Seminars and Colloquia by Series

The Seiberg-Witten equations with Lagrangian boundary conditions

Series
Analysis Seminar
Time
Wednesday, January 19, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tim NguyenMIT
The Seiberg-Witten equations, introduced by Edward Witten in 1994, are a first-order semilinear geometric PDE that have led to manyimportant developments in low-dimensional topology. In this talk,we study these equations on cylindrical 4-manifolds with boundary, which we supplement with (Lagrangian) boundary conditions that have a natural Morse-Floer theoretic interpretation. These boundary conditions, however, are nonlinear and nonlocal, and so the resulting PDE is highlyunusual and nontrivial. After motivating and describing the underlying geometry for the Seiberg-Witten equations with Lagrangian boundary conditions, we discuss some of the intricate analysis involved in establishing elliptic regularity for these equations, including tools from the pseudodifferential analysis ofelliptic boundary value problems and nonlinear functional analysis.

L^p Estimates for a Singular Integral Operator motivated by Calderón's Second Commutator

Series
Analysis Seminar
Time
Wednesday, December 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Eyvindur Ari PalssonCornell University
When Calderón studied his commutators, in connection with the Cauchy integral on Lipschitz curves, he ran into the bilinear Hilbert transform by dropping an average in his first commutator. He posed the question whether this new operator satisfied any L^p estimates. Lacey and Thiele showed a wide range of L^p estimates in two papers from 1997 and 1999. By dropping two averages in the second Calderón commutator one bumps into the trilinear Hilbert transform. Finding L^p estimates for this operator is still an open question. In my talk I will discuss L^p estimates for a singular integral operator motivated by Calderón's second commutator by dropping one average instead of two. I will motivate this operator from a historical perspective and give some comments on potential applications to partial differential equations motivated by recent results on the water wave problem.

Square function, Riesz transform and rectifiability

Series
Analysis Seminar
Time
Wednesday, December 1, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Svitlana MayborodaPurdue
The quest for a suitable geometric description of major analyticproperties of sets has largely motivated the development of GeometricMeasure Theory in the XXth theory. In particular, the 1880 Painlev\'eproblem and the closely related conjecture of Vitushkin remained amongthe central open questions in the field. As it turns out, their higherdimensional versions come down to the famous conjecture of G. Davidrelating the boundedness of the Riesz transform and rectifiability. Upto date, it remains unresolved in all dimensions higher than 2.However, we have recently showed with A. Volberg that boundedness ofthe square function associated to the Riesz transform indeed impliesrectifiability of the underlying set. Hence, in particular,boundedness of the singular operators obtained via truncations of theRiesz kernel is sufficient for rectifiability. I will discuss thisresult, the major methods involved, and the connections with the G.David conjecture.

Discrete Littlewood-Paley analysis and multiparameter Hardy spaces

Series
Analysis Seminar
Time
Wednesday, November 17, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Guozhen LuWayne State
In this talk, we will discuss the theory of Hardy spacesassociated with a number of different multiparamter structures andboundedness of singular integral operators on such spaces. Thesemultiparameter structures include those arising from the Zygmunddilations, Marcinkiewcz multiplier. Duality and interpolation theoremsare also discussed. These are joint works with Y. Han, E. Sawyer.

Weighted estimates for quasilinear equations with BMO coefficients on Reifenberg flat domains and their applications

Series
Analysis Seminar
Time
Wednesday, November 10, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Nguyen Cong PhucLSU
We discuss a global weighted estimate for a class of divergence form elliptic operators with BMO coefficients on Reifenbergflat domains. Such an estimate implies new global regularity results in Morrey, Lorentz, and H\"older spaces for solutionsof certain nonlinear elliptic equations. Moreover, it can also be used to obtain a capacitary estimate to treat a measuredatum quasilinear Riccati type equations with nonstandard growth in the gradient.

Exit times of diffusions with incompressible drifts

Series
Analysis Seminar
Time
Wednesday, November 3, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Andrej ZlatosUniversity of Wisconsin, Madison
We consider the influence of an incompressible drift on the expected exit time of a diffusing particle from a bounded domain. Mixing resulting from an incompressible drift typically enhances diffusion so one might think it always decreases the expected exit time. Nevertheless, we show that in two dimensions, the only simply connected domains for which the expected exit time is maximized by zero drift are the discs.

Rational Inner Functions in the Schur-Agler Class

Series
Analysis Seminar
Time
Wednesday, October 27, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Greg KneseUniversity of Alabama
The Schur-Agler class is a subclass of the bounded analytic functions on the polydisk with close ties to operator theory. We shall describe our recent investigations into the properties of rational inner functions in this class. Non-minimality of transfer function realization, necessary and sufficient conditions for membership (in special cases), and low degree examples are among the topics we will discuss.

Polya sequences, gap theorems, and Toeplitz kernels

Series
Analysis Seminar
Time
Wednesday, October 20, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mishko MitkovskiGeorgia Tech
A separated sequence of real numbers is called a Polya sequence if the only entire functions of zero type which are bounded on this sequence are the constants. The Polya-Levinson problem asks for a description of all Polya sequences. In this talk, I will present some points of the recently obtained solution. The approach is based on the use of Toeplitz operators and de Branges spaces of entire functions. I will also present some partial results about the related Beurling gap problem.

Sobolev orthogonal polynomials in two variables and partial differential equations

Series
Analysis Seminar
Time
Wednesday, October 6, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Miguel PinarDpto. Matematica Aplicada, Universidad de Granada
Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

Series
Analysis Seminar
Time
Tuesday, October 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sasha AptekarevKeldish Institute for Applied Mathematics
The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence Q_{n+1} = zQ_n - a_{n-p+1}Q_{n-p}, p \in \mathbb{N}, n\geq p, where z is a complex variable and the coefficients a_k are positive and satisfy the perturbation condition \sum_{n=1}^\infty |a_n-a|<\infty . Our results generalize known results for p = 1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal-Nevai class. As is known, for p\geq 2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Q_n satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Q_n in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl's famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szeg\"o functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite-Pad\'e approximants as well as to vector continued fractions.

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