50,000,000 prime numbers. Mathematical Intelligencer 0 (1977) 7-19.

David Terman: Mathematical Biology

Models for biological systems often exhibit a rich structure of dynamic behaviors. Examples include models for cardiac rhythms, tumor growth, the cell cycle, interacting species and neuronal activity patterns. In this module, we consider one of these systems in detail.

In the first lecture we discuss the biological system and methods for modeling that system. In the second lecture we study mathematical methods for analyzing the model. These may consist, for example, of geometric phase space methods for understanding nonlinear dynamical systems. The third lecture is devoted to computational methods. We will use interactive numerical software that solves and displays solutions to the model. This will allow us to introduce a number of important mathematical topics including bifurcation theory, oscillations and chaotic dynamics. Finally, in the last lecture we integrate the analytic and computational tools to study the mathematical model and understand what the mathematics has to tell us about the motivating biological problem.

VIGRE at The Ohio State University