Details of IMR 2004 activities  Mathematics Graduate Program 
GeneralitiesLectures 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. 
1. The Inverse Function Theorem^{ }TopicThe 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 infinitedimensional versions may surface. Professor Han has kindly written some notes.
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2. Linear algebra^{ }TopicA useful synopsis of linear algebra: vector spaces, linear transformations, eigenv{alueector}s, etc.
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3. Metric spaces: to R^{n} ... and beyond!TopicThe most common topological spaces are metric spaces and R^{n} 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 Buzz Lightyear's Pizza Planet is probably a metric space, in spite of the Evil Emperor Zurg.
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4. Manifolds^{ }TopicAny surface in R^{3} of the form z=f(x,y) is a manifold, and so is any curve in R^{n}, 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.
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5. The Axiom of Choice^{ }TopicThe 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.
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6. Ordinary differential equations^{ }TopicOrdinary 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 FrenetSerret equations classically use the curvature and torsion measurements of curves in R^{3} 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.
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7. The Classical Lie Groups^{ }Topic GL_{n}, SL_{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 occasion will be a friendly one.
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Tracking numbers^{ }TopicThe 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.
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2. PDEs as related to GeometryTopicA description will go here as soon as we figure out how to describe it.
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3. Correlations in discrete probability modelsTopicDiscrete 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.
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4. Groups acting on TreesTopicIf 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 nonminimal tree actions.
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5. Graph AlgebrasTopicThis will be about Algebras that arise from graphs, by a complicated method that is extremely unnatural.
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Administrative "stuff"
(Friday afternoon) ^{ }
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. At 3:00 (Friday), Teresa Delcorso will give a short presentation on applying for Competitive Fellowships and other pots of money.

Our computers & software^{ }TopicThis 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.
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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.Have fun, and please help one another. 
FourSquare^{ }Four Square rulesMake a square and number squares 14. 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 soccerstyle 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 nonpedestrian transportation will be arranged. 