# 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: **

#### 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.

### 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: **

**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:**

**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: ***I n 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: **