Details for IMR 2009
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Our computers & software
(Friday 3:30 PM)
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 | |
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Written Qualifying Exams
(Thursday and Friday 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 topologyTopicsMetric 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
from a past IMR lecture on metric spaces.
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2.The Inverse and Implicit Function TheoremsTopicsThe 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 Michael Vogelius |
3.Normal Forms for MatricesTopicsGiven 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 Eric Carlen |
4.The Axiom of ChoiceTopicsThe 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 |
5.ManifoldsTopicsAny 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 Chris Woodward |
6. Ordinary differential equationsTopicsAbstract: 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
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8. The Classical Lie GroupsTopics The symmetry groups of vector spaces preserving linearity and extra linear algebra data (bilinear forms, etc.) occur naturally in myriad locales in mathematics. These so-called classical groups SLn, On, Spn, and their variants are basic algebraic groups and Lie groups, with many applications.
Lecturer:
Professor Feng Luo
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Graduate Student Research Glimpses
Glimpse 1 | |
| Lecturer: | Amit Priyadarshi |
| Title: | Hausdorff Dimension of Fractal Sets |
| Abstract: | The definition of Hausdorff measure and Hausdorff dimension of subsets of a general metric space will be given. We will see a few examples of fractal sets. Many important examples are obtained as an "invariant set" of "iterated function systems" on a complete metric space. A general problem is to obtain theorems which allow accurate estimation of the Hausdorff dimension of such an invariant set. We use tools from functional analysis to obtain formula for Hausdorff dimension of invariant sets. No measure theory background will be assumed for this talk. |
Glimpse 2 | |
| Lecturer: | Jin Wang |
| Title: | Volatility Models in Mathematical Finance |
| Abstract: | Volatility is a measure of the amount of randomness of a financial instrument at any time. It is difficult to measure and even harder to forecast, however, it is one of the main inputs into option pricing models. In this talk, I will describe some of the important volatility models, including the Black-Scholes model, Dupire's local volatility model and Heston's stochastic volatility model. Additionally, I will introduce a volatility model for multi-dimensional stochastic process, and talk about some open problems. |
Glimpse 3 | |
| Lecturer: | Wesley Pegden |
| Title: | Playing randomly to avoid repetition: Thue games and the Local Lemma |
| Abstract: | It turns out that it is possible to have a sequence over three
symbols without any consecutive repeated subsequences. Such a
sequence was constructed by Axel Thue at the beginning of the last
century. (It's not too hard to see that such a sequence is not
possible over two symbols!) Thue's results began the study of
nonrepetitive sequences, a subject rich with diverse questions.
We focus on the subject of nonrepetitiveness in the setting of a game. Suppose two players make a sequence over some fixed set of symbols by alternatingly choosing the next symbol of the sequence. We want to know: can one of the players force the resulting sequence to be nonrepetitive? It turns out, in various senses, that the answer to this question is often yes. Surprisingly, we get proofs only by letting the player interested in nonrepetitiveness play randomly. It is *not* true in this case that the player is likely to win. However, using an extension of the Lovasz Local Lemma, it is possible to show that he *might* win---and this, we we will see, is enough to deduce that a perfect player in is situation would always win. We will discuss the main ideas of the proof, discuss the Local Lemma as a tool in Combinatorics, and play some games. |
Glimpse 4 | |
| Lecturer: | Debajyoti Nandi |
| Title: | Combinatorial identities using representation theory |
| Abstract: | We will talk about the connection between certain combinatorial identities and representation theory of a class of infinite dimensional Lie algebras (viz., affine). Modules over such algebras carry a structure of infinite dimensional graded vector space, which is a countable direct sum of fnite dimensional vector spaces. The notion of dimension for finite dimensional vector spaces can be generalized to the notion of graded dimension for such vector spaces. The basic idea is to compute the graded dimension in two different ways. Roger-Ramanujan type identities - one of which states that the number of partitions of a natural number into parts congruent to 1 or 4 modulo 5 is the same as the number of partitions into parts congruent to 2 or 3 modulo 5 - can be explained in this way. The talk will not assume any familarity with graded vector spaces or Lie algebras. |
Glimpse 5 | |
| Lecturer: | Dan Staley |
| Title: | Geometric Group Theory |
| Abstract: | Geometric group theory essentially uses the tools of geometry to study algebra. We take finitely generated groups, and find metric spaces on which they have an appropriate group action, called a "geometric" action. One of the first spaces we have with this group action turns out to be the group itself, and the closely-related Cayley graph of the group. It turns out you can familarity with graded vector spaces, or Lie algebras. take a lot of concepts from geometry and apply them to groups; for example, there is a notion of the "curvature" of a group. This ends up giving us a lot of algebraic information about the group. |
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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! | |
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Welcome Lunch
(Friday at 12:30)
Lunch
(Saturday & Sunday at 12:30)
All lunches in Hill 703 Friday's lunch will be a large gathering for new and continuing graduate students and faculty.
Saturday and Sunday 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".
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Four-SquareFour Square rulesMake 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. Sara Blight will lead the fun. |
AerobieAerobie 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.
James Dibble will be the convener.
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