Schedule
Friday, February 10, 2:00pm - 3:00pm, Keck 370
Antonino De Martino, Chapman University
Harmonic and polyanalytic functional calculi on the S-spectrum for bounded and unbounded operators
Abstract: Harmonic and polyanalytic functional calculi have been recently defined for bounded commuting operators. Their definitions are based on the Cauchy formula of slice hyperholomorphic functions and on the factorization of the Laplace operator in terms of the Cauchy-Fueter operator D and of its conjugate D. Thanks to the Fueter extension theorem when we apply the operator D to slice hyperholomorphic functions we obtain harmonic functions and via the Cauchy formula of slice hyperholomorphic functions we establish an integral representation for harmonic functions. This integral formula is used to define the harmonic functional calculus on the S-spectrum. Another possibility is to apply the conjugate of the Cauchy-Fueter operator to slice hyperholomorphic functions. In this case, with a similar procedure we obtain the class of polyanalytic functions, their integral representation and the associated polyanalytic functional calculus. The aim of this talk is to disucss about the harmonic and the polyanalytic functional calculi to the case of bounded and unbounded operators and to prove some of the most important properties. These two functional calculi belong to so called fine structures on the S-spectrum in the quaternionic setting. Fine structures on the S-spectrum associated with Clifford algebras constitute a new research area that deeply connects different research fields such as operator theory, harmonic analysis and hypercomplex analysis.
Friday, April 14, 2:00pm - 3:00pm, Keck 370
Fred Lin (University of Bonn)
Singular Brascamp-Lieb Inequality with Dimension 1,2,2, and 1
Abstract: There are numerous important inequalities in harmonic analysis falls in the class of Singular Brascamp-Lieb Inequality such as Hilbert transform, bilinear Hilbert transform, and twisted paraproduct, etc. Unlike the Brascamp-Lieb inequality which admits a unified method to characterized the Lp boundedness of all dimensions and multi-linearity, in singular Brascamp-Lieb inequality, the method to show the Lp boundedness varies in different cases. One of the famous conjecture is the the bounedness of triangular Hilbert transform. This form is a tri-linear form composed with three two dimension function and a one dimension kernel in a three dimension ambient space. If the dimension of the ambient space is clear, we express the rest dimension data as (2,2,2;1). In this sense, I will present the classification and boundedness(or unboundedness) of family of singular Brascamp-Lieb inequality with dimension data (1,2,2;1) in this talk.
Friday, April 21, 2:00pm - 3:00pm, Keck 370
Daniel Alpay (Chapman University)
Discrete first order systems and their characteristic spectral functions
Abstract: We define first–order discrete systems in the scalar and matrix–valued case. They are characterized by sequences of pair of matrices, called admissible sequences. We present two important examples of such sequences, called Szegö and Nehari sequences. We introduce the characteristic spectral functions associated to a first–order system. We define in particular the scattering function, the Weyl function and the reflection coefficient function and we study the relationships between these functions. This is joint work with Israel Gohberg.
Thursday, April 27, 10:00am - 11:00am, Keck 370
Marija Galić (University of Zagreb)
Analysis of a linear 3d fluid-composite structure interaction problem
Abstract: Fluid-structure interaction (FSI) problems are multi-physics problems which arise in many applications. The most known examples are aeroelasticity and biomedicine. The problem we study in this talk is motivated by the interaction between the blood flow in a coronary artery treated with a vascular stent. The vascular stent is a thin, metallic mesh tube which is inserted at the location of the narrowing of a diseased coronary artery in order to keep the passageway open. We consider a linear fluid-structure interaction problem between the flow of a viscous, incompressible fluid, and an elastic, composite structure. The fluid flow is modeled by the time-dependent Stokes equations, while the elastic structure consists of a cylindrical shell supported by a mesh-like structure. The fluid and the mesh-supported structure are coupled via the kinematic and dynamic boundary coupling conditions describing continuity of velocity and balance of contact forces at the fluid-structure interface. We prove the existence and uniqueness of a weak solution to this problem, and, moreover, show that such a weak solution possesses an additional regularity in both time and space variables for initial and boundary data satisfying the appropriate regularity and compatibility conditions imposed on the interface.
Friday, April 28, 2:00pm - 3:00pm, Keck 370
Oumar Wone (Chapman University)
Invariants of linear differential equations and applications
Abstract: We explain the theory of invariants of linear differential equations from basics and give possible applications to conformal geometry and differential Galois theory.
Tuesday, May 2, 10:00am - 11:00am, Keck 370
Stefano Pinton (Politecnico di Milano)
Quaternionic and Cliffordian fine structures on the S-spectrum
Abstract: The slice hyperholomorphic functions, thanks to their integral representation formula, turn out to be a suitable space of functions over which it is possible to construct a functional calculus, the so called S-functional calculus, for bounded quaternionic linear operators. The Laplace operator applied to a sclice hyperholomorphic function gives an axially monogenic function. Also these functions admit an integral representation formula that can be used to construct the so called F -functional calculus. Since the Laplace operator can be factorized by the Cauchy-Fueter operator and its conjugate, other two functional calculi can be defined. The first one, the so called Q- functional calculus, is defined over the space of axially harmonic functions. This set of functions is the image through the Cauchy-Fueter operator of the set of slice hyperholomorphic functions. The second one, the so called P2-functional calculus, is defined over the space of axially polyanalytic functions of order 2. This set of functions is the image through the conjugate of the Cauchy-Fueter operator of the set of slice hyperholomoprhic functions. We call the quaternionic fine structure the collection of all these functional calculi. Moreover, the quaternionic fine structure can be defined also for unbounded operators. In this talk I will give an overview of the definitions of all these functional calculi explaining some of their properties. I will also introduce the Cliffordian fine structure in dimension five. This talk is based on joint works with F. Colombo, I. Sabadini and A. De Martino.
Friday, May 5, 3:00pm - 4:00pm, Keck 370
Aaron Palmer (UCLA)
The sharp interface limit of a mean-field game phase transition model
Abstract: Mean-field games may exhibit phase transitions where the players exhibit qualitatively distinct behaviors in different parameter regimes. The 'Ising game' is a prototype inspired by the well-known 'Ising model' of statistical physics. A phase transition appears when the 'Ising game' admits two constant equilibrium solutions in the super-critical parameter regime. In this talk, we consider a macroscopic limit for which Nash-equilibrium solutions to the 'Ising game' concentrate on the two constant equilibria. A sharp interface forms and minimizes an energy functional related to an anisotropic space-time minimal surface. To prove this, we reformulate The 'Ising game' equilibria as critical points of an energy functional with non-local spatial interactions, a kinetic energy of the time gradient, and a double well potential. Our Gamma-convergence argument combines tools from both local and non-local phase transition models and handles the novel initial and terminal conditions from the mean-field game. Joint work with W. Feldman and I. Kim.