References for IMR 2007
Details of IMR 2007 activities
Mathematics Graduate Program



This page is under development. Some of the information on the page may be incorrect.

Get ID card, Advance to GO. (Thursday afternoon/Friday morning)  
We hope that most students will arrive early and have time to deal with some initial administrative details. These include getting keys to offices, registering for courses, filling out various financial forms, and getting a university ID card (which will also be a library card). We also hope to take a picture of each of you, for display in the department and possibly on the web.
  • Please see Donna Lapinski on the third floor to have your picture taken so we can begin to recognize you.
  • International students: Check in at the International Center on the College Avenue campus. Attend workshops as necessary (such as the Employment Workshop to learn about I-9 and W4 forms and to apply for a social security card). You also should find out about PALS (English as a Second Language) and possibly some meetings about TA training.
  • Check your department mailbox in Hill 315. You may also have a personal mailbox in the post office, but you are responsible for mail in your department mailbox. Tuition remission cards will be in your department mailbox by 8/31/2007.
  • See Risa Hynes in Hill 322 for a computer account.
  • See Lynn Braun in Hill 311 for a key to your office (you will need to give a five dollar deposit).
  • If you have a TA appointment please see Lynn Braun in Hill 311 and give her your payroll papers and learn about the Health Benefits workshop (and then turn in your health benefits papers to Ms. Braun).
  • If you are a Fellow turn in paperwork to Lynn Braun in Hill 311.
Just do what you can!
 

Our computers & software   (Friday at 4 PM)
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
Graduate 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 if you are an incoming students, know that you can take it as a "free shot" - because you are not officially a student until the semester starts.
 

The Lectures

1. Metric spaces and elementary topology

 Topic
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. For some important concepts, like completeness and contraction mapping theorem, metric spaces are the natural setting.

Lecture notes are here.

Lecturer
Professor Eugene Speer is an expert in Mathematical Physics, as well as how the Undergraduate Office works (he used to be the Undergrad Vice Chair).
 

2. 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, 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 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), and is scheduled to teach Math 518 in Spring 08.
 

3. Linear algebra 

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

Lecturer
Professor Vladimir Retakh Info to follow
 

4. 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 discuss some of them.
References: A "home page" for the Axiom of Choice; a systematic exposition by Professor Ken Ross.

Lecturer
Professor Simon Thomas works on the boundary between logic and group theory. He will be teaching Selected Topics in Algebra: Geometric Group Theory this fall.
 

5. Ordinary differential equations 

 Topic
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 Konstantin Mischaikow
 

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

7. Bilinear Forms  

Topic
In Physics, the gravitational tensor is most positively, definitely a bilinear form. I'm less positive about the ones popping up in Morse Theory (geometry) and Invariant Theory (algebra). They are used to define the Classical Groups of Lecture 7 too. Here are some Notes about them.

Lecturer
Professor Siddhartha Sahi
 

8. 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 J. B. Tunnell
 

9. Transforms  

Topic
This talk is about the basic conflict of philosopy versus technicality in mathematics. When you do research, it has to be about something interesting like:

What is Complexity? What is Radomness? What is perfect order? How close can we get to perfect order?

On the other hand, to say something NONTRIVIAL, you need technical tools like Fourier transform, an ability to visualize abstract things like groups, algebras, functions of complex variables, and so on.

To be a successful mathmatician means you find a balance between these two.

Lecturer
Professor József Beck an an expert in combinatorics, especially partial colorings, discrepancy theory, and positional games, among others.
 

The Glimpses

1. Mathematical Biology  

 Topic
I will start with some basic concepts in mathematical biology, then explain the importance of multistability in biology. Finally prove a stability result with applications to mitogen-activated protein kinase pathway using theories from geometric singular perturbation and monotone systems.

Lecturer
Liming Wang is a six-year graduate student, and will finish next year. Her thesis advisor is Eduardo Sontag.
 

2. Experimental Math and Ramsey Theory  

 Topic
When I first came to grad school, I has no clue how to do research on math. It actually does not always need to be hard and take three years to understand the language. I will talk a bit about a nice new result on Ramsey theory or Schur number to be specific.

Lecturer
Thotsaporn Thanatipanonda (Aek) is working with Doron Zeilberger, one of the greatest experts in experimental math. He thinks programming is fun. Apart from math, he likes watching reality TV shows, playing video games, cleaning house and eating food.
 

3. Hyperbolic surfaces  

 Topic
A round sphere has a metric of positive constant curvature. A cylinder has a metric of zero curvature. Most surfaces have metrics of negative constant curvature (hyperbolic metric). A hyperbolic surface can be constructed by gluing hyperbolic triangles or hexagons. All the marked hyperbolic metrics on a surface form the Teichmuller space.

Lecturer
Ren Guo is a five-year graduate student. He is interested in geometry and topology. His advisor is Prof. Feng Luo.
 

4. Graded dimensions and generating functions

 Topic
The graded dimensions of vector spaces with finite gradings can be expressed in terms of generating functions. We can find these generating functions for various interesting algebras. One reason we would like to find these functions is because by using related techniques we can consider the representations of the automorphism group of an algebra acting on the algebra. In this talk I will describe the basic techniques and apply them to one important example.

Lecturer
Colleen Duffy is a fifth year graduate student working with Professor Wilson in the area of noncommutative algebra. My research involves finding representations of certain algebras associated with graphs. When I am not doing math, I am most likely hiking, canoeing, playing aerobi, or doing something with my church.
 

5. Hyperbolic PDEs in Mathematical Physics 

 Topic
Hyperbolic partial differential equations are prominently featured in many of our most successful models of physical phenomena. Familiar examples of such phenomena that have been successfully modeled with hyperbolic PDEs include the flow of liquids and gases, the propagation of sound waves, and the propagation of electromagnetic waves (in the form of visible light, radio signals, microwaves, cell phone signals, etc.), described by Maxwell.s equations. More exotic examples include Einstein.s gravitational theory of General Relativity and the Born-Infeld model of electromagnetism, a nonlinear version of the classical Maxwell Equations. In this talk, I will describe some of the important qualitative features shared by many hyperbolic systems, including the finite speed of propagation of data, the definability of .energies,. the formation of singularities in the solutions, and the development of shock waves. Some of these properties will be illustrated with simple examples. Additionally, I will introduce a few of the systems mentioned above and describe some open questions concerning their behavior.

Lecturer
Jared Speck is a 5th year graduate student with two advisors: Michael Kiessling and Shadi Tahvildar-Zadeh. Michael researches statistical mechanics, the foundations of physics, and nonlinear PDEs, while Shadi is an expert in nonlinear hyperbolic PDEs. My current research interests are strongly influenced by their work; I have worked on the Born-Infeld model of nonlinear electromagnetism and various models concerning self-gravitating fluids. Being a music enthusiast, I appreciate that my academic interests overlap with the physics of sound.
 

Other Stuff in the Schedule

Administrative "stuff"  
Below is a list of administrative chores. You can do most of them any time during the week of August 27-31. Friday August 31 from 2:00 to 3:30 is set aside to do any that remain.
  • International students Check in at the International Center on the College Avenue campus. Attend workshops as necessary (such as the Employment Workshop to learn about I-9 and W4 forms and to apply for a social security card). You also should find out about PALS (English as a Second Language) and possibly some meetings about TA training.
  • All students
    • Check your department mailbox in Hill 315. You may also have a personal mailbox in the post office, but you are responsible for mail in your department mailbox.
    • Tuition remission cards will be in your department mailbox by 8/29/2007.
    • See Risa Hynes in Hill 322 for a computer account.
    • See Justin Gross in Hill 326 to have your picture taken. (This is for the department only; you will need to have your picture taken elsewhere for your Rutgers ID).
    • See Donna Lapinski in Hill 346 for a key to your office (you will need to give a five dollar deposit).
    • If you have a TA appointment please see Lynn Braun in Hill 311 and give her your payroll papers and learn about the Health Benefits workshop (and then turn in your health benefits papers to Ms. Braun).
    • If you are a Fellow turn in paperwork to Lynn Braun in Hill 311.
    • You will also need to register for classes. After registering for classes you will need get your term bill which will need to be paid for which you will need Tuition remission card mentioned above. The process of registering and paying bills will be covered at the university wide orientation for new students. If you have additional questions about this, you can ask Traci Cole in Hill 306.

     

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.)
Other ad hoc rules will be revealed by Paul Ellis at random moments, especially those about "offending the sensibilities."   ;)

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.
 

Move in; Relax  

 You may need this time to move into your lodgings.
Have fun, and please help one another.
 

Dinner in New Brunswick 

 Several faculty members and continuing grad students will help pay for dinner for the entering grad students! The location is likely to be the Harvest Moon Brewery. Please come. Arrangements will be made at IMR.
 

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