UM Student Analysis Seminar: Fall 2010

New date/room: Wednesdays, 4:10-5:00pm, Room 4096, East Hall
University of Michigan Department of Mathematics



PDE Reading Group: There is also a student-run PDE reading group that meets on Mondays and Thursdays. See the webpage of the group for more information.

Date SpeakerTitle
Thu, 9/16/2010Jen BeichmanSomewhere between local and global: Almost global existence and the wave equation
Thu, 9/23/2010no talkno talk (because of the Lazarsfeld lecture)
New date/time/room: Wed, 9/29, 4pm (4096 EH)Rafe KinseyThe Kakeya Needle Problem and Besicovitch Sets I: Construction of the Sets
Wed, 10/6Rafe KinseyThe Kakeya Needle Problem and Besicovitch Sets II: Applications to Combinatorics, Number Theory, Harmonic Analysis, and PDE
Wed, 10/13Jeff CalderExistence and uniqueness for the Catt\'{e} regularization of the Perona-Malik equation
Wed, 10/20Joe RobertsIntroduction to Universal Taylor Series
Wed, 10/27 (special room: 3866 EH)Updated: Jingchen (Chandler) WuNoncooperative Differential Games (joint with Student AIM seminar)
Wed, 11/3William GignacThe Ergodic Theorem
Wed, 11/10Andy ZimmerEntropy in Dynamics
Wed, 11/17Purvi GuptaNewton's Method and the Nash-Moser Implicit Function Theorem
Wed, 11/24n/aNo talk because of Thanksgiving
Wed, 12/1Rafe KinseyThe Uncertainty Principle in Harmonic Analysis
Wed, 12/8Nikolaos Pattakos (MSU)Continuity of Weighted Estimates in A_p Norm

Abstracts


Somewhere between local and global: Almost global existence and the wave equation, Jen Beichman (9/16/10): The basic questions in PDE revolve around existence and uniqueness. In this talk, we will start with the basic local existence theorem in PDE using the Banach fixed point theorem and explore the continuity arguments used to extend the existence times of solutions to inhomogeneous wave equations. The technique involves very basic analysis, and the style of argument can be generalized to showing long time existence for other PDE. This talk should be accessible to first-years, who are encouraged to come. (Graduate-level PDE experience will not be necessary.)

The Kakeya Needle Problem and Besicovitch Sets I: Construction of the Sets, Rafe Kinsey (9/29): What is the smallest set in the plane in which one can rotate a unit line segment 180 degrees? This problem, posed by Kakeya in 1917 and solved by Besicovitch in 1927, and related questions--some of them still open--turn out to have deep and important connections to combinatorics, number theory, harmonic analysis and PDE. In this talk, I will present a solution to the problem, an elegant geometric construction. (In next week's follow-up talk, I will talk about the many cool applications of these sets, and related conjectures.) The talk should be accessible without much background in analysis; the construction is mostly elementary geometry.

The Kakeya Needle Problem and Besicovitch Sets II: Applications to Combinatorics, Number Theory, Harmonic Analysis, and PDE, Rafe Kinsey (10/6). The Kakeya Needle Problem--what is the smallest set in the plane in which one can continuously rotate a unit line segment 180 degrees--turns out to have important applications to diverse areas of mathematics, including combinatorics, number theory, harmonic analysis, and PDE. I will talk about this problem and some of its applications, as well as related open conjectures. Among the applications I hope to present is Fefferman's counter-example to the ball multiplier conjecture in harmonic analysis. (The talk should be accessible to those who missed last week's talk, where I presented a construction of the solution to the Kakeya problem, provided you take the construction of the solution on faith.)

Existence and uniqueness for the Catt\'{e} regularization of the Perona-Malik equation, Jeff Calder (10/13). The Perona-Malik equation is arguably one of the most widely used partial differential equations (PDE) in image processing and computer vision. Although numerical schemes are (for the most part) stable, the continuum equation is an ill-posed forward/backward parabolic PDE. In an attempt to explain this paradoxical result and provide a rigorous theoretical foundation for the Perona-Malik equation, Catte\'{e}, Lions, Morel and Coll (1992) introduced a minor spatial regularization in the nonlinearity which allowed them to prove existence, uniqueness and $C^\infty$ regularity of solutions. In this talk, I will briefly explain why the Perona-Malik equation is ill-posed and then I will present the existence proof for the Catt\'{e} regularization (uniqueness and regularity are standard proofs). The proof is a very nice application of Schauder's fixed point theorem.

Introduction to Universal Taylor Series, Joe Roberts (10/20). In calculus we were warned that commutativity and associativity do not hold for conditionally convergent series. By reordering the terms or adding parentheses, a conditionally convergent series can be made to converge to any real number. More strikingly, there exists a power series that by adding parentheses it can be made to converge to any continuous function on [0,1] that vanishes at zero. These universal Taylor series have applications in complex analysis. I will present the construction of this series, and it will be accessible to anyone.

Noncooperative Differential Games, Jingchen (Chandler) Wu (10/27). Special room: 3866 EH. (Joint with Student AIM Seminar.) In game theory, differential games are a group of problems related with dynamical system and optimal control. But in differential game, there exist more than one objective function, which makes it different than normal optimal control problem. In this talk, I will introduce the definitions of equilibria in differential games and illustrate how to find equilibria in some problems (Cournot and Stackelberg equilibria). Basic concepts of optimal control such as HJB equation and Pontryagin’s maximum principle will also be mentioned in the talk.

The Ergodic Theorem, William Gignac (11/3). This talk will be an introduction to the Birkhoff ergodic theorem, the first major theorem in ergodic theory. I will discuss what it says, what it means, its strengths, and its limitations. There will be plenty of examples and applications.

Entropy in Dynamics, Andy Zimmer (11/10, 4pm, 4096 EH). This talk will be an introduction to entropy in dynamical systems. I will focus on the following three questions: what is entropy in dynamics, what does it measure, and why is it useful. Although this talk will focus on topological entropy, I will also briefly describe measure theoretic entropy and its relation to topological entropy. The talk should be accessible to anyone.

Newton's Method and the Nash-Moser Implicit Function Theorem, Purvi Gupta (11/17, 4pm, 4096 EH). The Nash-Moser theorem first appeared as a crucial technical step in Nash’s proof of his celebrated embedding theorem (1956). Later isolated by Moser, this result generalizes the classical implicit function theorem to a class of tame Frechet spaces, and has found wide applicability in geometric analysis and nonlinear partial differential equations. In this talk, I will state and prove a version of the Nash-Moser theorem for spaces of smooth functions, focussing mainly on Nash’s clever modification of Newton’s method from calculus. The talk should be accessible to anyone.

The Uncertainty Principle in Harmonic Analysis, Rafe Kinsey (12/1, 4pm, 4096 EH). The uncertainty principle really is a collection of principles along the following lines: a function and its Fourier transform cannot both be too localized. In this talk, I will discuss several manifestations of this principle within harmonic analysis, as well as (time permitting) its applications to PDE and its connection to the physical uncertainty principle of Heisenberg.

Continuity of Weighted Estimates in A_p Norm, Nikolaos Pattakos (12/8, 4pm, 4096 EH). In this talk a proof of the continuity of the norm of a general Calderon-Zygmund operator in a weighted L^p space as a function of the A_{p} norm of the weight is presented. The Riesz-Thorin interpolation theorem plays an important role in the proof. The sharpness of the estimate will also be discussed.
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Last modified: Mon Dec 20 14:40:00 EST 2010
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