Mathematics Department - IMR 2008 Details

# Details for IMR 2008

 Our computers & software   (Friday 3:30 PM) This will provide a brief overview of the computing environment of the Math Department and the University. Particular attention will be given to items of interest to math graduate students. LecturerGraduate student XXXX, a member of the department's Computing Committee, will be speaking.

 Written Qualifying Exams   (Thursday and Friday 9AM-noon) Mostly for second-year students but incoming students may take it without it counting towards their two attempts.

# Faculty Expository Lectures

These lectures are intended to provide a brief review or introduction to some important topics that are sometimes missed in undergraduate programs.

## 1.Metric spaces and elementary topology

Topics
Metric spaces, including Rn and the most common function spaces, provide a nice setting in which to discuss basic concepts of topology - continuity, compactness, connectedness, and so on. Metric spaces provide a natural setting for some fundamental concepts and results that appear throughout mathematics, such as completeness and the contraction mapping theorem, are the natural setting for.

Here are lecture notes from a past IMR lecture on metric spaces.
Lecturer: Professor Dan Ocone

## 2.The Inverse and Implicit Function Theorems

Topics
The Inverse and Implicit Function Theorems are fundamental tools in the study of many problems in geometry (the local structure of manifolds, and when is the hypersurface f=0 a manifold - advertising for lecture 6) and analysis (bifurcation problems and the solvability of ODEs and PDEs, advertising for lecture 4). The context of these theorems, their statements and a few typical applications will be provided. If there is unbounded time, infinite-dimensional versions may surface.

Lecturer: Professor Roger Nussbaum

## 3.Normal Forms for Matrices

Topics
Given an equivalence relation on matrices—similarity, congruence, etc.—one can often find a particularly simple and essentially unique representative from each equivalence class: a normal form for matrices under that equivalence relation. In an abstract setting the problem is to choose a basis in which the matrix representing a linear transformation or bilinear form is in this normal form. We will look at several instances and some applications.

Here are lecture notes .

Lecturer: Professor Eugene Speer

## 4.The Axiom of Choice

Topics
The Axiom of Choice and its familiar equivalents, such as Zorn's Lemma, are used in many "constructions" in analysis, algebra, and topology. This lecture will discuss some of them.
References: A "home page" for the Axiom of Choice; a systematic exposition by Professor Ken Ross.

Lecturer: Professor Hector Sussmann

## 5.Manifolds

Topics
Any surface in R3 of the form z=f(x,y) is a manifold, and so is any curve in Rn, but manifolds are much more. The Implicit Function Theorem lets you certify them. Stokes' Theorem, the classical groups, spheres and donuts all play a role here, and some are edible.

Lecturer: Professor Chris Woodward

## 6. Ordinary differential equations

Topics
Ordinary differential equations (ODE's) provide a pleasant application of the abstract Contraction Mapping Theorem on metric spaces (Lecture 1). A form of local existence and uniqueness can be efficiently verified. For some linear systems, linear algebra provides a way of explicitly writing solutions. ODE's are important in geometry: e.g. the Frenet-Serret equations classically use the curvature and torsion measurements of curves in R3 to give unique descriptions of curves (under suitable hypotheses), and since the equations satisfy global Lipschitz estimates, solutions must always exist.
The situation with partial differential equations is very different, as was verified by Hans Lewy. An exposition of Lewy's lovely and important example, using elementary complex analysis, is here.

Lecturer: Professor Michael Vogelius

## 7. Multilinear algebra

Topics
A brief introduction to tensors, tensor spaces, and tensor products, and exterior products, and other basic constructions in multilinear algebra, and where they arise in mathematics.

Lecturer: Professor Roe Goodman

## 8. The Classical Lie Groups

Topics    The symmetry groups of vector spaces preserving linearity and extra linear algebra data (bilinear forms, etc.) occur naturally in myriad locales in mathematics. These so-called classical groups SLn, On, Spn, and their variants are basic algebraic groups and Lie groups, with many applications.

Lecturer
Professor J. B. Tunnell

## 9. Integral Transforms

Topics
The method of integral transforms is one of the most powerful mathematical methods, arising in mathematical physics, probability and statistics, number theory, and combinatorics It allows us to find exact solutions of many problems in differential and integral equations and it lies in the foundation of Integral Geometry. Your MRI-tests are based on integral transform. Integral transforms are a type of "mathematical outsourcing": you have a problem, you transform it to another problem in a different area of mathematics, solve it there and find a way to transform your solution back. I will outline basic notions of the theory.
Lecturer:

## Glimpse 1. Asymptotic Enumeration

Abstract.
For large n, the number of triangle-free graphs on n vertices is approximately 2^{n^2/4+n}. The interesting thing about this is not the number itself, but the story behind it. What the theorem says, in fact, is that if we know a graph has no triangles, then it is very likely bipartite. We can extend this: if we know a graph is triangle-free, but not bipartite (as mentioned above, this is unlikely; but that doesn't mean it is impossible), then most of the time we can make it bipartite by removing just one vertex. The talk will start from here and head toward some original results on somewhat similar questions.

Lecturer
Liviu Ilinca

## Glimpse 2. Sieve Methods in Number Theory

Abstract.
Sieve methods have a wide variety of applications in analytic number theory. In this talk, I will present the basics of sieve methods. In addition, I will present a way of using Selberg's sieve to give a version of the large sieve inequality.

Lecturer
Sara Blight

## Glimpse 3. The mass of an asymptotically flat 3-manifold

Abstract.
Surely you know how to weigh a bag of apples, or even an elephant! But what about some weirdo called a noncompact Riemannian 3-manifold''? You'll see that it's pretty much weighable if it's not too fat''. You don't want your scale to go nuts, do you?

Lecturer
Luc Nguyen

## Glimpse 4. Introduction to Option Pricing and Black Scholes Formula

Abstract.
One of the main problems in Mathematical Finance is derivative pricing . I will start with pricing of an European call option under a two state economy . After that, Brownian motion and Ito lemma will be introduced, I will end up wi th the famous Black Scholes option pricing formula.

Lecturer
Ming Shi

## Glimpse 5. The binomial theorem, a.k.a. the automorphism property, a.k.a. the formal Taylor theorem, and its corollary the higher derivatives of a composite function à la Faà di Bruno: a hint to vertex operator algebras

Abstract.
We shall begin by recalling the ultra-classical combinatorial/algebraic observation: the binomial theorem, although in a form perhaps somewhat unfamiliar. We shall only be concerned with it algebraically. We next recast this result in a compact version which realizes the key property as an operator on polynomials acting very similar to an automorphism. We again recast the result showing how this "automorphism" operator may be represented as a formal translation. We use this result to give a one-line proof of Faà di Bruno's formula which is a calculation of the higher derivatives of a composite function. In some sense the idea behind this proof lies at the heart of the theory of vertex operator algebras.

Lecturer
Tom Robinson

 Breakfast! (Saturday to Monday mornings at 8:30) We will try to supply an agreeable breakfast (this means free food, which is usually interesting to graduate students). Wake up!

 Welcome Lunch  (Friday at 12:30) Lunch  (Saturday & Sunday at 12:30) We will try to supply an agreeable lunch (this means free food, which is usually interesting to graduate students). Discussion at lunch on Friday should include most students' advisors, who can help students decide on initial registration for courses.

## Four-Square

Four Square rules
Make a square and number squares 1-4. Get a ball that bounces well.
The game begins with player number one dropping the ball and hitting it politely into any of the other squares. The person standing in that square lets the ball bounce in their square, before hitting it to any other square. The game continues until the ball is hit out of bounds or a player can not retrieve the ball.
At Rutgers, a new number one comes into play each time. If the player in square number n loses, each of the players in squares less than n move up one square. (Other rules require player number one moves to square four.)
Here is a more detailed set of rules.   Sara Blight will lead the fun.

## Aerobie

Aerobie is a relaxed soccer-style game played with an aerobie (a plastic annulus with aerodynamic properties like a frisbee has). We play by any of several sets of rules. The grad students will recruit you to play regularly with the Rutgers Aerobie Team if you let them.

James Dibble will be the convener.

## Friday Night Dinner

David Duncan and other continuing grad students will lead a dinner for entering graduate students. There will be some funds to defer the cost. The location is likely to be the Harvest Moon Brewery. Please come. Arrangements will be made at IMR.

This page was last updated on August 01, 2017 at 11:00 am and is maintained by grad-director@math.rutgers.edu.