Details for IMR 2011
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Our computers & software
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
(Tuesday and Wednesday 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 TopologyTopics 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. 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
on metric spaces.
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2. Vector Spaces and Canonical FormsTopics Given 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 matricesunder 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 Anders Buch |
3. The Axiom of ChoiceTopics 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: "home page" for the Axiom of Choice; a systematic exposition by Professor Ken Ross. Here are lecture notes from a past IMR. Lecturer: Professor Simon Thomas |
4. Importing Tools into Discrete MathematicsTopics One of the appeals of discrete mathematics is that it deals with elementary mathematical structures (finite systems of finite sets), and many of its problem statements can be understood by high school students. Often these problems are accessible through clever but elementary and self-contained arguments. Other times these problems are solved unexpectedly by using methods from other fields of mathematics: linear, multilinear and commutatitve algebra,topology, geometry and analysis. In this lecture I'll present two or three examples of combinatorics problems which are solvedby methods imported from other branches of mathematics.Lecturer Professor Michael Saks |
6. Duality and Multilinear AlgebraTopics A brief introduction to tensors, tensor spaces, and tensor products, and exterior products, and other basic constructions in multilinear algebra, and where they arise in mathematics.Here are slides for this lecture.
Lecturer:
Professor Roe Goodman
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7. Fourier SeriesTopics Just like a Taylor series is a limit of polynomials, a Fourier series is a limit of trigonometric polynomials. Another way of writing a Fourier series is as the sum over all integers n of c_n exp(inx), where c_n are called the Fourier coefficients. The key to computing the Fourier coefficients of a given function is to consider an inner product operation that is an infinite-dimensional analog of the usual dot product of 3-dimensional vectors. This leads us to the converse question, under which conditions a Fourier series converges (in the sense of mean square convergence), and to the concept of a complete inner-product-space, also known as a Hilbert space. In the relevant Hilbert space, Fourier series correspond to using a particular orthonormal basis consisting of the functions exp(inx) (times a constant factor). The question also arises under which conditions a Fourier series converges pointwise or uniformly. I close with an application in quantum mechanics.Lecturer: Professor Roderich Tumulka |
9.The Inverse & Implicit Functions TheoremsTopics TBA
Lecturer:
Professor Christopher Woodward
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10. ManifoldsTopics Manifolds are spaces that are locally homeomorphic to R^n for some n. Obvious examples of manifolds are R^n , itself and S^n = {points of distance 1 from the origin in R^{n+1} }. Less obvious examples are SO(n) = { n x n invertible matrices of determinant +1} and the set of possible configurations of a wall-climbing robot's legs.In this lecture, we will give an overview of the current state of manifold classification and give examples where manifolds arise naturally in solving mathematical, physical, and engineering problems. Lecturer: Professor Steven Ferry |
Graduate Student Research Glimpses
Glimpse 1 | |
| Lecturer: | Tianling Jin |
| Title: | The Method of Moving Planes |
| Abstract: | We will discuss the method of moving planes through some examples. It is an elementary and powerful tool in PDEs. |
Glimpse 2 | |
| Lecturer: | Jay Williams |
| Title: | Descriptive Set Theory and Classification Problems |
| Abstract: | Descriptive set theory began as the study of "nice" sets of reals, but has since come up in a wide range of mathematical contexts. We will start by discussing the perfect set property and Borel sets. Then we will see how the tools of descriptive set theory can be used to compare the relative complexity of classification problems in many different fields of mathematics. |
Glimpse 3 | |
| Lecturer: | Camelia Pop |
| Title: | How Math Can Be Used On Wall Street |
| Abstract: | The aim of this talk is to give an intuitive introduction to one of the fundamental problems in mathematical finance - options pricing. To better illustrate the problem, suppose one owns 100 Google shares and fears that the current economic instability will render them almost worthless in one year, but it may as well happen to double value because of technological innovations. Then, what one can do is purchase a put option on Google with a certain strike price K, which will enable them in one year to sell the stocks at the strike price if the stock market value falls below K. The question we want to answer is: what is the fair price to pay today for such a derivative security? To do this, we use a stochastic model for stock prices, and depending on the model, the derivative security's price will be characterized as the solution to some very interesting PDE. I plan to describe the main ingredients from stochastic analysis and PDEs which center in solving this problem. |
Glimpse 4 | |
| Lecturer: | Jorge Cantillo |
| Title: | Zero-Density Estimates and Applications |
| Abstract: | In Number Theory, L-functions are introduced as an analytic tool to study many arithmetic problems, like the distribution of primes in arithmetic progressions. Knowledge about the distribution of nontrivial zeros of a family of L-functions can be translated into knowledge about their coefficients which by definition carry arithmetic information. The Generalized Riemann Hypothesis (GRH) already says a lot about the location of these zeros from which arithmetic implications for many applications follow. However, in the absence of a proof for GRH, weaker information about the distribution of the zeros is needed to obtain unconditional results. This information is provided by zero-density estimates which in some applications can produce results comparable to those implied by GRH. This talk presents the ideas above in the case of the Riemann zeta-function and the problem of counting primes in short intervals. |
Glimpse 5 | |
| Lecturer: | Jinwei Yang |
| Title: | Introduction to Vertex Operator Algebras |
| Abstract: | I will start with the concept of Lie algebra, illustrating it with a basic three-dimensional Lie algebra sl(2) and an infinite-dimensional Lie algebra sl(2)^. Then I will show how objects called "vertex operators" can be used to realize this Lie algebra. Such operators are related to string theory in physics. Then I will sketch the general concept of "vertex operator algebra" as an abstraction of algebraic properties of vertex operators. These algebraic structures are related to many ideas in mathematics as well as physics. In part of my own research, I have been studying representations of "strongly graded vertex algebras," and I will sketch some of this work and its motivation. |
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Breakfast!
(Friday and 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
(Wednesday at 12:30)
Lunch
(Friday, Monday and Tuesday at 12:00)
The lunches for Friday, Monday and Wednesday will be in Hill 703 and the lunch for Tuesday will be in 323. Wednesday's lunch will be a large gathering for new and continuing graduate students and faculty.
Friday, Monday, and Tuesday
lunches will be for participants in the mini-conference.
During lunch on Friday, 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. Robert McRae 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.
Thom Tyrrell will be the convener.
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