UM Student Analysis Seminar: Winter 2011

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

The Student Analysis Seminar features talks by and for grad students interested in analysis; all are welcome and encouraged to come, including first-years. The seminar is being organized by Rafe Kinsey; email him (rkinsey at umich dot edu) for more information. To join our mailing list, go to directory.umich.edu, search for "student-analysis," bind to your umich uniqname, and join the group.

PDE Reading Group: There is also a student-run PDE reading group that will be meeting Mondays at 4 and Wednesdays at 3. See the webpage of the group for more information. Also of interest are the other student seminars, especially the student applied and interdisciplinary math seminar.

Date SpeakerTitle
Wed, 1/5/2011n/aNo meeting
Wed, 1/12/2011Jeff CalderAn Introduction to Wavelets
Wed, 1/19/2011William GignacThe Measurable Riemann Mapping Theorem
Wed, 1/26/2011n/ano talk (see PDE reading seminar at 3 and student AIM seminar at 1)
Wed, 2/2/2011n/aPostponed due to snow
Wed, 2/9/2011Jen BeichmanThe Hilbert Transform and the Water Wave Equation
Wed, 2/16/2011Joe RobertsIntroduction to Pseudodifferential Operators and Applications to Elliptic PDE
Wed, 2/23/2011Andy ZimmerConvexity in Ergodic Theory
Wed, 3/2/2011n/aNo meeting (break)
Wed, 3/9/2011n/aNo meeting to avoid conflict with Ziwet Lectures
Wed, 3/16/2011Nate TotzConstructing Infinitely Many Conserved Quantities of the KdV Equation (rescheduled from 3/9)
Wed, 3/23/2011Tim FergusonPainlevé's Problem and Removable Sets for Analytic Functions
Wed, 3/30/2011Sara LapanLocal Holomorphic Dynamics in the Complex Plane
Wed, 4/6/2011William GignacHolomorphic Dynamics on the Riemann Sphere

Topics suggested that people would like to hear about: Choquet's Theorem (functional analysis/convex geometry); distintegration theorems (measure theory); when is a probability space a Lebesgue space; Fourier analysis on (Lie and other) groups; geometric measure theory; gamma convergence (with applications to homogenization theory of PDE); the Gomboc; pseudodifferential operators; paraproducts.

Abstracts


An Introduction to Wavelets, Jeff Calder (1/12/11, 4pm, 4096 EH). The theory of wavelets first appeared in the mid 1980s and was influenced by ideas from both pure and applied mathematics. Since then, wavelets have found applications in diverse areas ranging from signal processing to partial differential equations to astronomy. In this talk, I will introduce the wavelet transform and discuss some of the basic mathematical properties of wavelets. The talk will be accessible to everyone.

The Measurable Riemann Mapping Theorem, William Gignac (1/19/11, 4pm, 4096 EH). The measurable Riemann mapping theorem is a generalization of the Riemann mapping theorem we all learned in complex analysis to the setting of quasiconformal geometry. My hope is to spend the first part of the talk introducing the notion of a quasiconformal map, leading up to a statement of the theorem. After that, I will (briefly) sketch a proof, which involves using techniques from harmonic analysis to solve a PDE. This talk should be accessible to any grad student, as I myself have very limited knowledge of quasiconformal geometry, harmonic analysis, or PDEs.

The Hilbert Transform and the Water Wave Equation, Jen Beichman (2/9/11, 4pm, 4096 EH). (Rescheduled from last week due to snow.) In this talk, I will discuss the derivation of the Hilbert transform and its surprising application to a specific case of the water wave equation. This talk will be accessible to anyone who knows what a holomorphic function is; a passing familiarity with Navier-Stokes is helpful, but not necessary.

Introduction to Pseudodifferential Operators and Applications to Elliptic PDE, Joe Roberts (2/16, 4pm, 4096 EH). We all know that a constant coefficient differential operator corresponds to multiplication by a polynomial (called the symbol of the operator) in frequency space. Pseudodifferential operators generalize the class of symbols to include those corresponding to "fractional" differential operators as well as those with variable coefficients. More importantly, the inverse (modulo a smoothing operator) of such a pseudodifferential operator is also a pseudodifferential operator, leading to applications to solving PDE's. I plan to spend about half of the talk discussing basic facts about pseudodifferential operators, such as how they act on Sobolev spaces, how to compose them, etc., and then spend the other half discussing applications to elliptic partial differential equations. It should be accessible to everyone.

Convexity in Ergodic Theory, Andy Zimmer (2/23, 4pm, 4096 EH). Many dynamical systems preserve a natural measure. For instance, a Hamiltonian system preserves the Liouville measure. In this talk we will use tools from functional analysis to discuss the space of all measures preserved by a dynamical system. Under certain conditions this space will be both convex and compact and thus we are able to apply results such as the Krein-Milman theorem and Choquet's theorem. The latter implies that a dynamical system can be decomposed into ergodic components. The talk should be accessible to everyone.

Constructing Infinitely Many Conserved Quantities of the KdV Equation (rescheduled from 3/9), Nate Totz (3/16, 4pm, 4096 EH). A quantity involving the dependent variables of an evolution equation is called conserved if it is invariant in time along solutions to the equation. Many equations arising from physical systems have natural conserved quantites, such as those expressing the conservation of mass, momentum, energy, etc. However, in this talk, we will prove the surprising fact due to Gardner that the Korteweg-de Vries equation (KdV) has an infinite number of conserved quantities by giving a method for explicitly calculating the conserved quantities. This in turn will allow us to conclude that KdV is well-posed in Sobolev spaces of nonnegative index uniformly for all time.

Painlevé's Problem and Removable Sets for Analytic Functions, Tim Ferguson (3/23, 4pm, 4096 EH). I will discuss Painlevé's problem, which deals with removable sets for analytic functions. Basically, a compact set K is removable if any analytic function defined and bounded in U/K can be extended to be analytic on all of U, where U is some domain containing K. For example, a point is removable by Riemann's theorem on removable singularities, whereas a disc is not removable. I will state some general results about Painlevé's problem, and then will discuss its relation to Hausdorff measure.

Local Holomorphic Dynamics in the Complex Plane, Sara Lapan (3/30, 4pm, 4096 EH). This talk is the first in a two part series on complex dynamics. I will discuss the behavior of germs of holomorphic functions near a fixed point of the complex plane. I will show how the coefficient of the linear term in the power series expansion near the fixed point can determine the local dynamics of the system. This leads to a classifications of such germs. In particular, I will describe the intriguing case when the coefficient is 1. Such germs have repelling and attracting “petals” around the fixed point. There will be pictures with pretty colors. If time permits, I will discuss how these types of results are currently being expanded to higher dimensional dynamics.

Holomorphic Dynamics on the Riemann Sphere, William Gignac (4/6, 4pm, 4096 EH). My goal for this talk is to give an overview of the dynamics of rational iteration on the Riemann sphere, focusing on the dynamics on the Julia set, where things are the most interesting. Most of the time will be spent talking about the concept of "equidistribution," which is central to the theory. This is meant to be a "big picture" talk; we will talk more about results and techniques than working out technical details. Also, even though this is the second talk in a two part series on holomorphic dynamics, you don't need to have attended last week's talk in order to understand this one.
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Email rkinsey (at) umich (dot) edu with comments, questions, and corrections.
Last modified: Fri Aug 26 21:27:26 EDT 2011
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