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What is the Graduate Student Analysis Seminar?
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The GSAS is an analysis seminar in which the talks are given by and to graduate students. The GSAS has the aims of providing a broad view of analysis to its participants, providing opportunities to speak in a low pressure situation, and increasing communication and support between analysis students.
Unless otherwise noted, the GSAS will meet at 2pm every Thursday in PDL C401.
Please contact me if you're interested in talking or being added to the mailing list.
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Schedule for Spring 2012
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Thursday May 17, 2:00pm
Conformal Welding Maps for Plane Trees
by Joel Barnes
David Aldous's continuum random tree (CRT) is the scaling limit of uniform trees, and provides the underlying structure for important recent work on the scaling limit of random surfaces. The Hausdorff dimension of the CRT is 2, but there is no known natural embedding of the CRT into the complex plane. We describe the welding problem for the CRT, and a potential path to a solution. Along the way, we apply Ahlfors's theory of quasiconformal maps to give an algorithm to solve the welding problem for finite trees, which may have application in the computation of polynomials for Grothendieck's dessins d'enfants (children's drawings). Pictures included.
Thursday May 10, 2:00pm
Decoupling of Modes for the Elastic Wave Equation
by Justin Tittlefitz
We consider the elastic wave equation u" = A(x,D)u; u = (u_1, ...,
u_n). Elastic waves in isotropic media exhibit two modes of
propagation; one where the displacement is in the direction of
propagation (the P-mode) and one where displacement is orthogonal to
propagation (the S-mode). By examining the orthogonal projections
(Pi_P and Pi_S) onto these modes, one can effectively "diagonalize"
the operator A, writing A(x,D) = c_P^2(x) Laplacian Pi_P + c_S^2(x)
Laplacian Pi_S + lower order terms (c_P and c_S correspond to the
propagation speeds of the two modes). The question then arises: to
what extent are these modes preserved throughout a wave's evolution?
Specifically, for initial displacement f, if Pi_S f = 0, is Pi_S u = 0
(or vice-versa) for all t as well? If not, what can be said about this
interaction?
In this talk, we will discuss a result due to Brytik, deHoop, Smith
and Uhlmann, showing that the P <-> S interaction is smoothing of
degree 1. We will also cover some basic properties of
pseudodifferential operators in order to understand this problem, and
hopefully have time to discuss a related open question.
Thursday April 12, 2:00pm
Invariance principle and a local limit theorem for Reflected Brownian Motion
by Louis Fan
It is well known that Simple random walks in \epsilon Z^n converges to Brownian motion in R^n (the Donsker's Invariance Principle in 1950's). Moreover, some local limit theorem holds, which tells us that the convergence rate of the transition densities at any fixed point is about \epsilon^n. Do we have analogous result for Reflected Brownian Motion? We will discuss these results and illustrates standard techniques in obtaining heat kernel estimates. In particular, we will discuss the probabilistic interpretations of Isoperimetric inequality, Nash's inequality and Poincare inequality.
Thursday April 5, 2:00 pm
Tangent Measures and Geometry
by Stephen Lewis
One of the important tools in Geometric Measure Theory is the theory of tangent measures. We will talk about what it means for one measure to be "tangent" to another, and discuss some examples where the geometry behaves very nicely. A recent generalization called a pseudo-tangent measure will be explored near the end, as well as some recent work.
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Schedule for Autumn 2011
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Tuesday November 15, 3:30 pm
Thermoacoustic Tomography in Elastic Media
by Justin Tittelfitz
In this talk, we will discuss an emerging method of medical imaging known as thermoacoustic tomography. Mathematically, the problem is that of recovering the initial displacement $f$ for a solution $u$ of a wave equation in $[0,T] \times \mathbb{R}^3$, given measurements of $u$ on $[0,T] \times \partial \Omega$, where $\Omega \subset \mathbb{R}^3$ is some bounded domain containing the support of $f$.
This problem is relatively well-understood in acoustic media (i.e. for a scalar wave equation), but less is known about the problem in elastic media. In this talk, I will discuss this case; specifically, we assume $u = (u_1,u_2,u_3)$ solves the elastic wave equation \begin{align*} \partial_t^2u = \nabla \cdot \left( \mu(x) ((\nabla u) + (\nabla u)^T)\right) + \nabla (\lambda(x) \nabla \cdot u), \end{align*} and discuss sufficient conditions on the Lam\'e parameters $\mu$ and $\lambda$ to ensure recovery of $f$ is possible.
Tuesday November 1, 3:30 pm
Good Approximations and Parameterizations
by Stephen Lewis
In 1960, Reifenberg proved the "Reifenberg Topological Disk" theorem; that any compact set in RR^n "well approximated" by an m-plane at every point and scale is topologically an m-dimensional ball. Since then, many variations of this, employing different approximation strengths and approximating sets, have been proven in different areas, including minimal surface and harmoic function theory. In this talk, I will draw a lot of pictures, state some of these theorems and one of my own, and try to suggest a common theme and a couple of conjectures to put these into a larger theory.
Tuesday October 18, 3:00pm
CONCENTRATION-OF-MEASURE INEQUALITIES FOR DIFFUSIONS
by Andrey Sarantsev
Does a certain random variable have heavy or light tails? In other words, can this random variable be very large with a relatively high probability? We consider a diffusion process (the solution of a stochastic differential equation) and apply some techniques to figure this out for its maximum on a finite time interval. We also explore some connections with logarithmic Sobolev inequalities.
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Schedule for Spring 2011
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Thursday, June 2nd, 2:30 pm
Hybrid Inverse Problems : UM-EIT and PAT/TAT
by Ilker Kocyigit
There are different imaging modalities and each has its own advantages
and disadvantages, for example electrical impedance tomograpy (EIT)
safe, non-invasive and provides high contrast, whereas ultrasound
usually has high resolution. Hybrid methods aim to combine some of the
advantages of these different modalities. We will talk about two such
methods; Ultrasound Modulated EIT and Photoacoustic/Thermoacoustic
Tomography, and related inverse problems along with some of the known
results.
Thursday, May 26th, 2:30pm
What is the Black-Scholes model?
by Josh Tokle
To buy an option means to pay some money today in order to be able to buy shares of a stock at a predetermined price at some point in the future. What is a fair price for such a transaction? In 1997, Black and Scholes won a Nobel prize for their mathematical model for options pricing, which expresses the price of an option as the solution to an SDE. In this high-level talk, I will derive the Black-Scholes formula using as little financial jargon as possible.
Thursday, May 12th, 2:30pm in C401
General Exam
Louis Fan
Go to Louis' general exam.
Thursday, May 5th, 2:30pm
Stationary distribution and reflecting diffusions
by Mauricio Duarte
Brownian motion is the king of diffusions. Its smoothing properties and the connection with the laplacian have allowed mathematicians to study properties of the stationary regime for Brownian driven processes in detail. Nonetheless, interesting diffusions that are not driven by BM can arise and we need to take a different approach to study their stationary behavior. This is achieved for reflecting process by taking a look at the local stationary process. We'll present the so called Reflected BM with inert drift as an illuminating example and talk about how these ideas can be extended to more general cases.
Based on the work by Richard F. Bass, Krzysztof Burdzy, Zhen-Qing Chen, and Martin Hairer. Stationary distributions for diffusions with inert drift. Probab. Theory Related Fields, 146(1-2):1.47, 2010.
Thursday, April 28th, 2:30pm
Hydrodynamic Limit of Interacting Particle Systems
by Louis Fan
In scientific research, people are often interested in the macroscopic evolution of a large (typically of the order of $10^{23}$) number of objects which are governed by some microscopic rules. Some typical examples are chemically reacting
systems, the evolution of some thermodynamic characteristics of a fluid, population genetics, etc. The macroscopic evolutions are sometimes described by deterministic differential equations, called the hydrodynamic limits. e.g. For a large
number of RBMs on a bounded Lipschitz domain, the hydrodynamic limit is the heat equation with Neumann boundary condition.
In this talk, we will outline some hydrodynamic behaviors and derive rigorously the hydrodynamic limit of a typical interacting particle system.
Thursday, April 21st, 2:30 pm
Thermoacoustic and thermoelastic tomography
by Justin Tittlefitz
In this talk, we will discuss an emerging method of medical imaging known as thermoacoustic tomography. Mathematically, we investigate the problem of recovering the initial displacement $f$ for a solution $u$ of a scalar (acoustic) wave equation in $[0,T] \times \mathbb{R}^3$, given measurements of $u$ on $[0,T]^B \times \partial \Omega$, where $\Omega \subset ^Z \mathbb{R}^3$ is some bounded domain containing the support of $f$.
If time permits, we will also look at the problem of thermoelastic tomography, where we instead assume $u = (u_1,u_2,u_3)$ solves the elastic wave equation
\begin{align*}
\partial_t^2u = \nabla \cdot \left( \mu(x) ((\nabla u) + (\nabla u)^T)\right) + \nabla (\lambda(x) \nabla \cdot u),
\end{align*}
and discuss sufficient conditions on the Lam\'e parameters $\mu$ and $\lambda$ to ensure recovery of $f$ is possible.
Thursday, April 14th, 2:30 pm
Analysis on Domains with Rough Boundaries
by Matt Badger
Two classic settings for analysis are domains with smooth boundary (the upper half plane) or simple geometry (convex domains). However, there is a defect in only considering these simple models. As Mandelbrot says in his introduction to The Fractal Geometry of Nature: "clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line".
In this talk, I will survey different classes of "rough domains" where analysis can be carried out. As times permits, we will visit results from Hunt and Wheeden's study of harmonic functions on Lipschitz domains in the 1960s to Hofmann, Mitrea and Taylor's recent (2010) demonstration that the L^p Neumann problem is solvable on chord arc domains with small constant
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Schedule for Winter 2011
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Tuesday, March 8th, 4:00 pm
Stochastic Processes Meet Partial Differential Equations
by Andrey Sarantsev
It is possible to solve elliptic and parabolic PDE by means of probability theory. In fact, many concepts from PDE (Green's function, harmonic measure, fundamental solution) can be translated into the language of stochastic processes; so these two theories
are dual. For any second-order elliptic operator, there is a Markov process which corresponds to this operator. The simplest example, which is the Brownian motion and the Laplacian, can be substantially generalized.
Tuesday, February 22, 4:00 pm
The Geometry of Measures and Scaling
by Stephen Lewis
In this talk, we will investigate the following question: "What does the scaling of a measure tell us about the geometry of its support?" Obviously, this requires some explanation. We will make sense of this question, look at some pictures, and see one or two concrete and easily stated results.
Tuesday, February 15, 4:00 pm
An Inverse Source Problem in Radiative Transfer
by Mark Hubenthal
We consider the radiative transfer equation, which governs the behavior of photons in a medium in the presence of some unknown source f. For a generic set of scattering and absorption coefficients, it is shown that the direct problem is well-posed. It is also known that the outgoing intensities of photons on the boundary of the domain uniquely determine f if the background absorption and scattering parameters are known a priori. This has applications in optical molecular imaging and radiation detection.
Tuesday, February 8, 4:00 pm
Interacting particle systems, free boundaries, and weak solutions to a non-linear parabolic equation in the unit interval
by Joel Barnes
I will talk about a differential equation from my research and (hopefully) prove a uniqueness result for weak solutions. Along the way I will define the Stefan problem, discuss boundary conditions, and describe three isomorphic discrete Markov Processes if time allows. Comfort in L^2 is the only prerequisite for the main portion of the talk.
Tuesday, February 1, 4:00 pm
Reflecting Processes on Bounded Domains
by Mauricio Duarte
Reflecting processes are a natural object to study in bounded domains. In general, they behave as Brownian motion inside of the domain and reflect instantaneously at the boundary in a given angle, which varies along the boundary. We'll focus on describing the stationary distribution of such processes and will give a precise formula for its density in bounded planar domains.
Tuesday, January 25, 4:00pm
Brownian Motion and Stochastic Calculus
by Andrey Sarantsev
Brownian Motion is the most important process in the theory of continuous-time
stochastic (=random) processes. This random function has amazing and very irregular behaviour;
e.g. it is continuous but nowhere differentiable and has infinite variation.
Nevertheless, it is possible to develop differential and integral calculus for random functions related to this process.
The main result is Ito's Change-of-Variable Formula; it shows that there are effects in stochastic calculus
that do not have anything analogous in the classical calculus.
Tuesday, January 11, 4:00pm
Rectifiable Sets and the Besicovitch 1/2-conjecture
by Matt Badger
The Lebesgue density theorem says that for Lebesgue almost every point $x$ in a subset $A$ of $n$-dimensional Euclidean space, the density $\lim_{r\rightarrow 0} m(A\cap B(x,r)))/r^n$ exists and is equal to $m(B(0,1))$. For lower dimensional Hausdorff measures this result is typically false! This is exciting, because it means that using Hausdorff measures one can see a variety of behaviors. Rectifiable sets are those sets on which the density exists at almost every point. The Besicovitch 1/2-conjecture is an outstanding 73-year old conjecture about rectifiable sets.
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