## Analysis Seminar

**Next lecture:**

Patrick L. Combettes

Université Pierre et Marie Curie - Paris 6

Proximity in Banach spaces: conjectures and recent results

Wednesday, October 22, 2014

3:00 p.m. in ES-139

Abstract:

We review the notion of proximity in the classical sense, in the sense of Moreau, as well as in the sense of Bregman. Some open and recently closed conjectures are discussed,as well as applications in the area of splitting algorithms for inverse and learning problems.

**Upcoming talks:**

Ron Yang,

Projective spectrum and the kernel bundle.

Wednesday, October 29, 2014

3:00 p.m. in ES-139

Abstract: For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital algebra ${\mathcal B}$ over $\cc$, its {\em projective spectrum} $P(A)$ or $p(A)$

is the collection of $z\in \cc^n$, or respectively $z\in \pn$ such that $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible in ${\mathcal B}$.

In this talk we will address two issues. First, we will show that if ${\mathcal B}$ is a Banach algebra then the {\em projective resolvent set} $P^c(A):={\cc}^n\setminus P(A)$ is a union of domains of holomorphy.

Second, for certain tuples we can construct a holomorphic line bundle over $p(A)$. An elementary example is given to show that this bundle can be nontrivial. The talked is based on a joint work with Wei He from

Southeastern University in China.

Dinh Dung

Vietnam National University, Hanoi

The cardinality of high- and infinite-dimensional hyperbolic crosses and their applications.

Wednesday, November 5, 2014

3:00 p.m. in ES-139

Abstract:

We give a brief overview of the problem of estimation of the cardinality of high- and infinite-dimensional hyperbolic crosses and applications in estimation of n-widths and epsilon-dimensions of classes functions having mixed smoothness and of analytic functions, and in stochastic PDE's

Eyvindur A. Palsson

Williams College

Wednesday, November 12, 2014

2:30 p.m., room TBA

Variational bounds for a dyadic model of the bilinear Hilbert transform

Abstract:

Variation norms, which are stronger than supremum-norms, are at least as old as Wiener's paper on quadratic variation from the 1920s. D. Lepingle was the one though that pioneered variational estimates and proved them for martingales. Subsequently such estimates have been established for other families of operators in harmonic analysis such as families of averages and singular integrals. Examples of applications of these norms can be found both in ergodic theory and rough path analysis.

We will present variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele and Demeter. The proof uses analysis on the Walsh phase plane and two new ingredients: (i) a variational extension of a lemma of Bourgain by Nazarov--Oberlin--Thiele, and (ii) a variation-norm Rademacher--Menshov theorem of Lewko--Lewko.

This is joint work with Y. Do and R. Oberlin.

Li An D. Wang

Trinity College

Wednesday, November 19, 2014

3:00 p.m. in ES-139

Title and Abstract: TBA

Theresa Anderson

Brown University

Wednesday, December 10, 2014

3:00 p.m. in ES-139

A new framework for Calderon-Zygmund Operators and its applications

Abstract: TBA

**Past talks archive:**

Joshua Isralowitz

What is... the Bellman function method?

Wednesday, October 1, 2014

3:00 p.m. in ES-139

Abstract:

In the previous talk, we discussed what the dyadic Carleson embedding theorem is. Furthermore, we defined ``the" Bellman function $F(x, y, z)$ defined on the convex domain $D = \{(x, y, z) \in \mathbb{R}^3 : y^2 \leq x, \ 0 \leq z \leq 1\}$ and showed that this function satisfies certain range and convexity properties.

In part II, we will remarkably reverse this logic to prove the dyadic Carleson embedding theorem. More precisely, we will show that the dyadic Carleson embedding theorem is true with explicit bounds if we can find \textit{any} function $F$ on $D$ (which is often called ``a" Bellman function) satisfying the range and convexity properties described last time.

Furthermore, we will discuss the pros and cons of the ideas developed in these two talks, and time permitting, we will discuss the method that harmonic analysts in the past five years or so have often able to replace this Bellman function method with: classical stopping time arguments!

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Joshua Isralowitz

What is... the Bellman function method?

Wednesday, September 17, 2014

3:00 p.m. in ES-139

Abstract:

In the mid 1990's, F. Nazarov, S. Treil, and A. Volberg ingeniously used ideas originated by D. Burkholder in the 1980's to develop a method to solve certain outstanding problems in harmonic analysis. Roughly speaking, the idea is that to prove certain estimates in harmonic analysis and martingale theory, it is often enough to show that real valued functions with certain convexity properties defined on certain domains in $\mathbb{R}^n$ exist.

In this talk, we will explain what this method is, and more precisely we will explain how this method in conjunction with the function $F(x, y, z) = 4 \left(x - \frac{y^2 }{z + 1} \right)$ on the domain $\{(x, y, z) \in \mathbb{R}^3 : y^2 \leq x, \ 0 \leq z \leq 1\}$ can be used to prove a dyadic version of the classical Carleson imbedding theorem on the real line. Don't have a clue as to how or why these two are connected? Come to the talk and find out!

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Philip Goyal

University at Albany - Department of Physics

An Informational Approach to Identical Particles in Quantum Theory

Wednesday April 9, 3:00pm in ES 108

A remarkable feature of quantum theory is that particles with identical intrinsic properties must be treated as indistinguishable if the theory is to give valid predictions. In the quantum formalism, indistinguishability is expressed via the symmetrization postulate, which restricts a system of identical particles to the set of symmetric states (`bosons') or the set of antisymmetric states~(`fermions').

However, the precise connection between particle indistinguishability and the symmetrization postulate has not been clearly established. There exist a number of variants of the postulate that appear to be compatible with particle indistinguishability. In particular, the widely influential topological approach due to Laidlaw DeWitt and Leinaas Myrheim implies that its validity depends on the dimensionality of space. This variant leaves open the possibility that identical particles are generically able to exhibit so-called anyonic behavior in two spatial dimensions.

Here we show that the symmetrization postulate can be derived on the basis of a simple novel postulate. This postulate establishes a functional relationship between the amplitude of a process involving indistinguishable particles and the amplitudes of all possible transitions when the particles are treated as distinguishable. The symmetrization postulate follows by requiring consistency with the rest of the quantum formalism. The key to the derivation is a strictly informational treatment of indistinguishability which prohibits the labelling of particles that cannot be experimentally distinguished from one another. The derivation implies that the symmetrization postulate admits no natural variants. In particular, the possibility that identical particles generically exhibit anyonic behaviour is excluded.

References:

[1] "Informational Approach to Identical Particles in Quantum Theory", http://arxiv.org/abs/1309.0478

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Mihai Stoiciu

Williams college

Spectral Properties of Random Schrodinger Operators with Small Coupling Constants

Wednesday March 12, 3:00pm in ES 108

Abstract: We consider one-dimensional Schrodinger operators with random potentials cV, where V is a fixed random potential and c is a constant. We investigate the spectral properties of these operators for small values of c. In particular, we describe the behavior of the density of states and the transition in the microscopic eigenvalue statistics, as the coupling constant c approaches 0.

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Joshua Isralowitz

Dyadic models of singular integral operators and Toeplitz products, part II

Wednesday March 5,

3pm in ES 140

Abstract: In the first part of these two seminars, we discussed a dyadic model for the positive Bergman projection and began to discuss how this model allows one to prove a rather striking theorem of Aleman, Pott, and Reguera on Toeplitz products with analytic symbols. In this second seminar, we will discuss how the results of the first seminar combine to prove this result and we will discuss how this theorem itself allows one to disprove the ``Sarason conjecture" on the Bergman space.

Furthermore, we will briefly discuss what dyadic models exist for the Hilbert transform and general CZOs. Time permitting we will also discuss interesting open problems.

Joshua Isralowitz

Dyadic models of singular integral operators and Toeplitz products

Wednesday, Feb. 12, 2014

12:30 pm, room ES 135

ABSTRACT:

A deep philosophical idea in modern harmonic analysis is the following: to prove results about singular integral operators (such as the Hilbert transform), one can try to prove the desired result for a simpler class of appropriate dyadic model operators, and then use this to prove the desired result for the given singular integral operator.

Interestingly, this approach was very recently used by Sandra Pott and Maria Carmen Reguera to study the Bergman projection of the classical Bergman space $L_a ^2 (\mathbb{D}, dA)$. The purpose of this talk is to describe their simple model, and furthermore, to discuss how it was very recently used by Alexandru Aleman, Sandra Pott, and Maria Carmen Reguera to not only settle (in the negative) the ``Sarason conjecture" regarding Toeplitz products on $L_a ^2 (\mathbb{D}, dA)$, but to also prove some very striking and surprising results about Toeplitz products on $L_a ^2 (\mathbb{D}, dA)$ with analytic symbols.

Furthermore, this talk will gently introduce the audience to the kind of estimates often found in modern harmonic analysis, and will also introduce the important concept of ``averaging over random dyadic lattices". Time permitting, we will discuss what similar models exist for more classical singular integral operators and discuss open problems. This talk will require very little background from the audience and should therefore be accessible to graduate students.

Jing Zhang

Stein open subsets with Analytic Complements in Compact Complex Spaces

Wednesday, Dec. 4, 3:00pm in ES-140

Abstract:

In Dr. Ron Yang's talk, he established the relationship between the invertible property of a type of operators and a domain of holomorphy. Later in Dr. Michael Range's talk, a detailed introduction to domains of holomorphy was given, and in particular their origins, their importance, and their applications were discussed. Stein manifolds and Stein spaces with singularities are generalizations of domains of holomorphy. Stein manifolds have been intensively studied in both complex analysis and topology (for example, Eliashberg and R. Gompf's work on topological characterization of Stein manifolds). Motivated by Ohsawa's theorem on nonsingular surfaces and Forstneri$\check{\mbox{c}}$'s examples in ${\mathbb{C}}{\mathbb{P}}^2$, we prove the following theorem:

Let $Y$ be an open subset of an irreducible normal reduced compact complex surface $X$ such that $X-Y$ is the support of a connected effective Weil divisor $D$. If $D$ is a big divisor and $Y$ contains no compact curves, then there is a Weil divisor $P_1$ with support $X-Y$ such that for all irreducible curves $C$ in $X$, $P_1\cdot C>0$. If $X$ is nonsingular, then

$P_1$ is an ample divisor with support $X-Y$ and $Y$ is a Stein surface.

We will give an example to show that this theorem is not true for higher dimensional complex manifolds. For a domain $\Omega$ in ${\mathbb{C}}^n$, it is well-known that $\Omega$ is Stein if and only if $H^i(\Omega, {\mathcal{O}}_\Omega)=0$ for all $0<i<n$, where ${\mathcal{O}}_\Omega$ is the structure sheaf and cohomology is Cech cohomology (note that $H^n(\Omega, {\mathcal{O}}_\Omega)=0$ is always true for every domain $\Omega$ in ${\mathbb{C}}^n$ by a theorem of Siu.) Laufer generalized this to Stein manifolds and proved that an open subset $U$ of a Stein manifold $M$ is Stein if and only if $H^i(U, {\mathcal{O}}_U)=0$ for all $0<i<n$. This is no longer true for an open subset of a compact complex manifold, and we will discuss how sheaf theory, cohomology theory, and techniques from birational geometry can combine to give sufficient conditions for an open subset of a compact complex space to be Stein.

R. Michael Range

Complex Analysis: A Quick Visit to Higher Dimensions

Wednesday, Nov. 20, 3:00pm in ES-140

Abstract:

The projective spectrum of $n$ elements in a Banach algebra discussed in

Prof. Yang's recent talk lives in $\mathbb{C}^{n}$ or in the corresponding

complex projective space of dimension $n-1$. Basic concepts and results of

multidimensional complex analysis are important tools that are used to study

this spectrum. In this expository talk I discuss a few of the surprising

features that arise in dimension greater than one, such as simultaneous

holomorphic extension, domains of holomorphy, the $\overline{\partial }-$

equation, and generalizations of the Cauchy kernel. No prior knowledge of

complex analysis in several variables is assumed.

Jon Bannon, Siena College

The Kadison-Singer problem.

Wednesday, Nov. 13, **4:30pm in ES-153**

Abstract: In his $1930$ text on quantum mechanics P.A.M. Dirac assumed that the physics of a quantum system is completely determined by the collection of its simultaneously measurable quantities. Mathematically stated, Dirac assumed that a pure state on a maximal abelian self-adjoint $*$-subalgebra (MASA) of bounded operators on a Hilbert space $H$ should have a unique extension to a state on the algebra of all bounded operators on $H$. In $1958$, Kadison and Singer proved that this is not true for a certain ``diffuse" MASA.

They were unable to establish that Dirac's assumption was false for the ``discrete" diagonal MASA, and the question of whether every pure state on the diagonal MASA has a unique state extension to $B(H)$ is known as the Kadison-Singer problem. This problem is equivalent to conjectures in many areas of research ranging from Banach space theory to engineering. A few months ago, Marcus, Spielman and Srivastava proved the conjecture. My talk will outline how the main result in this groundbreaking paper implies the original statement of the Kadison-Singer problem.

Rongwei Yang

Projective spectrum in Banach algebras, part 2

Wednesday, Nov. 6, 3:00pm in ES-140

Abstract: For a tuple A=(A_1,\ A_2,\ ...,\ A_n) of elements in a unital algebra B, its projective spectrum P(A) is the collection of z\in C^n such that A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n in not invertible. This talk shows some geometric and topological properties of P(A). Many examples will be given.

Rongwei Yang

Projective spectrum in Banach algebras

Wednesday, Oct. 30, 3:00pm in ES-140

Abstract: For a tuple A=(A_1,\ A_2,\ ...,\ A_n) of elements in a unital algebra B, its projective spectrum P(A) is the collection of z\in C^n such that A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n in not invertible. This talk shows some geometric and topological properties of P(A). Many examples will be given.

Joshua Isralowitz

Compactness of operators on Bergman and Fock spaces.

Wednesday, Oct. 16

3:00 p.m. in ES-140

Abstract: In a deep paper of D. Suarez published in 2007, the classical Axler-Zheng theorem was extended to show that an operator in the Toeplitz algebra of the Bergman space on the unit ball is compact if and only if its Berezin transform vanishes on the boundary. Very recently, a small flurry of results (such as extensions to the Fock space, new essential norm estimates, and vast simplifications of his original proof) have appeared which extend this theorem.

In this talk, we will discuss these recent results and the techniques used to prove them. Furthermore, we will discuss some very interesting open problems related to these results. This is partly joint work with Brett Wick and Mishko Mitkovski.

Speaker: Jani Virtanen, University of Reading

Title: Toeplitz operators on Besov-Dirichlet spaces.

Abstract: TBA

Wednesday at 3:00pm in ES 140

Speaker: Isaak Chagouel, SUNY at Albany

Title: Joint spectra of normal compact operators (part 2)

3:40 on Wednesday, April 24, in the lounge

Speaker: Isaak Chagouel, SUNY at Albany

Title: Joint spectra of normal compact operators

3:40 on Wednesday, April 17, in the lounge

Speaker: Dan Stevenson, SUNY at Albany

3:30 on Wednesday, April 3, in ES135 (If it turns out that ES135 is too small, we shall do it in the lounge.)

Speaker: Ron Yang, SUNY at Albany

Title: An elementary inequality about Mahler measure

Abstract: Let p(z) be a degree n polynomial with leading coefficient 1. Assume z_1, z_2, ..., z_k are all the zeros of p whose modulous are >1, then the Mahler measure M(p) is the modulous of the product |z_1z_2...z_k|. The Lehmer's problem in number theory is wether there is an absolute constant c>0, such that for every integer-coefficient p, either M(p)=1 (a trivial case) or M(p)>=1+c. In this talk we will see a new attribute, namely total distance td(p), and show that it is equivalent to M(p). This gives rise to an equivalent statement of the Lehmer's problem. This is a joint work with Konstantin Stulov. The proof is absolutely elementary and is accessible to undergraduates.

3:30 on Wednesday, March 27, in ES135 (If it turns out that ES135 is too small, we shall do it in the lounge.)

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Speaker: Jing Zhang

Title: On the Complement of a Hypersurface in Compact Complex Space

3:30 on Wednesday, February 27, in ES135

Abstract: Levi problem is a very old and important problem in several complex variables.

Over the years, various generalizations of the Levi problem were proposed and investigated.

Nessim Sibony raised the following generalized Levi problem.

{\bf Problem [Nessim Sibony]} Let $M$ be a compact complex manifold. Find some appropriate conditions on $M$ such that if $U$ is an open subset of $M$ which is locally Stein, then $U$ is Stein.

It seems that Sibony's question is very hard. To see this, let $M$ be a smooth projective variety defined over $\mathbb{C}$, then Sibony's question is reduced to generalized Hartshorne's question in algebraic geometry: What nonaffine smooth quasi-projective varieties are Stein? We know little about this question. In this talk, we will discuss the history of Levi problem and report our following result: Let $X$ be a compact complex space of pure dimension $d\geq 1$ (with any singularities). Let $Y$ be a proper open subset of $X$ such that the boundary $X-Y$ is support of an effective Cartier divisor $D$ (then $Y$ is locally Stein). We show that $Y$ is a Stein space if $D$ is a big divisor on every irreducible component of $X$, $H^i(Y, {\mathcal{O}}_Y(-Z))=0$ for all $i>0$, where $Z$ is any hypersurface on $Y$.

Speaker: Michael Range

Title: Some Estimates in the Theory of the Cauchy-Riemann Equations on Weakly Pseudoconvex Domains

3:30 on Wednesday, February 13, in ES135

Abstract: We present some applications of the new integral kernels discussed by D. Smitas in his talk last Wednesday to representations of (0,q) forms and estimates related to the Cauchy-Riemann equations on (weakly) pseudoconvex domains. In particular, we shall discuss a pointwise analogon of the classical "basic estimate" in the L² theory of the complex Neumann problem and some preliminary results towards proving a long standing conjecture about Hölder estimates.

Speaker: Dan Smitas

Title: An Analog of the Cauchy Kernel for Weakly Pseudoconvex Domains

3:30 on Wednesday, February 6, in ES135

Abstract: The Bochner-Martinelli kernel is applicable to an arbitrary domain, but is not holomorphic and does not reflect the complex geometry of its singularity. The kernel of Henkin and Ramirez alleviates these concerns, but is only defined on strictly pseudoconvex domains. We will introduce a new kernel for weakly pseudoconvex domains that, while not holomorphic, still satisfies classical estimates.

Speaker: Professor Tim Grove, CNSE

Title: The optics of charged particle beams

3pm on Wednesday, November 28, in ES139

Abstract: Science and technology at the atomic scale of dimensions rely heavily on instruments built around beams of charged particles. For example, electron microscopes with spherical aberration correction can resolve single atoms at resolution around 0.06 nm. (The radius of the hydrogen atom is 0.053 nm). Electron energy loss spectroscopy probes energy levels of materials. The helium gas field ion microscope focuses a beam of helium atoms to a probe of about 1 nm in size. The Large Hadron Collider at CERN has a design energy of 7 TeV, and produces sub-nuclear particles, including the putative Higgs Boson. All of these instruments use electric and magnetic fields to accelerate and focus a beam of charged particles.

The optics of charged particle beams derives from classical mechanics (geometrical optics), and quantum mechanics (wave optics). The central dynamical problem can be stated as follows: given an initial state for a particle, calculate the state at all future times. Hamilton’s principle of least action holds that the classical physical trajectory, chosen among an infinite number of hypothetical trajectories, has an extremum of the action integral between the initial and final points. This forms the starting point for the quantum mechanical description as well. A close analogy exists between charged particle optics and light optics. A survey of these ideas will be presented.

The basic tool is the calculus of variations, in which the time integral of the so-called Lagrangian function has an extremum for the physically allowable path of motion. An associated differential equation follows immediately. This is solved (with given initial condition) to give an analytical expression for the path of motion. It is generalized to quantum mechanics by introducing further postulates.

Speaker: Josh Isralowitz, SUNY at Albany

Title: A characterization of shift-invariant subspaces via Beurling-Lax-Halmos Theorem

3pm on Wednesday, October 31, in ES139

Abstract: In this talk, we discuss the question of when the Berezin transform characterizes the compactness of a bounded operator on a wide class of weighted Fock spaces. Furthermore, we discuss the question of whether all compact operators on these weighted Fock spaces are necessarily in the Toeplitz algebra.

Speaker: Yueshi Qin, SUNY at Albany

Title: A characterization of shift-invariant subspaces via Beurling-Lax-Halmos Theorem

3pm on Wednesday, October 17, in ES142

Abstract: Shift invariant subspaces in the vector-valued Hardy space $H^2(E)$ play important roles in Nagy-Foias operator model theory. A theorem by Beurling, Lax and Halmos characterizes such invariant subspaces by operator-valued inner functions $\Theta(z)$. When $E=H^2(D)$, $H^2(E)$ is the Hardy space over the bidisk $H^2(D^2)$. This paper shows that for some well-known examples of invariant subspaces in $ H^2(D^2)$, the function $\Theta(z)$ turns out to be strikingly simple.

Speaker: Jani Virtanen, University of Reading, UK

Title: Eigenvalues of Toeplitz operators on Hardy spaces

3pm on Wednesday, October 10, in ES142

Abstract: I discuss the (non)existence of eigenvalues of Toeplitz operators acting on Hardy spaces via the Riemann-Hilbert problem. I also consider some related spectral problems of these operators and list some open problems. Part of the talk is based on joint work with L. Wolf.

Speaker: Ron Yang, SUNY at Albany

Title: A note on multivariable Berger-Shaw theorem

3pm on Wednesday, October 3, in ES142

Abstract: The classical Berger-Shaw theorem asserts that if T is a hyponormal operator with m cyclicity then the self commutator [T*, T] is trace class. Further, its trace is dominated by a constant multiple of m.

It is an interesting question whether there is a two variable analogue of this theorem. This talk will report on a recent progress on this topic. It is based on a joint work with Yixin Yang.

Josh Isralowitz

Boundedness of matrix valued dyadic paraproducts on matrix weighted L^p

3pm on Wednesday, September 5, at ES142

Abstract:

Weighted norm inequalities for singular integral operators acting on scalar weighted $L^p$ is a classical topic that goes back to the 70's with the seminal work of R. Hunt, B. Muckenhoupt, and R. Wheeden. On the other hand, weighted norm inequalities for singular integral operators with matrix valued kernels acting on matrix weighted $L^p$ is very poorly understood and results (obtained by F. Nazarov, S. Treil, and A. Volberg in the late 90's) are only known for the situation when the kernel is essentially scalar valued.

In this talk, we discuss weighted norm inequalities for matrix valued dyadic paraproducts, which are dyadic ``toy models" of singular integral operators. Furthermore, we briefly discuss the possibility of using our results and a recent result of T. Hyt\"{o}nen to obtain concrete weighted norm inequalities for singular integral operator with matrix kernels acting on matrix weighted $L^p$.

Finally, we very briefly discuss the possibility of applying some of the ideas in this talk to the study of the usual Bergman projection on the usual Bergman space of the unit disk $\mathbb{D}$. This is joint work with Hyun-Kyoung Kwon and Sandra Pott.

Speaker: Oleg Lunin, Physics Department, SUNY at Albany

Title: Conformal Field Theory on Orbifolds, Part 2

Wednesday, April 25, at 3:00 pm in ES153

Speaker: Oleg Lunin, Physics Department, SUNY at Albany

Title: Conformal Field Theory on Orbifolds

Wednesday, April 11, at 3:00 pm in ES153

Abstract: Two-dimensional conformal field theory (CFT) utilizes the

methods of complex calculus to solve various problems arising in

statistical mechanics and in string theory. Recently string theorists

uncovered some surprising relations between CFT and gravity, which

became known as AdS/CFT correspondence. In particular, this

correspondence predicts that quantum properties of black holes can be

extracted from the study of the CFT on an orbifold, which is

constructed by taking a quotient of a regular manifold.

This talk will review the main ideas of string theory, conformal field

theory and AdS/CFT duality. The new technique for studying CFT on

orbifolds will be introduced, and it will be used to evaluate

correlations functions, which encode the physics of quantum gravity.

Speaker: Ron Yang, SUNY at Albany

Title: Isometries in operator theory, part 2

Wednesday, February 29, at 1:40pm in ES142

Speaker: Ron Yang, SUNY at Albany

Title: Isometries in operator theory

Abstract: This talk first surveys some fundamental roles isometries play in operator theory.

Then it will report on the current status of research on commuting pairs of isometries.

Wednesday, February 22, at 1:40pm in ES138

Speaker: Professor Jing Zhang

Title: On a Question of Nessim Sibony

Wednesday, February 15, at 2:40pm in ES 153

Speaker: Professor Dan Willard, SUNY at Albany

Title: A Summary of Goedel's Incompleteness Theorem, Its

Significance and Boundary-Case Exceptions.

Wednesday, November 30, at 2:30 in ES 153

Abstract:

This talk will be addressed to an audience who has no prior knowledge about

mathematical logic or Goedel's Incompleteness Theorem. It will summarize

Goedel's formalism, its significance and our research about its permissible

boundary-case exceptions.

The Incompleteness Theorem is a 2-part result that was published by Goedel

in 1931. The first half of its results demonstrated that no mechanical

procedure can identify all the true statements of arithmetic. Its second

contribution demonstrated that conventional logics are unable to

corroborate their own consistency.

Part of the reason that the Second Incompleteness Theorem is puzzling is

that it also demonstrates it is awkward for conventional logics to

instinctively presume (without a formal proof) their own consistency. Our

research into Self-Justifying systems has formalized a partial resolution

to this riddle by showing how some unconventional logics can bolster a type

of instinctive faith in their own consistency.

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Speaker: Michael Range, SUNY at Albany

A Pointwise Basic Estimate for the Complex Neumann Problem

On Weakly Pseudoconvex Domains (continued)

Wednesday, November 16, at 3:00 in ES 153

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Speaker: Michael Range, SUNY at Albany

A Pointwise Basic Estimate for the Complex Neumann Problem

On Weakly Pseudoconvex Domains

Wednesday, November 9, at 3:00 in ES 153

After a brief review of classical integral representation techniques in

multidimensional complex analysis we introduce new kernels of Cauchy-Fantappié

type on arbitrary smoothly bounded pseudoconvex domains which reflect the

complex geometry of the boundary of the domain. A major application is a new

pointwise estimate for certain derivatives, which is the analogue of the classical

basic estimate of Kohn and Morrey in the L^2 theory of the d-bar Neumann

problem. These results are part of a program to make progress on the long-standing

problem of Hölder estimates for the Cauchy-Riemann equations on pseudoconvex

domains of finite type.

Speaker: Kehe Zhu, SUNY at Albany

Title: Maximal zero sets for Fock spaces

Wednesday, November 2, at 3:00 in ES 153

Speaker: Yueshi Qin, SUNY at Albany

Title: The maxmal invariant subspace of some operators

Wednesday, October 5, at 3:00 in ES 153

Marco Varisco (UAlbany)

An introduction to cyclic homology

Wednesday, April 27, at 1:30 in ES 153

Abstract: Cyclic (co)homology was introduced by Alain Connes at the

beginning of the 1980’s as a non-commutative analog of de Rham

cohomology, and since then it has played a major role in various areas

of mathematics, from non-commutative geometry to algebraic K-theory.

This talk will be a basic introduction to the definition and main

properties of cyclic homology.

Ivana Alexandrova (UAlbany)

Aharonov-Bohm Effect in Resonances of Magnetic Schrodinger

Operators with Potentials with Supports at Large Separation

Wednesday, April 13, at 1:30 in ES 153

Abstract: Vector potentials are known to have a direct significance

to quantum particles moving in the magnetic field.

This is called the Aharonov--Bohm effect

and is known as one of the most remarkable quantum phenomena.

Here we study this quantum effect through the resonance problem.

We consider the scattering system consisting of

two scalar potentials and one magnetic field

with supports at large separation in two dimensions.

The system has trajectories oscillating between these supports.

We give a sharp lower bound on the resonance widths

as the distances between the three supports go to infinity.

The bound is described in terms of the backward amplitude

for scattering by each of the scalar potentials

and by the magnetic field, and it also depends heavily

on the magnetic flux of the field.

Stuart White (University of Glasgow)

Perturbations of operator algebras

Wednesday, March 16, at 1:30 in ES 153

Abstract: In 1972, Kadison and Kastler equipped the set of all

C$^*$-algebras on a fixed Hilbert space with a natural metric and asked

whether sufficiently close operator algebras are spatially isomorphic. I'll

give a survey talk on this problem and it's connections to similarity

problems. If time allows, I'll also discuss some recent progress. No

prior knowledge about operator algebras will be assumed.

Young Joo Lee (Chonnam National University)

Sums of several Toeplitz products on the Hardy space

Wednesday, February 16, at 1:30 in ES 153

Abstract : On the Hardy space over the unit ball, we will consider

operators which have the form of a finite sum of products of several

Toeplitz operators. We will discuss recent results on characterizing

problems of when such an operator is compact or of finite rank. Some of

our results show higher dimensional phenomena.

Alfonso Rodrigues (University of Seville, Spain)

December 8 from 3pm to 4pm in ES 153

Abstract: In this talk we provide a precise description of the lattice of invariant

subspaces of composition operators acting on the classical Hardy

space, whose inducing symbol is a parabolic non-automorphism. This

is achieved with an explicit isomorphism between the Hardy space

and the Sobolev Banach algebra $W^{1,2}[0,\infty)$ that induces a

bijection between the lattice of the composition operator and the

closed ideals of $W^{1,2}[0,\infty)$. In particular, each invariant

subspace of parabolic non-automorphism composition operator always

consists of the closed span of a set of eigenfunctions. As a

consequence, such composition operators have no non-trivial reducing

subspaces.

Joint work with S. Shkarin and M. Ponce

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Francisco Manuel Canto-Martin

Infinite codimension of a subspace of L^infty (R) with connections to Ergodic Theory

December 1 from 3pm to 4pm in ES 153

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Antti Perälä (University of Helsinki)

Carleson measures, distributions and Toeplitz operators with radial symbols

November 17 from 4pm to 5pm in ES 153

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Speaker: Pat Cade, SUNY at Albany Title: A higher order trace-determinant formula 3:30pm 10/20/2010 at ES 153

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