# Functional Analysis and Mathematical Physics Interdepartmental Research Group (FAMP)

## Upcoming Colloquia

More coming in Fall 2021!

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

**Date and Time:** Friday, May 21, at 10 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker: **Viacheslav Rabanovich (Institute of Mathematics, National Academy of Sciences of Ukraine)

**Title:***On Decompositions of an Operator into a Sum of Projections*

**Abstract:** Linear combinations of orthoprojections (in short, projections), in particular their
sums, appear

in various problems of operator theory and its applications. A classical result is
the spectral theorem on decomposition of a self-adjoint operator with a finite spectrum
into a linear combination of projections onto the eigenspaces, the projections being
pairwise orthogonal. Excluding the orthogonality condition for projections leads to
interesting. For instance, every bounded self-adjoint operator is a linear combination
of 4 projections (or an integral combination of 5 projections). Also, operators from
a wide class can be decomposed into sums of a small number of projections. Such decompositions
are used in different numerical problems, especially where the calculation process
can be parallelized, in information theory (frames as codes), and in quantum information
theory. We discuss the known facts on sums of projections, describe general methods
of constructing various decompositions, and partially consider numerical applications.

## Recent Colloquia

**Date and Time:** Friday, May 7, at 12 PM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Saurya Das (University of Lethbridge, Canada)

**Title:***Quantum Raychaudhuri Equation: Implications for Spacetime Singularities and the Quantum
Origin of Lambda*

**Abstract:** The Raychaudhuri equation predicts the convergence of geodesics and gives rise to
the singularity theorems. The quantum Raychaudhuri equation (QRE), on the other hand,
shows that quantal trajectories, the quantum equivalent of the geodesics, do not converge
and are not associated with any singularity theorems. Furthermore, the QRE gives rise
to the quantum corrected Friedmann equation. The quantum correction is dependent on
the wavefunction of the perfect fluid whose pressure and density enter the Friedmann
equation. We show that for a suitable choice of the wave-function this term can be
interpreted as a small positive cosmological constant.

**Date and Time:** Friday, April 30, at 12 PM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Michael Dunia (Graduate Student, CSU, Fresno)

**Title:***Massive Photon, Magnetic Charge, and the Dirac Quantization Condition*

**Abstract:** In this talk, we correct previous work on magnetic charge plus a photon mass. We
argue that contrary to previous claims this system has a very simple, closed-form
solution which is the Dirac string potential multiplied by an exponential decaying
part. We present interesting features of this solution, namely: (i) the Dirac string
becomes a real feature of the solution; (ii) the breaking of gauge symmetry via the
photon mass leads to a breaking of the rotational symmetry of the monopole's magnetic
field; (iii) the potential alteration of the Dirac quantization condition.

**Date and Time:** Friday, April 23, at 12 PM

**Location: **Via Zoom on the FAMP Zoom link

**Speaker:** Carsten Trunk (TU Ilmenau, Germany)

**Title: ***Perturbations of Periodic Sturm-Liouville Operators*

**Abstract: **

**Date and Time:** Friday, April 16, at 12 PM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Pasquale Bosso (University of Lethbridge, Canada)

**Title:***Quantum Gravity Phenomenology from the Generalized Uncertainty Principle*

**Abstract:** One of the cornerstones of Quantum Mechanics (QM), Heisenberg's Uncertainty Principle
(HUP), establishes that it is not possible to simultaneously measure with arbitrary
precision both the position and the momentum of a quantum system. This principle,
however, does not prevent one from measuring with infinite precision the system’s
position. However, theories of Quantum Gravity, aiming to bridge between General Relativity
and QM, predict the existence of a minimal observable length - a minimal uncertainty
on the position generally of the order of the Planck length. This prediction results
therefore in a contradiction with HUP, requiring a modification of the principle.
This need gave rise to the Generalized Uncertainty Principle (GUP). In this talk,
we show how the GUP can change known aspects of standard QM, leading to ways to test
Quantum Gravity.

**Date and Time:** Friday, April 9, at 11 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker: **Jean Renault (Institut Denis Poisson Université d'Orléans Rue de Chartres, France)

**Title:***The C*-Algebra of a Twisted Groupoid Extension*

**Abstract:** A strong motivation for the study of C*-algebras is the theory of unitary representations
of locally compact groups. In the seventies, M. Rieffel and P. Green developed respectively
the theory of induced representations and Mackey normal subgroup analysis in a C*-algebraic
framework. We discuss how Mackey's virtual groups program also fits into this framework
and illustrate it by the example of projective representations of locally compact
abelian groups.

**Date and Time:** Friday, March 26, at 12 PM

**Location:** Via Zoom on the FAMP Zoom link

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

**Title:***On Spectral Mapping Theorems and Asymptotics of Scalar Type Spectral C0-Semigroups*

Abstract: We establish spectral inclusion and mapping theorems for scalar type spectral
operators and thereby extend a *weak spectral mapping theorem* and a *generalized Lyapunov stability theorem*, known to hold for the C_{0}-semigroups of normal operators on complex Hilbert spaces, to the more general case
of the C_{0}-semigroups of scalar type spectral operators on complex Banach spaces. For such semigroups,
we also obtain a spectral mapping theorem for point spectrum, exponential estimates,
and an analogue of the *Gearhart-Prüss-Greiner* characterization of the uniform exponential stability for C_{0}-semigroups on complex Hilbert spaces.

**Date and Time:** Friday, March 19, at 12 PM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Sven-Ake Wegner (University of Hamburg, Germany)

**Title:***Port-Hamiltonian Differential Equations on Infinite Networks*

**Abstract: **

**Date and Time:** Friday, March 12, at 12 PM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Edward Sichel (FAMP, CSU Fresno)

**Title:***On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators*

**Abstract: **

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

**Date and Time:** Friday, May 14, at 10 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Jean Renault (Institut Denis Poisson Université d'Orléans Rue de Chartres, France)

**Title:***The C*-Algebra of a Twisted Groupoid Extension*

**Abstract:** A strong motivation for the study of C*-algebras is the theory of unitary representations
of locally compact groups. In the seventies, M. Rieffel and P. Green developed respectively
the theory of induced representations and Mackey normal subgroupvanalysis in a C*-algebraic
framework. We discuss how Mackey's virtual groups program also fits into this framework
and illustrate it by the example of projective representations of locally compact
abelian groups.

**Date and Time:** Friday, May 7, at 10 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Michael Gekhtman (University of Notre Dame)

**Title:***Generalized Cluster Structures and Periodic Difference Operators*

**Abstract:** We will consider a construction that ties together of several diverse notions including
spaces of periodic difference operators, Poisson sub manifolds of a Drinfeld double
of GL(n) and subsets of Grassmannians stable under the action of powers of a cyclic
shift. The theory of generalized cluster algebras serves as a unifying theme. Time
permitting, we will also discuss potential applications to representation theory

of quantum affine algebras at roots of unity.

Based on a joint work with M. Shapiro and A. Vainshtein and an ongoing project with
C. Fraser and K. Trampel.

**Date and Time:** Friday, April 30, at 10 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Michel L. Lapidus (UC, Riverside)

**Title:***Can One Hear the Shape of a Fractal Drum?*

**Abstract:** A well-known problem in mathematics and physics consists in understanding how the
geometry (or shape) of a musical instrument affects it sound. This gives rise to two
related types of mathematical problems: direct spectral problems (how the shape of
a drum affects its sound) and inverse spectral problems (how one can recover the shape
of a drum from its sound). Here, we consider both types of problems in the context
of drums with fractal (that is, very rough) boundary. We show, in particular, that
one can “hear” the fractal dimension of the boundary (a certain measure of its roughness)
and, in certain cases, a fractal analog of its length. In the special case of vibrating
fractal strings (the one-dimensional situation), we show that the corresponding inverse
spectral problem is intimately connected with the Riemann Hypothesis, which is arguably
the most famous open problem in mathematics and whose solution will likely unlock
deep secrets about the prime numbers. In conclusion, we briefly explain how this work
eventually gave rise to a 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.

**Date and Time:** Friday, April 23, at 10 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Michael Maroun (Independent Researcher, Boston, MA)

**Title:** Generalized Nonlinear Schrödinger Equation: Self-Dynamics by Convolution

**Abstract:**

**Date and Time:** Friday, April 16, at 10 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Oleksandr Baranovskyi (Institute of Mathematics, National Academy of Sciences of
Ukraine)

**Title:***Fractal Analysis of Singular, Nowhere Differentiable, and Nowhere Monotonic Functions*

**Abstract:** We give an overview of research topics at the Department of Dynamical Systems and
Fractal Analysis. In particular, various systems of encoding (representation) for
real numbers with finite and infinite alphabet are discussed. We use these systems
for analytical definition and studying some mathematical objects with complicated
local structure. Singular probability distribution functions, nowhere differentiable
functions, and nowhere monotonic functions are among them. We study fractal properties
of such functions, i.e., fractal properties of their level sets, graphs, and other
related sets.

**Date and Time:** Friday, March 26, at 9 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Vesselin G. Gueorguiev (Institute for Advanced Physical Studies, Bulgaria, Ronin
Institute for Independent Scholarship, USA)

**Title:***Revisiting the Cosmological Constant Problem within Quantum Cosmology*

**Abstract:** A new perspective on the Cosmological Constant Problem (CCP) is proposed and discussed
within the multiverse approach of Quantum Cosmology. It is assumed that each member
of the ensemble of universes has a characteristic scale a that can be used as integration
variable in the partition function. An averaged characteristic scale of the ensemble
is estimated by using only members that satisfy the Einstein field equations. The
averaged characteristic scale is compatible with the Planck length when considering
an ensemble of solutions to the Einstein field equations with an effective cosmological
constant. The multiverse ensemble is split in Planck-seed universes with vacuum energy
density of order one; thus, Λ̃≈8π in Planck units and a-derivable universes. For a-derivable
universe with a characteristic scale of the order of the observed Universe a≈8×10^{60}, the cosmological constant Λ=Λ̃/a2 is in the range 10^{−121} − 10^{−122}, which is close in magnitude to the observed value 10^{−123}. We point out that the smallness of Λ can be viewed to be natural if its value is
associated with the entropy of the Universe. This approach to the CCP reconciles the
Planck-scale huge vacuum energy-density predicted by QFT considerations, as valid
for Planck-seed universes, with the observed small value of the cosmological constant
as relevant to an a-derivable universe as observed.

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

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Selden Crary (Independent Researcher, Palo Alto, California, USA)

**Title:***The Algebra and Geometry of Twin-Point Designs in Optimal Design of Computer Experiments*

**Abstract:** We continue our introduction to optimal clustered designs in statistical design of
computer experiments. (1) New experimental evidence: The emergence of a hierarchy
of clustered designs, such as pairs of twin-point designs. (2) New theory: A theorem
stating the integrated-mean-squared-prediction-error objective function is a low-degree-truncated,
rational, generalized function; the connection of this “Nuclass” of function with
total positivity and of total positivity with the Riemann hypothesis; and a theorem
stating that the assumption of invariance of the algebra of twins, over their design
domain, leads to curvature of the domain. (3) Speculated applications, beyond design
of experiments: Superconductivity, topologically protected states of matter, biological
regeneration, and anti-senescence.

**Date and Time:** Friday, March 19, at 10 AM

**Location:** Via Zoom on the FAMP Zoom link

**Speaker:** Volodymyr Koshmanenko (Institute of Mathematics, National Academy of Sciences of
Ukraine, Interdisciplinary Research Center for Complex Systems, National Pedagogical
University, Kyiv)

**Title:***Setting of the Conflict Problem in the Framework of Functional Analysis*

**Abstract:** We give an introduction to the mathematical setting of the conflict problem in terms
of constructions in a Hilbert space. The opponents, as the main objects of conflict
phenomena, will be presented by operators and vectors in a Hilbert space or by probability
distributions on a joint resource space for opponents. The conflict phenomenon is
mathematically reflected as the intersection or overlap of operator domains and spectral
measure supports. The confrontation between opponents is described by the conflict
composition (a transformation in terms of spectral measures). It generates the conflict
dynamical system. Thus, there emerges the fundamental question of how to redistribute
the starting intersections and overlaps.

The resolution of a conflict problem means arriving at equilibrium states (fixed points) for the conflict dynamical system.

**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: **

**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: **

##### 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.

**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: **