Functional Analysis and Mathematical Physics Interdepartmental Research Group (FAMP)

Upcoming Colloquia

Date and Time: Friday, March 12, at 10 AM

Location: Via Zoom on the FAMP Zoom link

Speaker: Marat V. Markin (California State University, Fresno)

Title:On the Smoothness of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

Abstract:   Abstract for 03-12-2021 talk


Part of the Functional Analysis, Mathematical Physics, and Dynamical Systems (FAMPDS) Joint American-Ukrainian Virtual Colloquium Series

Functional Analysis, Mathematical Physics, and Dynamical Systems (FAMPDS) Winter 2021 Virtual Workshop

The event, featuring four talks, will take place on March 5, 2021, from 10 AM-3 PM.

The workshop is intended to create and consolidate contacts and foster collaborative opportunities between the scholars and graduate students involved in research in functional analysis, mathematical physics, dynamical systems, fractal, arithmetic, and noncommutative geometry, number theory, as well as other areas of interest. The talks are to be kept at the level accessible to the graduate students and non-experts.

Schedule 

Recent Colloquia

Part of the Functional Analysis, Mathematical Physics, and Dynamical Systems (FAMPDS) Joint American-Ukrainian Virtual Colloquium Series

Date and Time: Friday, February 26, 2021, at 10:00 AM

Location: Zoom at Zoom Link for FAMP

Speaker: Dr. Anatoly N. Kochubei (Institute of Mathematics, National Academy of Sciences of Ukraine)

Title:Non-Archimedean Radial Calculus

Abstract: Abstract for FAMPDS Feb. 26 talk

 


Date and Time: Friday, February 26, 2021, at 12:00 PM

Location: Zoom at Zoom Link for FAMP

Speaker: Michael Maroun, Ph.D. (Independent Researcher, Boston, MA)

Title:Exact Solutions of the Local Linearization Associated with the One-Dimensional Nonlinear Schrödinger Equation

Abstract: Abstract for Feb. 26 talk


Date and Time: Friday, February 19, 2021 at 11:00 AM

Location: Zoom at Zoom Link for FAMP

Title: Perfectoid Modular Curves and Langlands Correspondence

Speaker: Shanna Dobson (California State University, Los Angeles)

 Abstract:  


Date and Time: Friday, December 11, 2020 at 11:00 AM

Location: Zoom at Zoom Link for FAMP

Title: From Rainbows to Resurgence: Asymptotics of the Airy Function

Speaker: Will Hoffer (University of California, Riverside)

Abstract: In this talk, we take a modern perspective on the problem of finding the Stokes behavior of the Airy function through Borel resummation of its asymptotic expansion. In particular, we find that an ordinary asymptotic power expansion (when the parameter approaches infinity along the positive real axis) is missing exponentially small terms. Notably, these exponential terms become dominant as the phase of the parameter changes, and this switching on is directly responsible for the Stokes phenomenon. The primary result, then, is that the full analytic behavior of the Airy function resurges from the original expansion on the positive real line. This perspective can be thought of as resurgence analysis on a perturbative approach to the problem.  


 

Date and Time: Friday, December 4, 2020 at 11:00 AM

Location: Zoom at Zoom Link for FAMP

Title: The Energy Eigenvalue for the Singular Wave Function of the Three Dimensional Dirac Delta Schrödinger Potential via Distributionally Generalized Quantum Mechanics

Speaker: Dr. Michael Maroun (Independent Researcher, Boston, MA)

Abstract: Unlike the situation for the 1d Dirac delta derivative Schrödinger pseudo potential (SPP) and the 2d Dirac delta SPP, where the indeterminacy originates from a lack of scale in the first and both a lack of scale as well as the wave function not being well defined at the support of the generalized function SPP; the obstruction in 3d Euclidean space for the Schrödinger equation with the Dirac delta as a SPP only comes from the wave function (the 𝐿 2 bound sate solution) being singular at the compact point support of the Dirac delta function (measure). The problem is solved here in a completely mathematically rigorous manner with no recourse to renormalization nor regularization. The method involves a distributionally generalized version of the Schrödinger theory as developed by the author, which regards the formal symbol “𝐻𝜓” as an element of the space of distributions, the topological dual vector space to the space of smooth functions with compact support. Two main facts come to light. The first is the bound state energy of such a system can be calculated in a well-posed context, the value of which agrees with both the mathematical and theoretical physics literature. The second is that there is then a rigorous distributional version of the Hellmann-Feynman theorem.


 

Date and Time: Friday, November 20, 2020 at 2:00 PM

Location: Zoom at Zoom Link for FAMP

Title: An Introduction to Complex Dimensions: The Case of Fractal Strings

Speaker:  Michel L. Lapidus, Distinguished Professor of Mathematics, Burton Jones Endowed Chair in Pure Mathematics (Department of Mathematics, University of California, Riverside)

Abstract: We provide an introduction to the mathematical theory of complex fractal dimensions (developed by the author and his collaborators), which captures the vibrations that are intrinsic to both fractal geometries and the prime numbers. We focus here on the case of fractal strings, or one-dimensional drums with fractal boundaries. Complex dimensions are the poles of suitably defined geometric zeta functions associated to fractal strings. Intuitively, their real and imaginary parts correspond respectively to the amplitudes and the frequencies of “geometric waves” traveling through the space of scales associated with the fractal string. Explicit formulas, significantly extending Riemann’s original explicit formula for the prime number counting functional and the Riemann zeros, enable us to express very precisely the oscillations intrinsic to fractal and arithmetic geometries, via the  underlying complex dimensions. Key examples of such formulas are fractal tube formulas and spectral asymptotic formulas with complex exponents, along with formulas for the prime orbit counting functions of certain dynamical systems generalizing the dynamical counterpart of the Prime Number Theorem. We will illustrate aspects of the theory by means of the Cantor string as well as via self-similar strings, the complex dimensions of which happen to exhibit generically very intriguing quasiperiodic patterns, which we conjecture to form (generalized) quasicrystals. In the next lecture, we plan to discuss the higher-dimensional theory of complex dimensions, and of the corresponding fractal zeta functions. The main reference for this talk is M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions, Springer Monographs in Mathematics, Springer, New York, 2013 (second revised and enlarged edition). 


Date and Time: Friday, November 13, 2020 at 11:00 AM

Location: Zoom at Zoom Link for FAMP

Title: In How Many Dimensions Do We Live?

Speaker: Dr. Piero Nicolini (Frankfurt Institute for Advanced Studies, Institute for Theoretical Physics, Goethe University Frankfurt, Germany)

Abstract: In this talk, we introduce the concept of dimension. We are to see that dimensions depend on the process of measurement and on the dynamics of a physical system. Then, we analyze the reasons that support the existence of a higher dimensional Universe. Finally, we discuss the alternative scenario of dimensionally reduced spacetimes. 


Date and Time: Friday, November 6, 2020 at 11:00 AM

Location: Zoom at Zoom Link for FAMP

Title: Asymptotic Analysis of the Boltzmann Equation for Dark Matter Relic Abundance

Speaker: Jaryd Ulbricht (University of California, Santa Cruz)

Abstract: A solution to the Boltzmann equation governing the thermal relic abundance of cold dark matter is constructed by matched asymptotic approximations, using a uniform WKB method for large temperatures. The approximation of the relic density is an asymptotic series valid when the abundance does not deviate significantly from its equilibrium value until small temperatures. Resonance and threshold effects are taken into ac-count at leading order by approximating the thermally averaged cross section when the temperature is small compared to the mass of the dark matter particle. We compare our results to a numerical determination of the relic abundance using a benchmark model and find a fantastic agreement.  


Date and Time: Friday, October 30, 2020 at 11:00 AM

Location: Zoom at Zoom Link for FAMP

Title: Spectral Bounds for Damped Systems

Speaker: Dr. Carsten Trunk (TU Ilmenau,  Germany)

Abstract: