## Analysis Seminar

Speaker: Isaak Chagouel, SUNY at Albany

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

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

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Speaker: Isaak Chagouel, SUNY at Albany

Title: Joint spectra of normal compact operators

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

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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.)

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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.

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Speaker: Kehe Zhu, SUNY at Albany

Title: Maximal zero sets for Fock spaces

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

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Speaker: Yueshi Qin, SUNY at Albany

Title: The maxmal invariant subspace of some operators

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

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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.

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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.

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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.

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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.

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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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