References for IMR 2004

# Details of IMR 2004 activities

## Generalities

Lectures will contain simple examples and will stress the foundational importance of the topics. Connections between varied areas of mathematics and some applications will be emphasized whenever possible. Ideally the presentations will be neither too technical nor too rapid. We believe that all of the lectures will be useful to all of the students, and ideally, therefore, all entering students should attend all presentations. "Glimpses" will be given by graduate students of their work in several prominent research areas of the department. We also hope students will attend and enjoy the recreational opportunites that are provided.

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## 1. The Inverse Function Theorem

Topic
The Inverse and Implicit Function Theorems are fundamental tools in the study of many problems in geometry (the local structure of manifolds, 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, the infinite-dimensional versions may surface.
Professor Han has kindly written some notes.

Lecturer
Professor Zheng-chao Han studies non-linear partial differential equations which arise from geometric problems. He recently taught Math 501 (Real Analysis) and Math 517 (PDE's). This semester he is explaining how to solve PDE's in Methods of Applied Math.

## 2. Linear algebra

Topic
A useful synopsis of linear algebra: vector spaces, linear transformations, eigenv{alue|ector}s, etc.

Lecturer
Professor Robert Wilson has received awards recognizing his teaching, research, and service to the university. One of his major achievements, jointly with Richard Block, was classifying the mod p simple Lie algebras. He is now working on quasideterminants.

## 3. Metric spaces: to Rn ... and beyond!

Topic
The most common topological spaces are metric spaces and Rn is a nice example of a metric space. We'll review some definitions such as compact, connected, complete, and continuous (and anything else beginning with c) and examples.
Buzz Lightyear's Pizza Planet is probably a metric space, in spite of the Evil Emperor Zurg.

Lecturer
Professor Feng Luo studies low-dimensional topology. Although Professor Luo does "pure" (!?) mathematics, please note that one of his thesis students who studied folding of polyhedral surfaces got a patent from this work, and applied it to industrial packaging problems.

## 4. Manifolds

Topic
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 Steve Ferry is an expert on classification problems in manifolds and shapes. He will not tell you about Hilbert cube manifolds and homology manifolds unless you ask. But I definitely think you should ask him about the BC Blues.

## 5. The Axiom of Choice

Topic
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 choose some of them.
References: A "home page" for the Axiom of Choice; a systematic exposition by Professor Ken Ross.
Professor Ocone's notes are here.

Lecturer
Professor Daniel Ocone will be the Undergraduate Vice-Chair during 2004-2005. His main research interest is stochastic processes, and he has investigated applications in mathematical biology and mathematical finance.

## 6. Ordinary differential equations

Topic
Ordinary differential equations (ODE's) provide a pleasant application of the abstract Contraction Mapping Theorem on metric spaces. 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 Roger Nussbaum has recently been studying solutions of differential-delay equations and dynamical systems. He wrote the book on Functional differential equations (FDE's) and has studied ODE's and PDE's with the help of Functional Analysis. He is also famous for his book The fixed point index and some applications.

## 7. The Classical Lie Groups

Topic    GLn, SLn, On, Un, etc.
These groups of linear transformations occur and reoccur in almost all areas of mathematics. You should meet them now so when you re-encounter them the occasion will be a friendly one.

Lecturer
Professor Chris Woodward is an accomplished teacher and researcher. Ask him about honeycombs... or Wireless networks.

## Tracking numbers

Topic
The hilbert polynomial of a graded module carries a lot of information about the algebra. We will have a look at the coefficients of the the hilbert polynomial and attempt to extract some information from them.

Lecturer
Kia Dalili is a sixth-year graduate student at Rutgers working with Wolmer Vasconcelos. He obtained his bachelor's degree from Sharif University, and will defend his Rutgers thesis next Spring.

## 2. PDEs as related to Geometry

Topic
A description will go here as soon as we figure out how to describe it.

Lecturer
Aobing Li grew up and went to University in the Jianxi Province of China. She just received her PhD on the topic Conformally invariant nonlinear elliptic PDEs on manifolds. Beginning next month, she will be a member at the Institute for Advanced Study in Princeton. Beginning in Fall 2005 she will be an Assistant Professor at the University of Wisconsin. Not bad.

 This is not a PDE! What is it?

## 3. Correlations in discrete probability models

Topic
Discrete probability models such as the Ising and Potts models are of interest in combinatorics, probability, and statistical physics. My research involves problems concerning correlations between pairs of events in these models. I will describe some of these problems and try to give an idea of the difficulties involved in solving them.

Lecturer
Nicholas Weininger was an undergraduate at Macalester College and spent one semester studying in Budapest. He then worked for two years "on real-time operating systems and formal software verification" at Honeywell Technology Center. Now he is a fifth-year graduate student at Rutgers. His thesis adviser is Jeff Kahn.

## 4. Groups acting on Trees

Topic
If X is a graph, its universal cover X is a tree. The fundamental group G of X acts on this tree. If G is a subgroup of finite index in G and a finite subtree T of X is a fundamental domain for this action, we can label the vertices and edges of T with their stabilizer subgroups - this is the paradigm of a group acting on a tree. Since T is a fundamental domain, the action is clearly minimal. After explaining what this really means, I will talk about how to characterize non-minimal tree actions.

Lecturer
Laura Ciobanu was a graduate student in Romania. Her main interests are computational, combinatorial and geometric group theory. Her thesis advisor is Professor Sims, and this talk is about work with her coauthor, Professor Carbone.
Ms. Ciobanu was the coordinator of the REU summer program in 2002. She also won our TA teaching Excellence Award in 2002. Oh, and she is a great speaker.

## 5. Graph Algebras

Topic
This will be about Algebras that arise from graphs, by a complicated method that is extremely unnatural.

Lecturer
David Nacin is a graduate student working with Professor Wilson in the field of non-commutative geometry. He likes to play math songs on the guitar. His favorite math songs include YACINE, about a former math graduate student, and Mr. Graves, about the unheralded man who discovered the octonions a few months before Cayley.

## Our computers & software

Topic
This presentation will give 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.

Lecturer
Sam Coskey is a second-year graduate student. He was an undergraduate at the University of Washington. His nickname is Spaceghost. More relevantly, he is the current graduate student representative to the Math Department's Computing Committee.

## Lunch

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.

## Move in; Relax

You may need this time to move into your lodgings.

## 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 underhanded into any of the other squares. The person standing in that square lets the ball bounce in their square, before hitting it underhanded 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.)

## 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. Aaron Lauve will recruit you to play regularly with the Rutgers Aerobie Team if you let him.

## Barbecue Party at Buell Apartments

A party, whose attendance is almost restricted to graduate students. Since many incoming students will be moving into Buell apartments, this seems like a good location. It will start at 6PM. Go to Buell and look for the party on the lawn.

## Backyard Barbecue (6PM-)

An exciting celebration at the home of the wonderful graduate director (what are his motives?) will be given. This is located in East Brunswick, one mile from the nearest campus bus stop (Henderson apartments on Douglass, G bus). We will start the wood fire before 6PM, and food will be cooked. Maps will be distributed and non-pedestrian transportation will be arranged.

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