Analysis Seminar Archives

Analysis Seminar Archives

Fall  2017

Joint Analysis/Applied Math seminar
Date: Wednesday, November 29, 2017
Time: 11am-12noon
Place: Rome 204
 
Speaker: Michael Frazier, University of Tennessee

Title:  Green's functions and the existence of the gauge for local and
non-local time-independent Schrodinger equations

Abstract:  We discuss a series of papers with Igor Verbitsky (and in one case, with Fedor Nazarov), dealing with time-independent Schrodinger operators L = - Laplacian -q, where the potential q is non-negative.  The "gauge," as it is known in the probability literature, is the solution of Lu=0 on a domain, with u=1 on the boundary of the domain.  We obtain estimates for the Green's function of L, and conditions for the existence of the gauge, under very general conditions on the potential q.  The conditions, which describe how rapidly q can blow up near the boundary, are close to being necessary.  We also discuss related results when the Laplacian is replaced
by the fractional Laplacian.  The conclusions are based on a general estimate for the kernel of a Neumann series (I-T)^{-1} associated with an integral operator T of norm less than 1.

Fall 2016

Title: Ball Intersection properties  in Banach spaces

Speaker: Sudeshna Basu, GWU 

Date and Time:  October 28, 2016Friday, 1:00-2:00pm

Place:   Rome 771

Abstract:  We know from Hahn Banach Theorem that a point outside a convex set can be separated from the latter by a hyperplane. The  question whether this separation can be done in terms of disjoint balls  continue to interest mathematicians. In this work, we study such Ball intersection properties  which are closely related to the geometric nature of the Banach space.We  certain stability results for these  properties in Banach Spaces leading to a discussion in the context of operator spaces and relate it to Radon Nikodym Properties in Banach spaces.

Spring 2016

Speaker:   Professor Diane Holcomb, University of Arizona. (Diane graduated from GW in 2008, got a PhD from Wisconsin, and is now a postdoc at Arizona.)

Date: Tuesday, April 5 at 1:00 PM

Where: Rome 351

Title:  Local limits of Dyson's Brownian Motion at multiple times. 

Abstract:  Dyson's Brownian Motion may be thought of as a generalization of  Brownian Motion to the matrix setting. We can study the eigenvalues of a Dyson's Brownian motion at multiple times. The resulting object has different "color" points corresponding to the eigenvalues at different times. Similar to a single time, the correlation functions of the process may be described in terms of determinantal formulas. We study the local behavior of the eigenvalues as we take the dimension of the associated matrix to infinity. The resulting limiting process in the bulk is again determinantal and is described with an "extended sine kernel." This work aims to give an alternate description of the limiting process in terms of the counting function. In this seminar I will go over the the description and methods for finding such a limit. This is work in progress and is joint with Elliot Paquette (Weizmann Institute).

Speaker:  Professor David Drasin, Purdue University

Date: Wednesday, March 23 at 2:00 PM

Where: Monroe Hall B32

Title:  A geometric function theory for dimensions greater than two

Abstract:  Classical complex analysis is a high point in pure mathematics, but at the same time has had remarkable applications in theoretical physics, engineering etc.  In many ways it is a jewel in itself, which in many respects is special for two dimensions. For example, only when n=1 or 2 can a multiplication be introduced on Euclidean n-space to create a field.

An important property of the function w=f(z) being analytic is that in general it induces a conformal map: if w_0=f'(z_0)\neq 0, then angles are preserved at z_0 and (infinitesimal) circles centered at z_0 are mapped to (infinitesimal) circles centered at w_0=f(z_0). 

J. Liouville showed that this is a very special property of two dimensions: if F is a mapping on n-space (n > 2), then F will not be conformal unless it is a M\"obius transformation (it is easy to see that M\"obius transformations exist in all real dimensions).

One of the most compelling classical achievements is Picard's theorem: if f is  analytic, nonconstant and defined on the entire complex plane, then the range of f can omit at most one point; the exponential function w=e^z shows Picard's result sharp (note that this function is also everywhere (locally) conformal but is not a M\"obius transformation).

There seems to be no obvious way to extend the rich classical theory on one complex dimension to higher (real or complex) dimensions.  The most immediate choice would be to let w=f(z) be a mapping of complex n-dimensions where f has a convergent power series.  Several complex variables has become a well-established subject, but the range of an entire function can omit an open set, and unless f is linear, it will never be conformal.

I will discuss a less obvious generalization, in which power series no longer operate, but which preserves an important feature of one complex variable: the image of a small ball centered at z_0 is sent to a nice set containing f(z_0).  If we require this image to be a ball, Liouville's theorem would show that f must be a M\"obius transformation, and thus the key insight in defining a quasiregular mapping is to allow the image to be an (infinitesimal)_ ellipsoid of uniformly bounded
eccentricity. 

We will define this class of mappings and indicates a few of their general properties and pathologies.

A deep result in the theory of quasiregular mappings, due to Seppo Rickman is that the range of a nonconstant quasiregular mapping which is defined on all of n-space can omit an at most finite set of points, and in a remarkable example of 30 years vintage he showed that the complement of the range can have cardinality as large as desired when n=3.

If time permits, I will indicate some steps which Pekka Pankka and I introduced to give a new approach to Rickman's construction which shows the sharpness of his result in all dimensions.


Speaker:  Van Cyr, Bucknell University

Date: Thursday, March 3 at 2:00 PM

Where: Room 771

Title:  Complexity, periodicity, and applications to zero entropy symbolic dynamics 

 

Abstract: The automorphism group of a symbolic dynamical system (X; ) is the group of homeomorphisms of X that commute with . For many natural systems, this group is extremely large and complicated (e.g. a theorem of Boyle, Lind, and Rudloph shows that if X is a topologically mixing SFT, then Aut(X) contains isomorphic copies of all nite groups, the free group on two generators, and the direct sum of countably many copies of Z). This can be interpreted as a manifestation of the\high complexity" of these shifts. In this talk I will discuss recent joint work with B. Kra which places restrictions on the automorphism group of any subshift of \low complexity." This class is quite general and contains any minimal subshift whose factor complexity function has stretched exponential growth (with stretching exponent < 1=2). I will also discuss how the same proof techniques can be used to study the probability measures on X that are invariant under a subgroup of Aut(X).

Spring 2015

Speaker: Robbie Robinson

Date and Time: Wednesday, April 29, 3:00

Place: Seminar Room 267

Title: Stationary guassian Processes IV

Abstract: Stationary Gaussian processes are classical probabilistic models used in applications such as digital signal processing, machine learning and quantum mechanics. This is the second talk in this series.  In the first talk I set up the measure theoretic background for studying probability theory and stochastic processes, including a discussion the Kolmogorov consistency theorem.  In the second talk,  I will also describe classical random variables, and the special properties of Gaussian random variables that make Gaussian processes nice. We will discuss both the real and less well known complex cases


Speaker: Robbie Robinson

Date and Time: Wednesday, April 8, 3:00

Place: Seminar Room 267
 
TitleStationary guassian Processes III
 
Abstract: Stationary Gaussian processes are classical probabilistic models used in applications such as digital signal processing, machine learning and quantum mechanics. This is the second talk in this series.  In the first talk I set up the measure theoretic background for studying probability theory and stochastic processes, including a discussion the Kolmogorov consistency theorem.  In the second talk,  I will also describe classical random variables, and the special properties of Gaussian random variables that make Gaussian processes nice. We will discuss both the real and less well known complex cases

Speaker: Robbie Robinson

Date and Time: Wednesday, March 25, 3:00
Place: Seminar Room 267
TitleIntroduction to Gaussian Processes Part 2
 

Abstract: Stationary Gaussian processes are classical probabilistic models used in applications such as digital signal processing, machine learning and quantum mechanics. This is the second talk in this series.  In the first talk I set up the measure theoretic background for studying probability theory and stochastic processes, including a discussion the Kolmogorov consistency theorem.  In the second talk,  I will also describe classical random variables, and the special properties of Gaussian random variables that make Gaussian processes nice. We will disccuss both the real and less well known complex cases.


Date: Friday, March 20

Time: 4:00PM
Place: Seminar Room 267

Speaker: Professor Jane Hawkins, University of North Carolina, Chapel Hill
Title: Elliptic Functions, Lattices, and Dynamics

We give some connections between classical elliptic functions (doubly periodic meromorphic functions) and dynamical systems.  These functions give rise to interesting rational maps of the sphere as well as exhibit a wide range of dynamics when they are iterated.  We discuss dependence of the dynamics on the underlying period lattice and the important role played by a the lemniscate constant (a classical transcendental number).  The lecture will be largely self-contained with a fair amount of expository material, but will also include some recent work.

 


Speaker: Robbie Robinson

Date and Time: Wednesday, February 25, 3:00
Place: Seminar Room 267
TitleIntroduction to Gaussian Processes Part 2
 
Abstract: Stationary Gaussian processes are classical probabilistic models used in applications such as digital signal processing, machine learning and quantum mechanics. This is the second talk in this series. In the first talk I set up the measure theoretic background for studying probability theory and stochastic processes, including a discussion the Kolmogorov consistency theorem. In the second talk, I will also describe classical random variables, and the special properties of Gaussian random variables that make Gaussian processes nice. We will disccuss both the real and less well known complex cases.

Title: Introduction to Gaussian Processes
Speaker: Robbie Robinson, George Washington University
Date and Time: WednesdayFebruary 18, 3:00-4:00pm

Place: Monroe 267

Abstract: Stationary Gaussian processes are classical probabilistic models used in applications such as digital signal processing, machine learning and quantum mechanics. In the 1950s Girsanov introduced stationary Gaussian processes into ergodic theory as a class of examples with spectral properties that could, to some extent, be prescribed. Since then, Gaussian processes seem to have resurfaced in ergodic theory about once a decade to solve new problems. Yet from the ergodic theory point of view, Gaussian processes remain a somewhat peculiar class, still somewhat poorly understood, and largely distinct from the typical examples.

In this first talk, in what will be an intermittent set of talks, I will set up the measure theoretic background for studying probability theory and stochastic processes in general, and in the context of ergodic theory, including a discussion the Kolmogorov consistency theorem. I will also describe the special properties of Gaussian distributions that make Gaussian processes a nice kind of stochastic process to study. Later talks will discuss topics like Gaussian Hilbert spaces, spectral theory via Fock spaces and Weiner chaos, and the problems associated with moving from discrete to continuous time, and from real to complex valued random variables.


Title: The Flipped Continued Fraction

Speaker: Professor Karma Dajani, University of Utrecht
Date and Time: FridayFebruary 20, 4:00-5:00pm

Place: Monroe 267

Abstract: Srating with the regular continued fraction map, if one now performs the simple operation of flipping some of the branches completely or partially, then one gets a rich family of new continued fractions containing many well known expansions such as Nakada’s α-expansions, the backward continued fraction, the continued fraction expansion with odd partial quotients (and the continued fraction with even partial quotients), and many more. Each member of this new class is indexed by a Borel subset D of the unit interval. We call such expansions D- expansion or flipped continued fractions. We show that on the expansion level, the D-expansion and the regular continued fraction expanion are related via two operations known as singularization and insertion. This allows us to prove that for any Borel set D, quadratic irrationals have ultimately periodic D-expansions. We also study the ergodic properties of D-expansions. Due to results by R ́enyi, Thaler, and Zweimüller, for each Borel set D ⊂ [0,1] there exists a TD-invariant measure μD on [0,1) absolutely continuous with respect to Lebesgue measure. Under this measure, the dynamical system ([0, 1), μD, TD)
forms an ergodic system. In case D contains a neighborhood of 1, this measure μD is σ-finite, infinite, while it is a finite measure otherwise.


Date: Friday 6 Feb at 4:00PM

Location: Seminar Room, Monroe 267
Speaker: Joseph Auslander, University of Maryland
 
Title: "Equicontinuity, distality, regional proximality,, and the Veech relation"
Abstract: See attached pdf file.

Fall 2014

Title: Rigidity Sequences for Rationally Ergodic Maps
Speaker: Kelly Brooke Yancey of the University of Maryland
Date/Time: 3:00 Friday October 24 in the seminar room.
 
Abstract: The generic transformation in the set of all invertible transformations that preserve a finite measure is both rigid and weakly mixing. There have been many recent developments toward characterizing the possible rigidity sequences for weakly mixing maps.

In joint work with Rachel Bayless, we are interested in what happens when you move to preserving an infinite measure. I will discuss some new results from our paper. In particular, I will give a construction to show compatibility of rigidity with rational ergodicity and also analyze rigidity sequences for rationally ergodic, rigid transformations.

Title: Automorphisms of nonorientable surfaces
Speaker: Professor Bruce Kitchens of IUPUI, currently at NSF.
Date/Time: 11:30 am Wednesday, November 19 in the seminar room.


Abstract: The (well-studied) family of birational maps $f_{a,b}(x,y) = (y, \frac{y-b}{x-a})$ can be thought of as defining maps from the real projective plane to itself. Some of these maps give rise to automorphisms of nonorientable surfaces. The dynamics of these maps are interesting. Some of them are zero entropy and can be understood as piecewise rotations and translations. Others can be seen to have positive entropy because of their action on the homology but the dynamics is not easy to understand. It appears that some have contain strange atttractors and others "standard" horseshoes.

This is joint work with Roland Roeder.


Date: Tuesday, December 2 at 3:00PM in the seminar room.
Speaker: Prof. Anima Nagar/Indian Institute of Technology, New Dehli

Title: Dynamics of Induced systems.

Abstract: We look into the dynamics of the system induced on the hyperspace of non empty closed subsets of a compact metric space. We explore some interesting properties they display


Spring 2014

Speaker: Irina Popovici, US Naval Academy

Title: Border-Collision Bifurcations in A Piece-Wise Smooth Planar Dynamical System Associated with Cardiac Potential

Date/Time: Tuesday, April 8, 3:00 PM

Place: 1776 G #148

Abstract: The talk addresses the bifurcations of a two-dimensional non-linear dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation. The dynamical behavior of this continuous (but only piece-wise smooth) model transitions from simple (a unique attracting cycle) to complicated (co-existence of stable cycles) as the stimulus period is decreased from large towards zero.

The first bifurcation, of discontinuous period-doubling type, results from the collision of two cycles with a switching manifold. For stimuli periods just shorter than collision time, of the two cycles about to collide, the 2:1 escalator is stable and the alternans solution is unstable; with those co-exists a stable 1-escalator whose orbit lays away from the switching manifold. The resulting dynamical systems associated with these collisions are being described for some classes of parameters.


 Speaker: Flavia Colonna, George Mason University

Title: Norm and essential norms of weighted composition operators acting on reproducing kernel Hilbert spaces of analytic functions

Date/Time: March 21, 2014 3:00 PM

Place: Seminar room 267

Abstract: Let $\psi$ be an analytic function on the open unit disk D and let $\phi$ be an analytic self-map of D. The weighted composition operator with symbols $\psi$ and $\phi$ is defined on the space of analytic functions on D as$W_{\psi, \phi} f = \psi \cdot (f\circ \phi)$. Let H be a reproducing kernel Hilbert space of analytic functions on the unit disk. In this talk, we determine conditions on H and its kernel K which allow us to characterize the bounded and the compact weighted composition operators from H into weighted-type Banach spaces. We obtain an exact formula for the operator norm and an approximation of the essential norm of the operators mapping into the space $H_{\mu}^\infty$ consisting of the analytic functions of whose modulus is $O(\mu)$, where the weight $\mu$ is a positive continuous function on D. We obtain an exact formula of the essential norm for a large class of weighted Hardy Hilbert spaces. We also discuss the case when the Hilbert space H is replaced by a general Banach space of analytic functions such that all point evaluations are bounded linear functionals. This is joint work with Maria Tjani.


Speaker: Terry Adams, US DoD

Time: March 7, 2014 3:00PM

Room: Monroe 267, Seminar Room

Title: A Case of Anything Goes in Infinite Ergodic Theory

Authors: Terry Adams and Cesar Silva

Abstract: Suppose $T$ is an invertible finite measure preserving transformation. If $T\times T$ is ergodic, then $T^p \times T^q$ is ergodic for all nonzero integers $p$ and $q$. The transformation $T$ is weakly mixing, and $S\times T$ is ergodic for any ergodic finite measure preserving transformation $S$. The situation is different for infinite measure preserving transformations. See Kakutani/Parry (1963) for early examples demonstrating a difference. In the case of product powers $T^p\times T^q$, we show anything goes. There exists a class of rank-one infinite measure preserving transformations such that given any subset $R\subset \mathbb{Q} \cap (0,1)$, there exists $T$ in the class such that $T^p\times T^q$ is ergodic if and only if ${p} / {q} \in R$. Also, the same class is rich in conservative product powers. We make a connection with recent work of Johnson/Sahin on $\mathbb{Z}^d$-directional recurrence.


 

Speaker: Tyler White, Northern Virginia Community College

Time: 3:00 , Friday, February 21

Room: Monroe 267, Seminar Room

Title: Topological Mixing Tilings of $\mathbb{R}^2$ Generated by a Generalized Substitution

Abstract: Kenyon, in his 1996 paper, gave a class of examples of tilings of \mathbb{R}^2 constructed from generalized substitutions. These examples are topologically conjugate to self-similar tilings of the plane (with fractal boundaries). I have proven that an infinite sub-family of Kenyon's examples are topologically mixing. These are the first known examples of topologically mixing substitution tiling dynamical systems of \mathbb{R^2}.


Speaker: Irina Popovici, US Naval Academy

TBS

Title:Border-Collision Bifurcations in A Piece-Wise Smooth Planar Dynamical System Associated with Cardiac Potential

Abstract: The talk addresses the bifurcations of a two-dimensional non-linear dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation. The dynamical behavior of this continuous (but only piece-wise smooth) model transitions from simple (a unique attracting cycle) to complicated (co-existence of stable cycles) as the stimulus period is decreased from large towards zero.

The first bifurcation, of discontinuous period-doubling type, results from the collision of two cycles with a switching manifold. For stimuli periods just shorter than collision time, of the two cycles about to collide, the 2:1 escalator is stable and the alternans solution is unstable; with those co-exists a stable 1-escalator whose orbit lays away from the switching manifold. The resulting dynamical systems associated with these collisions are being described for some classes of parameters.

 


Fall 2013

When: Oct 25, 3:00-4:00 pm Speaker: Jane Hawkins - NSF and UNC Chapel Hill

Where: Monroe 267 (Seminar room) (map) Title: Connections between Complex Dynamics and Ergodic Theory

Abstract: While the Julia sets of rational maps of the sphere usually conjure up images of interesting topological features, they also possess many measure theoretic properties worth studying. Every rational map has several distinguished invariant measures: one is the unique invariant measure of maximal entropy and the other is a more geometric measure called conformal measure. Only in rare instances do they coincide. There is often a nonatomic invariant measure equivalent to conformal measure, sometimes infinite and sometimes finite. We give families of examples of these. We also mention one-sided Bernoulli properties and which maps rule out one-sided Bernoulli behavior.


When: Oct 11, 3:00-4:00 pm Speaker: Joseph Rosenblatt, University of Illinois, Urbana

Where: Monroe 267 (Seminar room) (map) Title: Norm Approximation in Ergodic Theory

Abstract: Classical ergodic averages give good norm approximations, but these averages are not necessarily giving the best norm approximation among all possible averages. We consider 1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function, 2) when these optimal Cesaro norm approximations are not comparable to the norm of the usual ergodic average, 3) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average,and 4) the oscillatory nature of optimal norm approximations as well as the norm of the usual ergodic average.


When: Oct 4, 3:00-4:00 pm
Speaker: Robbie Robinson, GWU

Where: Monroe 267 (Seminar room) (map)
Title: Topological transitivity for interval maps

Abstract: Topological transitivity is one of the three pillars of Bob Devaney's definition of chaos, along with sensitive dependence on initial conditions, and dense periodic points. In this talk we study the dynamics of piecewise continuous, piecewise monotonic maps of the interval. We discuss several notions of topological transitivity, including an unusual one from the 1960's due to Bill Parry, and discuss the inter-relations between these various notions.


Summe 2013

Analysis Seminar by Joe Jerome, Northwestern

When: May, 28

Where: Monroe 267 (Seminar room) (map)

Title: Kato Estimates for NLS, L^2 and H^1 solutions.


Analysis Seminar by Joe Jerome, Northwestern

When: May, 21

Where: Monroe 267 (Seminar room) (map)

Title: Strichartz Estimates for NLS.


Analysis Seminar by Joe Jerome, Northwestern

When: May, 15

Where: Monroe 267 (Seminar room) (map)

Title: Local and global existence in nonlinear Schrodinge qequation.


Spring 2013

Analysis Seminar by Anudeep Kumar (Grad stdent GWU)

When: April, 28

Where: Monroe 267 (Seminar room) (map)

Title: Expotenntial bases and frameson 2-dimensional trapezoids.


Analysis Seminar by Yen Do (Yale)

When: Thu, February 28, 2:30pm – 3:30pm

Where: Monroe 267 (Seminar room) (map)

Title: Quantitative convergence of Fourier series in weighted settings. Abstract: In this talk I will describe a recent joint result with Michael Lacey, where we obtain more quantitative information about convergence of Fourier series in weighted settings.


Fall 2012

Analysis Seminar by Pierre Emmanuel Jabin (UMD)

When: Fri, November 2, 11am – 12pm

Where: Monroe 267 (map)

Description: Title: Optimal regularity estimates for non linear continuity equations Abstract: We prove compactness and hence existence for solutions to a class of non linear transport equations. The corresponding models combine the features of linear transport equations and scalar conservation laws. We introduce a new method which gives quantitative compactness estimates compatible with both frameworks.


Friday, October 26 at 3:45 in the Seminar Room 267

Speaker: Tyler White, Northern Virginia Community College

Title: Topological Mixing Tilings of $\mathbb{R}^2$ Generated by a Generalized Substitution

Abstract: Kenyon, in his 1996 paper, gave a class of examples of tilings of \mathbb{R}^2 constructed from generalized substitutions. These examples are topologically conjugate to self-similar tilings of the plane (with fractal boundaries). I have proven that an infinite sub-family of Kenyon's examples are topologically mixing. These are the first known examples of topologically mixing substitution tiling dynamical systems of \mathbb{R^2}.


Friday, October 19; NOON

Place: Seminar Room

Speaker: Maria Gauladani, GW

Title: A Discrete Bernoulli Free Boundary Problem
Abstract: We consider a new type of free boundary problem for the p-Laplace operator. This problem is strictly related to the well-known Bernoulli problem: in this new formulation, the classical boundary gradient condition is replaced by a condition on the distance between two different level surfaces of the solution. Under suitable scaling this new formulation converges to the classical Bernoulli problem. We establish existence and qualitative theory in convex and non-convex regions. This is a joint work with M.d.M Gonzalez and H. Shahgholian.


The first analysis seminar of the year will be taken place:

Friday, October 12, 2012

Time: 3:45

Room: Monroe 267

Speaker: Flavia Colonna from George Mason University

Title: WEIGHTED COMPOSITION OPERATORS BETWEEN M¨OBIUS-INVARIANT ANALYTIC FUNCTION SPACES

Abstract: See the abstract here


Spring 2009

Date: January 13, 2009, 2:20 - 3:35 Seminar Room 267

Speaker: Alphonso Montes Rodriguez


Tuesday, March 3, 2pm

Speaker: Tyler White

Title: Ergodic Properties of Chacon's Transformation

Abstract: Chacon's transformation is known as one of the first examples of an ergodic system which is weakly mixing,

but not strongly mixing. In this talk we will show that Chacon's transformation has two other interesting properties by

using the techniques of joinings, which was originally developed in probability theory. We will show that Chacon's

transformation has minimal self-joings (i.e., it has no ergodic measures on its product space with itself, other than the

diagonal measures). This implies Chacon's transformation has no square roots or factors and commutes with almost

nothing. We also show that Chacon's transformation is not isomorphic to its inverse.

Note: This talk will last 1 hour and is the first part of Tyler White's Specialty Exam. All are invited to attend. The talk

will be followed at 3pm by a question and answer period, which is also public. The examination committee is

Robinson (Advisor), Conway & Yi.


Tuesday April 21, 2009, 3:30pm

Speaker: Robbie Robinson

First talk:

"A new kind of Science"

Abstract: The title of this talk is the same as the title of the provocative and controversial 2002 book by Mathematica

founder Steve Wolfram. In his book, Wolfram suggests that, in many cases, extremely simple discrete computational

models may better explain complex physical phenomena than models based on the real numbers and calculus. Many

of the models that Wolfram considers come from the realm of symbolic dynamics, including cellular automata,

substitutions, and subshifts of finite type. Of special importance to Wolfram is the question of which classes of models

can support a universal turing machine--a property Wolfram sees, if the answer is positive--as legitimizing that class.

Whether or not one shares Wolfram's philosophy, the questions from dynamics and logic that are raised are of

independent interest (and in many cases predate Wolfram's interest in them). In this talk we will consider various

models that fit into the broad category of subshifts of finite type. We will discuss the sharp difference between the one

dimensional case and the multi-dimensional case. We will discuss some of the logical problems that arise. We will

formulate these questions, borrowing terminology--existence and uniqueness--from differential equations.


Tuesday April 28, 2009 4:00pm

Speaker: Robbie Robinson

Second talk:

"The Wang tiles of Kari and Culick"

Abstract. In this talk we concentrate on a particular model for subshifts of finite type: Wang tiles. We first discuss

Wang's 1961 conjecture about the decidability of the tiling problem. This conjecture was shown to be false in 1966 by

Wang's student Berger, who did this by constructing the the first "aperiodic" set of Wang tiles, a set of 20,00 Wang

tiles that tile the plane, but only aperiodically. After this, an "arms race" set in, with various researchers attempting to

find smaller aperiodic Wang tile sets. Some contributions came from such famous people as Roger Penrose

(Penrose tilings) and Donald Knuth (author of TeX). The current record of 13 is a 1996 example due to Kari and

Culick. In this talk we will show how this example works and see that it is based on a

completely different idea than any of the previous examples. This example has two different interpretations: one as a

kind of arithmetic computation on an unusual digital numeration of a real number. But it can also be explained

(modeled) by a certain type of dynamical system. To do either of these, however, one needs...real numbers. We the

problem of finding "aperiodic" Wang tile sets (set of tiles that tile the plane, but only aperiodically)


Tuesday April 7, 2009 2:00 - 3:35pm

Speaker: Hugo Junghenn

Title: Some Dynamical Aspects of Semigroups of Operators on Banach Spaces.

Part 2

Tuesday, March 31, 2009 2:00 - 3:35pm

Speaker: Sudeshna Basu

Title: ON P-ADIC HILBERT SPACES

Abstract: In this talk, we introduce the p-adic Hilbert spaces and give several examples. We also talk about the

operators on these spaces and study their properties governed by the p-adic structure of the underlying space.

Part 1

Tuesday, March 24, 2009 2:00 - 3:35pm

Speaker: Sudeshna Basu

Title: ON P-ADIC HILBERT SPACES

Abstract: In this talk, we introduce the p-adic Hilbert spaces and give several examples. We also talk about the

operators on these spaces and study their properties governed by the p-adic structure of the underlying space.


Fall 2009

Speaker: Robbie Robinson

Time: Monday 11/23 Monroe Hall 267 4 pm

Title: Entropy and Rank 1

Abstract: I will discuss ergodic group actions, the idea of an amenable group using Folner sequences. The definition

of rank 1makes sense even for groups that are not amenable, but most properties we generally associate with rank 1

require amenability and a Folner sequence of towers. Then, getting more specific, I'll talk about directional entropy for

Z^d actions, an invariant that John Milnor invented to study cellular automata. Finally I'll discuss directional entropy

results for rank 1 Z^d actions, and how they depend on a strengthening of the Folner property.


Speaker: Paul Wright/University of Maryland

Tuesday, Nov. 3, 2009 Monroe Hall #267 4:00pm-5:00pm,

Title: Some rigorous results for the periodic oscillation of an adiabatic piston

Abstract: A simple model of an adiabatic piston consists of a heavy piston of mass M that separates finitely many

ideal, unit mass gas particles moving inside two gas containers. Averaging techniques, used to study the motion of

the slow-moving piston in the limit where M tends to infinity, suggest that the piston should oscillate periodically. For

one-dimensional chambers, the effects of the gas particles are quasi-periodic and can be essentially decoupled, and I

will show that we recover a strong law of large numbers that is characteristic of classical averaging over just one fast

variable: the deviation of the piston from its averaged behavior is no more than O (M ^ {-1/2}) on a time scale O (M ^

{1/2}) . I will also show that for a very general gas chamber in higher dimensions, the actual motions of the piston

converge in probability to the averaged behavior on that time scale, although a strong law is no longer possible. I

learned about this problem from the papers of Neishtadt and Sinai, who derived the averaged equations and pointed

out that an averaging theorem due to Anosov could be extended to this case.


Speaker: Joseph Herning

Tuesday, Oct 27, 2009 Monroe Hall #267 4:00pm-5:00pm,

Title: In next week's Analysis Seminar, which is Tuesday due to the Colloquium, I will conclude the discussion of the

paper by Sadun, and Solomyak on conditions for topological mixing for substitutions on two letters.


Speaker: Joseph Herning

Tuesday, Oct 20, 2009 Monroe Hall #267 4:00pm-5:00pm,

Title: In next week's analysis seminar I will continue Tyler's two-week discussion of the paper by Kenyon, Sadun, and

Solomyak on conditions for topological mixing for substitutions on two letters.


Speaker: Robbie Robinson

Monday, Oct 12, Monroe Hall #267 4:00pm-5:00pm,

Title: "The entropy and spectrum of rank-1 actions in one or more dimensions, Part II"

Abstract: After a brief diversion to explain the notion of "the spectrum" as it applies to ergodic theory, I will define rank

1 transformations and show that they have simple spectrum. This will allow us to conclude that the entropy of a rank

1 transformation is zero. But the proof comes at great expense (the spectral theorem & Sinai's theorem). Then I will

give a direct proof that rank 1 transformations have entropy zero.

This proof will be a prelude to the main goal of these lectures (that, alas, will come in a later talk): studying the

directional entropies for rank 1 actions of Z^d.


Speaker: Robbie Robinson

Date: Tuesday, October 6, 2009, Monroe Hall #267 4:00pm-5:00pm

Title: "The entropy and spectrum of rank-1 actions in one or more dimensions"

Abstract: Rank-1 transformations play a central role in elementary ergodic theory. Two well known facts are that rank-

1 transformations have simple spectrum and rank-1 transformations have entropy zero. The first result implies the

second. But a more direct proof of entropy zero, while not hard, is difficult (perhaps impossible) to locate in the

literature.

In this first talk (of what will likely be series of talks), I will explain these ideas, and present a simple proof of entropy

zero for rank 1 transformations. Later we will move on to the Z^d case, where even the definition of rank-1 becomes

less clear. But this lack of clarity, once properly understood, helps explain an old construction called "funny rank 1"

and also an old example of Dan Rudolph, which shows that directional entropy need not always be zero.


Speaker: Tyler White (I will be presenting part 2 of my talk).

Date: Monday, September 28, Monroe Hall #267

Title: Topological Mixing of

Z and R -Actions of Symbolic Dynamical Systems That Arise From Substitutions

Abstract: The goal of this talk is to present results from Kenyon, Sadun, and Solomyak. These results provide

necessary and sufficient conditions for flows (R –actions) and discrete time( Z – actions) symbolic dynamical systems which arise from substitutions on two letters to be topologically mixing. I will begin by providing a brief introduction to subshifts (including their associated topology) that come from substitutions and show how tiling dynamical systems

of R can come naturally from such subshifts. Next, I will provide the definition of a topologically mixing dynamical system and provide equivalent definitions of topological mixing in subshifts and tiling spaces. I will conclude by stating and proving a theorem by Kenyon, Sadun, and Solomyak that relates topologically mixing substitutions (resp. tilings) to the lengths of the substitution (resp. tilings).


Speaker: Tyler White

Date: Sept. 21, 2009 4:00-5:00, Monroe Hall #267 (seminar room)

Title: Topological Mixing of Z and R -Actions of Symbolic Dynamical Systems That Arise From Substitutions

Abstract: The goal of this talk is to present results from Kenyon, Sadun, and Solomyak. These results provide

necessary and sufficient conditions for flows (R –actions) and discrete time( Z – actions) symbolic dynamical systems which arise from substitutions on two letters to be topologically mixing. I will begin by providing a brief introduction to subshifts (including their associated topology) that come from substitutions and show how tiling dynamical systems of R can come naturally from such subshifts. Next, I will provide the definition of a topologically mixing dynamical system and provide equivalent definitions of topological mixing in subshifts and tiling spaces. I will conclude by stating and proving a theorem by Kenyon, Sadun, and Solomyak that relates topologically mixing substitutions (resp. tilings)to the lengths of the substitution (resp. tilings).


Spring 2010

Speaker: Katie Quertermous /University of Virginia

Time: April 5, 2010 Monroe Hall 267, 1:30-2:30pm

Title: Composition Operators Induced by LInear-fractional Transformations: Interesting Problems Resulting From Simple Maps

Abstract: In this talk, we'll consider composition operators on the Hardy space of the disk that are induced by linearfractional, non-automorphism self-maps of the unit disk. Specifically, we will investigate the structures of the unital C*-algebras generated by collections of these composition operators or by collections of composition operators and compact operators. We will study several examples, including one whose structure is related to the algebra of almost periodic functions on the real line and one that is isomorphic, modulo the ideal of compact operators, to the unitization of a crossed product by the integers. We will use the structure results to determine spectral information for algebraic combinations of composition operators and their adjoints.


Speaker: John Conway

Time: March 29, 2010 Monroe Hall 267, 1:30-2:30pm

Title: Powers and direct sums of operators.

Abstract: This is a preliminary report of some results on the problem of characterizing those operators $A$ such that $A^2$ is similar to $A \oplus A$. This is joint work with Alejandro Rodriguez of Abu Dhabi.

Speaker: Tepper L Gill/Departments of E&CE, Mathematics, Physics/Howard University

Time: March 1, 2010 Monroe Hall 267, 1:30-2:30pm

Title: Extension Of The Spectral Theorem*

Abstract:

In this talk, I show how the polar decomposition property and a few elementary results for vector measures and vector-valued functions leads an interesting extension of the spectral theorem for all closed densely defined linear operators on Hilbert spaces. In addition, by using recent results on the existence of an adjoint for bounded linear operators on separable Banach spaces, I show how to obtain an extension for a restricted, but very useful class of operators even in this case.

* (This talk is based on joint work with F. Mensah and W. W. Zachary)


Speaker: Svetlana Roudenko

Time: Feb. 22, 2010 Monroe Hall 267, 1:30-2:30pm

Title: The Hilbert transform, weighted function spaces and wavelets

Abstract: Calder\'on-Zygmund operators (e.g. the Hilbert Transform) are one of the major objects of Harmonic Analysis and its boundedness on various function spaces has been a central question for many years.

For various application problems it is necessary to consider function spaces with scalar and even matrix weights and study CZOs on them. I will show how the wavelet-like (aka the Littlewood-Paley) approach helps to resolve the

question of boundedness of CZOs on weighted function spaces.


Date: Wednesday Oct 20, 3 pm

Speaker: Robbie Robinson, GWU

Room: Mon 267

Title: Maps of the interval and number theoretic expansions

Abstract: Many simple ergodic maps of the interval are related to number theoretic expansions like continued fractions, base 2, etc. In this lecture we will discuss this theme, which goes back to Kakeya in the 1920s and Renyi in the 1960s. In particular, we will discuss the problem of finding absolutely continuous ergodic invariant measures.


Fall 2010

Date: Wed Dec 1

Time: 3:30-4:30pm

Place: Monroe Hall 267

Speaker: Kasso Okoudjou, University of Maryland

Title: Gabor frames and Wiener's Lemma

Abstract: In this talk, I'll introduce a generalization of Wiener's 1/f theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space , the corresponding frame operator is invertible on this space.

Therefore, for such a Gabor frame, the canonical dual inherits the properties of the generator. This talk is based on a joint work with I.Krishtal.


Date: November 3, 2010

Time: 3:30pm, Monroe Hall 267

Speaker: Manoussos Grillakis,

University of Maryland

Title: On the evolution of fluctuations for a large number of weakly interacting Bosons


Speaker: Chi-Kwong Li, College of William & Mary and GWU

Date/Time: September 15, 2010, Monroe Hall 267, 3:30-4:30pm

Title: Higher rank numerical ranges and quantum error correction

Abstract:

The study of the higher rank numerical range of operators was motivated by the construction of quantum error correction codes, and is connected to many interesting problems in operator theory. We will described recent results and open problems on the topic.


Fall 2011

Monday, October 10, 2:20PM

Seminar Room 267

Speaker: E A Robinson

Title: Kakeya's Theorem and its generalizations

Abstract: In 1924 Soichi Kakeya published a paper describing a far reaching generalization of both the decimal and continued fraction representations of real numbers in the unit interval, but this work was largely forgotten in the chaos surrounding WWIi. Similar results were later independently obtained by Bissinger (1944), Everett (1946), Renyi (1957) and Parry (1964). In this talk we will give a proof of Kakeya's theorem, which is essentially just calculus, and

discuss some of its generalizations, including Parry's version of the theorem, which relates to some modern ideas about maps of the interval.


Date: Wed, Nov 2

Time: 2:20pm-3:35pm

Place: Monroe Hall 267

Speaker: Sergey Suslov, Arizona State University

Title: On Integrability of Non-autonomous Nonlinear Schroedinger Equations

Abstract: We show, in general, how to transform the non-autonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is outlined. A few simple integrable systems are discussed.


Date: Mon, Nov 7

Time: 2:20pm

Place: Monroe Hall 267

Speaker: Irina Mitrea, Temple University

Title: Harmonic Analytic and Geometric Measure Theoretic methods in Several Complex Variables

Abstract: Practice has shown that the combination of Harmonic Analysis, Geometric Measure Theory and Complex Analysis is an extremely fertile and potent mix in the complex plane, with many notable achievements whose degree of technical sophistication is breathtaking. In sharp contrast with these successes the case of Several Complex Variables has been very little explored from the perspective of the latest advances of Harmonic Analysis and Geometric Measure Theory. The aim of this talk is to discuss some recent progress in a program whose goal is to study the extent to which tools and methods from Harmonic Analysis and Geometric Measure Theory may yield a qualitative upgrade of some of the most fundamental results in Several Complex Variables.


Spring 2011

Date: Wed, March 9

Time: 3:45pm-4:45pm

Place: Monroe Hall 267

Speaker: Don Hadwin, University of New Hampshire

Title: A General View of Multipliers and Composition Operators

Abstract: I will discuss a general setting in which multiplication operators and composition operators can be studied. This general framework includes a vast array of examples, some where these ideas have been studied in great detail and some in which the concepts are new. We prove some results in this new setting that include many classical results and which include new results even in the classical settings. This is mostly joint work with Eric Nordgren and some with Zhe Liu.


Date: Fri, Feb 18

Time: 3pm-4pm

Place: Monroe Hall 251

Speaker: Joseph Rosenblatt, University of Illinois

Title: Rigidity along Sequences in Dynamical Systems


Date: Wed, Feb 16

Time: 11am-12noon

Place: Monroe Hall 267

Speaker: Michael Frazier, University of Tennessee, Dept. Head

Title: Estimates for Green's functions via Neumann series

Abstract: Neumann series are a standard tool in operator theory for inverting operators which are sufficiently small perturbations of the identity. We will be concerned with integral operators with positive, symmetric kernels. Under certain conditions, we obtain estimates for the kernel of the Neumann series in terms of the kernel of the original operator. These estimates can be used to obtain estimates for Green's functions of Schrodinger operators and other applications.


Date: Wed, Jan 19

Time: 3:30-4:30pm

Place: Monroe Hall 267

Speaker: Justin Holmer, Brown University

Title: Blow-up solutions on a sphere for the 3d quintic NLS in the energy space

Abstract: Solutions to the focusing nonlinear Schroedinger (NLS) equation i\partial_t u + \Delta u + |u|^{p-1}u = 0 for nonlinearities between mass-critical (p = 1 + 4/d) and energy-critical (p = (d + 2)/(d - 2)) can blow-up in finite time. In the mass-critical setting, the blow-up occurs on a discrete (dimension zero) set whereas in the mass-supercritical (p > 1 + 4/d) setting, the blow-up can occur on sets of positive dimension.

Using microlocal methods, we first prove that the log-log blow-up solutions studied by Merle-Raphael (2001-2005) with single blow-up point to the mass-critical equation remain regular in the energy space away from the blow-up point, resolving a conjecture of Raphael-Szeftel (2008). We are thus able to insert such solutions into higher-dimensional equations under symmetry assumptions; such equations will be mass-supercritical. In particular, we construct a large class of radial solutions that blow-up on a sphere for the three-dimensional energy-critical NLS. This is joint work with Svetlana Roudenko. We also discuss some other recent work in the field.


Spring 2012

Speaker: Loredana Lanzani, University of Arkansas andNSF

Title: ``The Cauchy Integral in C^n$

Thursday, April 26, 2:20-3:20pm unless otherwise specified

Time: 2:20

Room: Monroe 267

Abstract: The classical Cauchyintegral is a fundamental object of complex analysis whose analytic propertiesare intimately related to the geometric properties of its supporting curve.

In this talk I will begin byreviewing the most relevant features of the classical Cauchy integral. I willthen move on to the (surprisingly more involved) construction of the Cauchyintegral for a hypersurface in $\mathbb C^n$. I will

conclude bypresenting new results joint with E. M. Stein concerning the regularityproperties of this integral and their relations with the geometry of thehypersurface. (Time permitting) I will discuss applications of theseresults to the

Szeg\H o and Bergman projections (that is, the orthogonalprojections of the Lebesgue space $L^2$ onto, respectively, the Hardy andBergman spaces of holomorphic functions).


April 12

Room: Monroe 267 Time: 2:20

Speaker: Stephen Casey, American University

Title: Deconvolution and Sampling on Non-Commensurate Lattices via Complex Interpolation Theory

Abstract

Multichannel deconvolution was developed by C. A. Berenstein et al. as a technique for circumventing ill-posedness. Solutions to the analytic Be- zout equation associated with certain multichannel deconvolution problems are

interpolation problems on unions of non-commensurate lattices. These solutions provide insight into how one can develop general sampling schemes on such sets. We give solutions to deconvolution problems via complex interpolation theory. We then give specific examples of non-commensurate lattices, and use a generalization of B. Ya. Levin’s sine-type functions to develop interpolating formulae on these sets


March 19

Room: Seminar room (Monroe 267) Time: 2:30-3:30

Speaker: Luiz Farah, Universidade Federal de Minas Gerais, Brazil

https://sites.google.com/site/lgfarah/

Title: The supercritical generalized KdV equation: Global well-posedness in the energy space and below

Abstract: We consider the generalized Korteweg-de Vries (gKdV) equation u_t + u_xxx + mu (u^{k+1})_x = 0, where k>=5 is an integer number and mu =1 or -1. In the focusing case (mu=1), we show that if the initial data u_0 belongs to the Sobolev space H^1(R) and satisfies E(u_0)^{s_k} M(u_0)^{1-s_k} < E(Q)^{s_k} M(Q)^{1-s_k}, E(u_0) >= 0, and |(u_0)_x |_{L^2}^{s_k} | u_0|_{L^2}^{1-s_k} < |Q_x |_{L^2}^{s_k} |Q|_{L^2}^{1-s_k}, where M(u) and E(u) are the mass and energy, then the corresponding solution is global in H^1(R). Here, s_k=(k-4)/2k and Q is the ground state solution corresponding to the gKdV equation. In the defocusing case (mu=-1), if k is even, we prove that the Cauchy problem is globally well-posed in the Sobolev spaces H^s(R) for s>4(k-1)/5k. This is a joint work with Felipe Linares (IMPA) and Ademir Pastor (UNICAMP), and supported by FAPEMIG, CNPq and CAPES/Brasil.


Wednesday, Feb 22

Time: 4:00 Room: Duques 250

Speaker: Professor Tom Meyerovitch from the University of British Columbia
Title: Limiting entropy and independence for high-dimensional isotropic shifts

Abstract: In statistical mechanics, many types of systems are known or believed to exhibit "mean field behavior" in high dimensions. One could consider a sequence of systems (subshifts) $(X^{\otimes 1},X^{\otimes2},\ldots,X^{\otimes d},\ldots)$ obtained by imposing the same fixed set of constraints along any direction, where $X^{\otimes d}$ is a $d$-dimensional shift. I will present the following result, conjectured by Louidor, Marcus and Pavlov : For any set of constraints, the limit of the $d$-dimensional topological entropies of the resulting shift as $d$ tends to infinity is equal to the "independence entropy" of the constraints. In particular, this implies this limit is easy to explicitly compute, and is equal a rational multiple of log(n), for some natural number n. This is based on joint work with Ronnie Pavlov.


Thursday, Feb 16

Time: 2:20 Room: Monroe 267

Speaker: Robbie Robinson

Title: Absolute continuity vs total singularity (Part 2)

Abstract: What could be simpler than a strictly-increasing continuous function on the unit interval? Well, even though such a function is uniformly continuous and has bounded variation, it may fail --- in a spectacular way --- to be

absolutely continuous. In the first part we will prove that a strictly-increasing continuous function is absolutely continuous if and only if it inverse has nonzero derivative almost everywhere. We will also discuss implications of this lemma for maps of the interval and numeration. In the second part, we will look at examples of strictly-increasing continuous functions that have an almost everywhere zero derivative. In particular, we will describe a class of examples due to Raphael Salem whose graphs are a type of fractal related to the Koch snowflake. We will also discuss the famous "question mark" function ?(x) of Dirichlet, and the "box" function \box(x) of John H. Conway. These functions provide examples of monotonic bijections between the rationals and dyadic rationals in the unit interval.


Thursday, Feb 9

Time: 2:20 Room: Monroe 267

Speaker: Keri Kornelson, University of Oklahoma,
http://www.math.ou.edu/~kkornelson/

Title: Fourier bases on fractal Hilbert Spaces

Abstract: The study of Bernoulli convolution measures, which are supported on Cantor subsets of the real line, dates back to the 1930's, and experienced a resurgence with the connection between the measures and iterated function systems. We will use this IFS approach to consider the question of Fourier bases on the L^2 spaces with respect to Bernoulli convolution measures. There are some interesting phenomena that arise in this setting, e.g. one can sometimes scale or shift the Fourier frequencies of an orthonormal basis by an integer and obtain another ONB. We also describe properties of the unitary operator mapping between two such Fourier bases. This operator exhibits a fractal-like self-similarity, leading us to call it an "operator-fractal".


Thursday, Feb 2

Time: 3:30-5:00 Room: Government Hall 102

Speaker: Robbie Robinson

Title: Absolute continuity vs total singularity

Abstract: What could be simpler than a strictly-increasing continuous function on the unit interval? Well, even though such a function is uniformly continuous and has bounded variation, it may fail --- in a spectacular way --- to be

absolutely continuous. In the first part we will prove that a strictly-increasing continuous function is absolutely continuous if and only if its inverse has nonzero derivative almost everywhere. We will also discuss implications of this lemma for maps of the interval and numeration. In the second part, we will look at examples of strictly-increasing continuous functions that have an almost everywhere zero derivative. In particular, we will describe a class of examples due to Raphael Salem whose graphs are a type of fractal related to the Koch snowflake. We will also discuss the famous "question mark" function ?(x) of Dirichlet, and the "box" function \box(x) of John H. Conway. These functions provide examples of monotonic bijections between the rationals and dyadic rationals in the unit interval.


Thursday, January 19, 2012 Time: 2:20 Room: Monroe 267

Speaker: Professor Joshua Michael Lansky, American University

http://www.american.edu/cas/faculty/lansky.cfm

Title: Representations and Liftings for Finite Reductive Groups NOTE: This is a combination of analysis and algebra.

Abstract: We give a basic introduction to the theory of reductive groups over finite fields and their representations. The aim of the talk is to present a construction for a lifting of classes of representations in a fairly general setting.

Fall 2008

Date: September 15, 2008, 2:20 - 3:35 Seminar Room 267

Speaker: Gabriel Prajitura

Title: The geometry of orbits

Abstract: We will discuss several geometric aspects of orbits of linear

operators, all related to the concept of hypercyclicity.

Date: Sept. 29, 2008 2:20 - 3:35 Seminar Room 267

Speaker: Nathan Feldman, Washington & Lee

Title: Invariant Subspaces in the Hardy Space of the Slit Disk

Abstract: The Hardy Space on a region G in the complex plane is a Hilbert space of analytic functions satisfying a

certain growth restriction.


Date: Nov 10, 2008 2:20 - 3:35 Seminar Room 267

Speaker: Tom Kriete, University of Virginia

Title: Spectral theory for algebraic combinations of Toeplitz and composition operators.

Abstract: The spectrum of a linear operator on a finite dimensional Hilbert space is just the collection of eigenvalues

of the corresponding matrix. Things are more interesting on infinite dimensional spaces. One of the most satisfying

chapters in operator theory is the spectral theory of Toeplitz operators with continuous symbols. The most elegant

approach to this is via an examination of the C*-algebra that these operators generate. This talk will discuss spectral theory in a larger C*-algebra obtained by adding more generators, each of which is a composition operator.