head> References for IMR 2006

Details of IMR 2006 activities


Mathematics Graduate Program


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 8/31/2005.
  • 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 Paul Raff will be speaking, substituting for graduate student Sam Coskey, who is a member of the department's Computing Committee.
 

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 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.
 

3. 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. Last year, he was the acting chair of the Math Department.
 

4. 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 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.
 

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 taught the graduate Financial Mathematics course last year, and will be teaching the Undergraduate Math Finance course this Fall. His main research interest is stochastic processes, and he has investigated applications in mathematical biology - and mathematical finance.
 

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 Peter Landweber is also a former Mathematics Graduate Director. He studies cobordism and other homology theories. Ask him how to use a pair of pants to study topological quantum field theories.
 

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.
 

8. Bilinear and Quadratic Forms  


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 Friedrich Knop studies symmetry in geometry and analysis. He also knows about teaching calculus at Rutgers, since he is in charge of one of the very large beginning calculus courses.
 

9. Transforms  


Integral transforms take functions to functions to functions, typically they are of the form Tf = ∫ab K(X,t)f(t) dt. Although they are linear, it isn't so useful to write them in matrix form. My favorite is Tf=0; it takes witchcraft to invert that one!

Fourier transforms have K=exp(-ixt) and are useful in number theory and probability theory. Discrete Fourier transforms are used in signal processing and combinatorics. Laplace transforms, where K=exp(-xt), are incredibly useful in building bridges, probability theory and Wi-Fi systems. The Mellin transform, in which K=xt-1, was named for a Finnish mathematician and not Carnegie-Mellin.

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

The Glimpses

1. Differential Delay Equations 

 Topic
This talk is about differential equations that relate a function's derivative at time t to its value at time (t-1):  x´(t) = f(x(t-1)).
I will discuss why such equations are of scientific interest and describe some ways in which they differ from ordinary differential equations; I'll also sketch some of the ideas from complex analysis, fixed point theory, and dynamical systems that go into the study of these equations. We'll look at lots of instructive pictures.

Lecturer
Ben Kennedy is a fifth-year graduate student in Functional Analysis, and will defend his Rutgers thesis next Spring. His thesis advisor is Roger Nussbaum
 

2. Discrepancy Theory 

 Topic
Discrepancy theory is a relatively new branch of mathematics dealing with irregularities in the distribution of points in some ambient space. Discrepancy theory comes into play when one has to choose data points for high-dimensional numerical integration or pick a small representative sample from a large collection.

The game-theoretic variant of a notoriously difficult open problem in discrepancy theory considers two players, Maker and Breaker, who take turns colouring the integers from 0 to N with their own colours. Maker wins if his/her lead on some homogeneous arithmetic progression exceeds a pre-specified target, and Breaker wins otherwise. Assuming both players play perfectly, who wins?

The answer will be revealed! (It depends upon the target, of course.)

Lecturer
Sujith Vijay is working with József Beck. Is there a discrepancy between this blurb and the actual lecture? You will have to see for yourself!
 

3. Symplectic Geometry 

 Topic
In Lagrangian mechanics we have a phase space consisting of position vectors q and momentum vectors p; the symplectic form is the quadratic form p.q. A subspace L (such as the subspace where q=0) is a Lagrangian if the quadratic form vanishes.

This gets jazzed up when positions lie on a manifold Q, in which case the phase space M is the cotangent manifold; it is a special case of a symplectic manifold. Symplectic 4-manifolds are important in Donaldson theory. This will all become much clearer when you hear Ms. Mau explain it.

Lecturer
Sikimeti Mau Her thesis advisor is Chris Woodward.
 

4. Schrödinger's Windows

 Topic
Many simulations in Quantum Mechanics are like video games in that the shock waves bounce off the edges of the playing field. Why can't they get it right? Chris will explain, and show movies too.

Lecturer
Chris Stucchio is a fifth-year graduate student in Mathematical Physics, and will defend his Rutgers thesis next Spring. His thesis advisor is Avi Soffer. He currently holds the University's Bevier Research Fellowship. He insists that this is all a mistake, but he'll take the money.
 

5. Gradient estimates for PDE's 

 Topic
Given an educated guess about what may solve a given PDE, the gradient points the way downhill to a solution of the PDE. We hope it doesn't point the way to a cliff!

Lecturer
Ellen Bao is a fourth-year graduate student working with Yanyan Li.
 

Other Stuff in the Schedule

Administrative "stuff" (Friday afternoon)  
During this time we hope that most students will deal with more 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.
  • 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 8/29/2006.
    • See Risa Hynes in Hill 322 for a computer account.
    • See Donna Lapinski in Hill 346 for a key to your office (you will need to give a five dollar deposit). Also have your picture taken so we can begin to recognize you.
    • 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 and have your photograph taken for your Rutgers identification card. For this you will need the Tuition remission cards mentioned above.
Is this enough?
 

Applying for Fellowships (Fri afternoon)  
At 2:00 (Friday), Teresa Delcorso will give a presentation on applying for Competitive Fellowships and other pots of money.
Please attend; it could be worth $10,000 to you!

More Administrative "stuff" (Monday morning and afternoon)  
Surely you haven't done all the red tape yet. Take another whack at it!

Breakfast! (Friday 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!
 

Lunch  (Friday to Monday 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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