Mathematics Department - IMR 2010 Details

Details for IMR 2010


Mathematics Graduate Program

Our computers & software  
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.

Lecturer
Graduate student Kellen Myers , a member of the department's Computing Committee, will be speaking.
 

Written Qualifying Exams   (Monday and Tuesday 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.

Here are lecture notes on metric spaces.
Lecturer: Professor Daniel 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 5) and analysis (bifurcation problems and the solvability of ODEs and PDEs, advertising for lecture 9). 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.

Here are notes for this lecture.

Lecturer: Professor Zheng-Chao Han

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 from a past IMR.

Lecturer: Professor Robert Wilson

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: "home page" for the Axiom of Choice; a systematic exposition by Professor Ken Ross.

Lecturer Professor Gregory Cherlin

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 Feng Luo

6.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.
Here are some lecture notes on integral transforms.
Lecturer: Professor Roe Goodman

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.

Here are slides from a past IMR lecture.

Lecturer: Professor Lev Borisov
 

8. The Classical Groups  

Topics    The symmetry groups of the Classical Geometries on finite-dimensional vector spaces - or, one step away, on projective spaces. The word "geometry" should be broadly interpreted and included various types bilinear forms -as well as other things. Thus symmetric forms lead to the orthogonal groups O(n), alternating forms to the symplectic groups Sp(n), etc. Theses groups share many basic properties among themselves, but each type maintains its individuality. The Classical Groups and minor variants of them occur in myriad locales in mathematics. A companion to this phenomenon is the fact that the Classical Groups provide most of the building blocks for all groups satisfying certain finiteness conditions (e.g. Lie groups, algebraic groups, finite groups).
Here are some lecture notes on classical groups.

Lecturer: Professor Richard Lyons
 

9. Ordinary differential equations 

 Topics    Differential equations are closely related to geometry and physics. In this talk, I will introduce some simple differential equations arising from differential/algebraic geometry and general relativity. I will also explain how such equations can be solved using geometric information.

Lecturer: Professor Jian Song
 

Graduate Student Research Glimpses


Glimpse 1
Lecturer: Brent Young
Title:Deriving Fluid Models from Discrete Particle Dynamics in Classical Physics
Abstract: To come.

Glimpse 2
Lecturer: Sushmita Venugopalan
Title:Vector bundles and connections
Abstract: I'll describe vector bundles over manifolds. To do calculus over these, we need something called a 'connection'. I'll try to tell you why a vector bundle over the sphere S2 doesn't admit a non-trivial flat connection, whereas a torus does.

Glimpse 3
Lecturer: Humberto Montalvan-Gamez
Title: Super-Uniformity of Billiard Paths and other Point Sets and Curves
Abstract: To come.

Glimpse 4
Lecturer: Gabriel Bouch
Title:The Speed of Sound and Statistical Mechanics
Abstract: To come.

Glimpse 5
Lecturer: Yusra Naqvi
Title: Group Actions on Buildings
Abstract: To come.



Breakfast! (Saturday and Sunday 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  (Tuesday at 12:30) Lunch  (Saturday and Sunday at 12:30, Monday at 12:00)

All lunches in Hill 703

Tuesday's lunch will be a large gathering for new and continuing graduate students and faculty.

Saturday, Sunday and Monday lunch will be for participants in the mini-conference. During lunch on Sunday, some continuing graduate students will lead an informal discussion about: "What every math grad student should know".
 

Grad Student Welcome Meeting

The current math graduate students are organizing a welcome meeting for the entering Ph.D. students on Saturday Night. Time and place will be announced. (Priyam Patel is the contact person.)


 

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.   Bobby Demarco 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.

Thom Tyrrell will be the convener.
 

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