Mathematics Department - References for IMR 2003

Details of IMR 2003 Activities


Mathematics Graduate Program

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 review 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 teach Math 501, Real Variables, in the fall 2003 semester. His principal research interest is stochastic processes, and he has investigated applications in mathematical biology and mathematical finance.
 

Metric spaces: the example of Rn 

 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.

Lecturer
Professor Feng Luo studies low-dimensional topology. He will be teaching a graduate course on the topology and geometry of 3-manifolds this semester. 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.
 

Linear algebra 

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

Lecturer
Professor Robert Wilson is returning to teaching and research after a number of years as a major administrator in the university, first as Chair of the Math Department, and then as a Dean in the Faculty of Arts and Sciences. Professor 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.
 

The classical linear groups 

 Topic
GL(n), O(n), U(n), etc. These groups of linear transformations occur and reoccur in almost all areas of mathematics. You should meet them now so when you reencounter them the occurence will be friendly.

Lecturer
Professor Roe Goodman is an accomplished teacher and researcher and bassoonist. Therefore he will spend most of his time as Acting Chair of the Math Department during the fall 2003 semester. He has also coauthored a very big book about the classical groups.
 

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 Stephen Greenfield has been graduate director several times. Now he is not. He seems to like COLORS. He is interested in lots of mathematics and applications of mathematics. He has studied several complex variables and PDE's and a little bit of Lie theory, and some combinatorics and even cryptography.
He is chiefly responsible for most of this (the IMR web stuff).
 

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 analysis ( bifurcation problems and the solvability of PDEs). I will discuss the set up of these theorems, provide a few sample applications, and perhaps indicate where and how their infinite dimensional versions arise.
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 and this semester he is teaching an introductory graduate course on PDE's.
 

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 Charles Weibel is the Director of the Mathematics Graduate Program. He writes: "Professor Weibel studies Motives. They were introduced by Cézanne, and mostly live in Algebraic Geometry these days. What are his motives?"
 

Administrative "stuff" 

 During this time we hope that most students will deal with 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.
  • 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 8/27/2003.
    • See Risa Hynes in Hill 322 for a computer account.
    • See Grace Kurkowski in Hill 346 for a key to your office (you will need to give a five dollar deposit).
    • Please see Chuck Weibel or Carla Ortiz in the Graduate Office (Hill 306) to 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.
Is this enough?
 

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
Stephen Hartke is currently working with Fred Roberts on topics in graph theory. He received his Bachelor's degree from the University of Dayton. He is the graduate student representative to the Math Department's Computing Committee. And yes, he wants two hams for dinner.
 

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.
 

Scaling Equations for Multiresolution Analyses: A Characterization Problem 

 Topic
Functions can be decomposed in many ways. For example, sound waves can be decomposed into pure tones, represented by sine and cosine waves of different frequencies (thus "harmonic analysis"). A useful technique for many modern applications is to decompose functions by scale (think of different resolutions for digital images). I will give a concise introduction to this idea, then describe the problem I am working on and some of the interesting turns it has taken.

Lecturer
Eva Curry is a fifth-year graduate student at Rutgers. Her thesis advisor is Dick Gundy. She is a member of the Math Department's Graduate Student - Faculty Liaison Committee, which provides some useful services to math grad students. She began her undergraduate career at Bowdoin College, but completed it at the University of Maine. Eva also likes to sing, but is not as good at it as Mr. Weininger. She is an avid outdoorswoman, when she has the time.
 

L-functions and their zeros 

 Topic
Today L-functions are playing the central role in number theory. I will give an easy introduction to them with explanations about why they are important. The glimpse into my research will amount to talking a bit about statistics of zeros of L-functions.

Lecturer
Matthew Young obtained his bachelor's degree at the University of Minnesota. He is currently working with Henryk Iwaniec. A semi-false biography can be obtained here.
 

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 fifth-year graduate student at Rutgers working with Wolmer Vasconcelos. He obtained his bachelor's degree from Sharif University.
 

Restricted permutations 

 Topic
I will briefly recount the history of the study of restricted permutations. Then I will talk about some stuff I've done.

Lecturer
Vince Vatter, a graduate of Michigan State, is a third-year student working with Doron Zeilberger. He believes that the AL Rookie-of-the-year award should go to Angel Berroa, and hopes that his Tigers make it to 121.
 


The "management" thanks Mr. Vatter for volunteering very recently to be a speaker. Mr. Weininger was supposed to give the (now virtual!) talk below but an illness in his family has prevented this.

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 current 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 fourth-year graduate student at Rutgers. His thesis adviser is Jeff Kahn. He likes to sing." I believe the last "he" applies to Mr. Weininger, not to Professor Kahn.
 

From the Laplace equation to harmonic maps: the impact of topology 

 Please use this link.
 

Move in 

 You may need this time to move into your lodgings. Have fun, and please help one another. There is also a free opera performance nearby :

FRIDAY, August 29, at 8:00 p.m. - Buccleuch Park, New Brunswick, NJ - Puccini's Turandot with Joseph Colaneri conducting; Eva Urbanova' (Turandot), Jennifer Check (Liu`), Eduardo Villa (Cala`f), Andrew Gangestad (Timur), Kim Josephson (Ping), Michael Forest (Pang), Charles Reid (Pong), Charles Anthony (Emperor Altoum), Alfred Walker (Mandarin), and Belinda Oswald and Elaine Flynn Young (Handmaidens). (Raindate: Sunday, August 31)
This is one of " The MET IN THE PARKS series of free outdoor concert performances by The Metropolitan Opera".
 

Barbecue 

 An exciting celebration at the home of the wonderful graduate director (what are his motives?) will be given. Maps will be distributed and transportation will be arranged.
 

Party 

 A party, whose attendance is almost always restricted to graduate students, is usually given by the (now) second-year students for the entering and now first-year students. I hope that more information will follow about this.
 

Hike 

 The actual hike
The hike will take us along a section of the Delaware and Raritan Canal on what used to be a tow path for the canal. The path is flat, reasonably wide, and has a packed earth surface. "Hike" may be a bit of a misnomer, since our pace will be more of a comfortable stroll. We may walk as far as seven miles (eleven km), or we may turn around sooner depending upon the mood of the participants. Further information about the canal is available, including maps and history.

Bring a water bottle full of good drinking water, since it will still be warm here at the beginning of September. Dress in comfortable clothing. Hiking boots are not necessary, but some sort of comfortable footwear is a good idea (there may be a few roots sticking up, so sandals might not be desirable). While the trail is mostly shaded by big trees, I do recommend sunscreen. You may wish to wear bug spray or some other sort of insect deterrent, as well.

Background
If the weather is suitable, students will be led on a long walk from Hill Center. Although New Jersey is a very densely populated part of the U.S., beautiful areas are available for recreation. During the hike, watch out for poison ivy and other menaces, such as skunks, poisonous snakes, snapping turtles, and faculty. Also don't forget West Nile virus and Lyme disease and rabies and ... also black bears, oh my!

Some of the rules suggested by the State of New Jersey for encounters with bears may also be useful when graduate students meet faculty members. Appropriately, the most relevant rules are the odd-numbered ones (merely replace bear by faculty member in what follows):

  1. Remain calm. Most bears are easily frightened into leaving.
  1. Make sure the bear has an escape route.
  1. If a bear stands on its hind legs, it is merely trying to get a better view rather than acting in a threatening way.
  1. If you encounter a bear at close range, remain standing upright, avoid direct eye contact, back up slowly and speak in a calm, assertive voice.

Added 8/13/2003 Two cases of Rocky Mountain Spotted Fever were recently diagnosed in Hunterdon County. Diseases spread by ticks can be serious. 

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