Fridays in Skye 284
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Social Time: 3.30-4p
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Talk: 4-4.50p
Upcoming Talk
09 June 2023
Practice Defense
In this talk I am going to present the basics and development of fractal geometry. I am also going to present the explicit formula for fractal strings as well as the Taylor series expansion that represents it. I am then going to discuss certain types of fractal strings (those that induce smooth functions) as well as the expansions that represent those. I am going to discuss the Weierstrass fractal in more detail as well, and the fractional Taylor series that generates it.
Scheduled Talks
Spring 2023
09 June 2023
Practice Defense
In this talk I am going to present the basics and development of fractal geometry. I am also going to present the explicit formula for fractal strings as well as the Taylor series expansion that represents it. I am then going to discuss certain types of fractal strings (those that induce smooth functions) as well as the expansions that represent those. I am going to discuss the Weierstrass fractal in more detail as well, and the fractional Taylor series that generates it.
02 June 2023
Introducing Acute Myeloid Leukemia Modeling through ODE and PDE Models
Hematopoietic stem cells (HSCs) are characterized by their ability of self-renewal to replenish the stem cell pool and differentiate to more mature cells. The subsequent stages of progenitor cells also share some of this dual ability. It is yet unknown whether external signals modulate proliferation rate or rather the fraction of self-renewal. Three multicompartment models, which rely on a single external feedback mechanism, have been recently proposed. I compare these multi-compartment models with our updated multi-lineage ODE model. We adapt this multilineage model to have 2 versions of AML models, working towards understanding the progression and development of Acute Myeloid Leukemia cells. I will then expand from the ODE model and introduce a PDE advection diffusion reaction model for Acute Myeloid Leukemia.
26 May 2023
Intro to Regularity Structures
Martin Hairer's theory of Regularity Structures is useful to solving semi linear stochastic partial differential equations of the form Lu = F(u,\xi) where \xi is some noise term (not necessarily Gaussian). There are a number of innovative ideas presented in the theory to overcome the classical ill-posedness of these types of previously inaccessible SPDE. The famous examples of its usage is in solving the KPZ equation and many euclidean field theories. During the talk, we will familiarize ourselves with some of the machinery involved.
12 May 2023
Virtual Excessive Homology on Surface bundles
In this talk we will talk about virtual excessive homology on surface bundles. Begin the talk by defining surface bundles, Mapping class group, monodromy representation and virtual excessive homology. Then introduce corresponding theorems from Dehn-Nielsen-baer, Eels-Earle- Schatz and Birman. Next I will introduce some relatively recent results of virtual excessive homology on 3-manifolds, specifically surface bundle over circle (Mapping torus) and mention briefly about how to generalize these results in the future.
5 May 2023
Brownian Motion in Q_p and Q_p^d
We give a crash course on the Wiener process in real spaces as a model for Brownian motion. We introduce the space Q_p and a direct analogue of the Wiener process. Certain qualitative differences are exhibited, and future directions will be outlined.
28 April 2023
A Very Brief Introduction to Quantum Computing
Quantum computing is a rapidly growing area that has found applications to many different fields such as cryptography and computational chemistry. In this talk, we will go over the basics of quantum mechanics and how to analyze quantum logic gates and circuits. We will also cover some famous quantum algorithms, namely the quantum Fourier transform, Shor's algorithm, and Grover's algorithm.
21 April 2023
Why did the optimization problem cross the road?
To get to the other side, of course! But the other side of what? In this gentle talk, we'll talk about both sides of this metaphorical 'road', and why people who solve optimization problems - optimists, if you will - cross it pretty frequently. We'll start on the first side of the road with monotone inclusion problems and how they are formulated via monotone operators, one of the basic objects of the field. We'll cross the road and talk about fixed-point theorems in general, and how and why monotone inclusion problems are intimately connected to fixed point problems. We'll wrap it up with a real-life, practical example from my own research and see explicitly just how the famous Halpern iteration makes 'crossing the road' worth it for optimists in the present day.
7 April 2023
In Search for the Driving force of Life
In this talk we will try to describe the underlying driving forces in the growth of pollen tubes using a system of ODEs derived from Reaction Kinetics. This is a classic style of systems in Applied Math Biology, which can sometimes be significantly easier to approach using basic theory from Dynamical Systems. We will introduce the following concepts, the Law of Mass Action, Limit Cycles, and Phase Plane Analysis, to help us on our endeavor. Our discussion will be primarily constrained to the existence of limit cycles, as much of this problem is still open research.
Winter 2023
17 March 2023
Why I Will Never Be a Number Theorist
In this final talk of the quarter, I wish to move away from complicated and abstract topics and discuss the story of the time that I did some research in Elementary Number Theory only to discover that the results were not new. I continued trying in the hopes that my work would eventually produce something new. I will let you guess how that went. Join me as we derive the results that I found using nothing more than elementary algebra and number theory.
10 March 2023
An introduction to Hierarchically Hyperbolic Spaces
When looking at finitely generated groups acting on metric spaces, the case of a group acting on a hyperbolic metric is exceptionally nice: the hyperbolicity condition gives rise to many nice statements about the group action, including statements on growth and solvability of the word problem.
Unfortunately, not all metric spaces are hyperbolic [citation needed], however, many groups, such as mapping class groups and right angled Artin groups, act on metric spaces which have many properties which are similar to hyperbolic metric spaces called hierarchically hyperbolic spaces (HHS). In brief: a hierarchically hyperbolic space is a metric space whose coarse geometry is controlled by a family of related hyperbolic spaces.
In this (hopefully accessible) intro talk, I'll start by introducing some nice examples of HHSs, talk about some of the relevant axioms needed for understanding HHSs, then describe some of the most salient properties shared by all HHSs.
03 March 2023
Introducing Acute Myeloid Leukemia Modeling through ODE and PDE Models
Hematopoietic stem cells (HSCs) are characterized by their ability of self-renewal to replenish the stem cell pool and differentiation to more mature cells. The subsequent stages of progenitor cells also share some of this dual ability. It is yet unknown whether external signals modulate proliferation rate or rather the fraction of self-renewal. Czochra proposes three multicompartment models, which rely on a single external feedback mechanism. I compare Czochra's multi compartment model with our updated FACS gene ODE model. I will then expand from the ODE model and introduce our PDE advection diffusion reaction model.
24 February 2023
Street Fighting Mathematics: Using Computers for Fun and Profit
When taking math classes, a lot of time is spent on how we can prove true things, but how can you figure out what's true to begin with? Often it's useful to try lots of examples and intuitions, moving fast with very little emphasis on the "rigor" that is typically expected of a working mathematician. In this talk we'll show how to use sagemath, a computer algebra system, to make wild guesses and computations relating to various problems I've worked on over the years, many of which led me to a solution. To quote the author of Streetfighting Mathematics: "[come to this talk], and you'll be dangerous".
17 February 2023
Snow-White Light and the Seven Elementary Catastrophies
Once upon a time, physicists developed a theory of optics in which light is described by rays which undergo refraction and reflection. In this theory, optical causticsâ including rainbows, webs of light made by rippling water, and bright patterns made by glass cupsâ are described by envelopes of rays along which the light is brightest.
Mathematically, these caustics can be defined as sets of critical points of a suitable function that implicitly defines the trajectories of the light rays. Structurally stable caustics (under perturbations near critical points) are called elementary catastrophies. In the 1960s, René Thom proved that there are seven types of elementary catastrophies of codimension at most four. Consequently, Thom's theorem may be used to classify the possible shapes of optical caustics.
This expository talk, supported by the Fay Fellowship, is the first of the ``Big C Seminar" talks delivered to prepare graduate students for the Victor Shapiro Distinguished Lecture in Mathematics to be given by Sir Michael Berry on March 16, 2023.
10 February 2023
Representation stability for FI-modules and some analogues
The notion of representation stability emerged in the last decade as a means to understand the limiting behavior of certain naturally-arising sequences of representations of finite groups. The idea is to describe such a sequence as a single algebraic object, namely a âcategory moduleâ for a suitably chosen category. The limiting behavior of this sequence may then be explained in terms of a single finite generation property of the corresponding category module. I will give an overview of the classical case where the category in question is that of finite sets and injective maps (FI), highlighting the implications for homological stability of the symmetric groups with twisted coefficients. Then, I will discuss ongoing work extending these results to other categories and analogous twisted group homologies.
03 February 2023
Towards Defining DAHAs
A double affine Hecke algebra (DAHA) is an incredibly powerful tool that is now considered by many to be the "missing link" between special functions and representation theory. These insanely technical objects have more recently found a widespread use of applications in many different fields of mathematics as well as physics. My goal for this talk is to attempt to define double affine Hecke algebras and give some sort of feel for their uses by examining them from a couple different viewpoints with as little technical information as possible.
27 January 2023
Functions that Count
What's better than the prime numbers? A function that counts them! It turns out that adding extra structure to sequencesâ such as counting functions, Dirichlet series, and zeta functionsâ can help us to better understand the original sequence.
Allow me to invite you into the high society of mathematics for one hour only as we study the couture of sexy primes and million dollar questions. Be sure to pack your logarithms and log-log-logarithms!
20 January 2023
On Investigating chemical signaling networks in the Drosophila wing disc
During development of the Drosophila (fruit fly) wing disc, the regulation of cell height and tissue curvature is crucial in developing correct tissue shapes. This requires the interplay between mechanical forces and chemical signaling pathways, at both the cell and tissue levels. Although there are many known components in this process, there is still much work to be done. In this talk we will walk through the new signaling regulations identified through a combination of experimental and modeling approaches, as well as the process of using these models to investigate the robustness of the proposed networks.
Fall 2022
02 December 2022
On Asymptotic Expansions with Complex Exponents and their Applications
What do fractals and prime powers have in common? Complex exponents! Specifically, functions which describe these objects (the size of an epsilon neighborhood of a fractal and the summatory von Mangoldt function, as respective examples) admit asymptotic expansions which look like power series, but with complex numbers as exponents.
In this talk, we discuss formalization of such complex exponent power series/asymptotic expansions in order to more systematically study them and extend existing analytic machinery. In particular, we extend the Borel transform to classes of such expansions suitable for these applications and discuss Borel summation of such complex-powered asymptotics expansions.
04 November 2022
Logic in Color
Category theory can be understood as a vast generalization of what we mean by "logic". I will present this language of "generalized logic", also known as the "coend calculus". Together with a complementary visual language of colors, strings, and beads, I propose that anyone can learn category theory on a much deeper level than current material/books. I'll demonstrate by presenting the concept of limit.
28 October 2022
An Introduction to the Nonabelian Hodge Correspondence
The Narasimhan-Seshadri Theorem establishes a correspondence between stable vector bundles and unitary representations of the fundamental group of a compact Riemann surface. The Nonabelian Hodge Correspondence expands this to homeomorphisms between the moduli spaces of representations, flat connections on the bundle whose monodromy comes from the representation, and Higgs bundles for a KĂ€hler manifold. In this talk we shall restrict to the case of Riemann surfaces where general conditions on the Higgs field are trivially satisfied, making the correspondence easier to understand. We will give an overview of the correspondence including the Corlette-Donaldson theorem and then shall look at some properties of the Dolbeault Moduli space.
14 October 2022
Projective Points Over Matrices and Their Separability Properties
We will consider topological quotients of real and complex matrices by various subgroups and their connections to spacetime structures. These spaces are naturally interpreted as projective points. In particular, we will look at quotients of nonzero 2 x 2 matrices by the subgroups GL, SL, O, and SO and study various results about their topological separability properties. We will discuss the interesting result that, as the group we quotient by gets smaller, the separability properties of the quotient improve.
07 October 2022
A Rigidity Condition for Composition in an Augmented Span Double Category
Augmented generalized span categories provide a compositional framework for mathematical models of simple open systems in classical mechanics. The idea of augmenting an object with an element of a vector space extends the earlier idea of an augmentation to a framework that includes Riemannian metrics and symplectic 2-forms. The procedure for constructing augmentations on composite spans was previously ad-hoc, but a rigidity condition in the extended setting forces a unique compositional rule for augmentations on pullbacks. The procedure involves defining appropriate functors to $k$-\textbf{Vect} and constructing action categories from these functors. The framework we explore should allow the compositional principle to be applied more broadly and with more intrinsic rigidity.
30 September 2022
Stated Skein Modules and DAHAs
Many knot invariants come from looking at highly noncommutative associated groups. As these groups can be incredibly difficult to work with, one can instead consider corresponding commutative algebras and representations. However, if you want to still extract knot invariants you need to quantize these algebras and skein theory provides a breathable way to understand these deformations. In 2012, Berest and Samuelson provided a geometric way to understand these invariants and in the process uncovered certain defining modules for a bigger underlying beast, double affine Hecke algebras. I will discuss these module structures and how they act in the context of a newer, more general skein theory recently established by Thang LĂȘ. This new theory, called stated skein theory, provides significant additional algebraic structure to these algebras and modules and will hopefully lead to more insights into a nicer presentation of these DAHAs.
The UC Riverside Math Graduate Student Seminar (GSS) is brought to you by the UCR student chapter of the American Mathematics Society. GSS is organized by the officers of the chapter.
Officers
Will Hoffer, President
Jacob Garcia, Vice President
Bill Terry, Secretary
Rahul Rajkumar, Treasurer