Fridays in Skye 284
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Social Time: 12.30-1p
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Talk: 1-1.50p
Upcoming Talk
03 May 2024
Practice defense
In this talk I am going to present the basics and development of fractal geometry. I will introduce the definitions of both ordinary fractal strings and generalized fractal strings. I am also going to present the explicit formula for generalized fractal strings as well as the Taylor series expansion that represents it. We can recover a fractal via the explicit formula if we know its complex dimensions and given that our string is languid. In order to represent the explicit formula as a fractional Taylor series, I used the Schwartz definition of fractional derivative of distributions. I will present an example of a fractal, called the Cantor fractal, for which we can extract the fractal for its known complex dimensions via the explicit formula since the Cantor fractal is languid. In the next part of the talk, I am going to present a new topic called fractal cohomology. This is particularly important for realizing the Weierstrass curve as a two-dimensional fractal. I am going to introduce the iterative function system that generates this curve as well as the fractal power series that represents it. The fractal power series has its roots in fractal cohomology and is generated through the iterated function system. I will present on this as well.
Scheduled Talks
07 June 2023
TBA
TBA
31 May 2024
TBA
TBA
24 May 2024
TBA
TBA
17 May 2024
TBA
TBA
10 May 2024
TBA
TBA
03 May 2024
Practice defense
In this talk I am going to present the basics and development of fractal geometry. I will introduce the definitions of both ordinary fractal strings and generalized fractal strings. I am also going to present the explicit formula for generalized fractal strings as well as the Taylor series expansion that represents it. We can recover a fractal via the explicit formula if we know its complex dimensions and given that our string is languid. In order to represent the explicit formula as a fractional Taylor series, I used the Schwartz definition of fractional derivative of distributions. I will present an example of a fractal, called the Cantor fractal, for which we can extract the fractal for its known complex dimensions via the explicit formula since the Cantor fractal is languid. In the next part of the talk, I am going to present a new topic called fractal cohomology. This is particularly important for realizing the Weierstrass curve as a two-dimensional fractal. I am going to introduce the iterative function system that generates this curve as well as the fractal power series that represents it. The fractal power series has its roots in fractal cohomology and is generated through the iterated function system. I will present on this as well.
26 April 2024
Surfaces in R^3 with Prescribed Extrinsic and Intrinsic Curvatures
The question of whether a manifold can be isometrically embedded into some Euclidean space is a long-standing problem in differential geometry. In this talk, we will look at some background and context on isometric embeddings, and then we will study isometrically embedded surfaces of revolution with different prescribed extrinsic/intrinsic curvatures. In particular, we prescribe the induced metric, the mean curvature with conformal metric, the second fundamental form, and the conjugate momentum, in order to study the resulting nonlinear ODE's.
19 April 2024
Higher Teichmuller Theory
Teichmuller theory began as the classification of complex structures on a Riemann surface. The space of all such structures is homeomorphic to $\mathbb{R}^{6g-6}$ and it became known as Teichmuller space. From another point of view, it is the moduli space of all surface group representations that are discrete and faithful into the Lie group $PSL(2, \mathbb{R})$. Higher Teichmuller theory searches for generalizations into higher rank Lie groups and one of its aims is to identify connected components of the moduli space consisting solely of such ‘nice’ representations. Many tools are used in this study, and this talk will highlight the role played by the theory of Higgs bundles in higher Teichmuller theory.
12 April 2024
Dissipative Solutions to the Surface Quasigeostrophic Equations
In a friendly expository manner, we'll elaborate on some basic techniques useful for solving PDEs. The Surface Quasigeostrophic equation (SQG) occurs naturally as a model for surface ocean currents and mathematically constitutes the 2D equivalent of the Navier Stokes Equations, featuring many of the same types of singularities. In this talk, we'll discuss ideas from harmonic and fourier analysis that will be useful to prove the well-posedness and smoothness of solutions to the equation with initial data valued in L^\infty(\mathbb{R}^2). These results follow from my most recent research in trying to prove the well-posedness of SQG with initial data valued in uniformly local L^p spaces.
Winter 2024
15 March 2024
Surgery on Manifolds and Scalar Curvature
The space of all Riemannian metrics on a given manifold is an infinite dimensional Frechet manifold. This enormous space is, in fact, convex and so not topologically interesting. But things might become much more complicated when we put a geometric condition on our metric. As an example, the space of positive scalar curvature metrics on the 7-sphere has infinitely many distinct path-components. The main tool for studying the topology of the space of positive scalar curvature metrics is the surgery theorem. This theorem was proved by Gromov and Lawson and, independently, by Schoen and Yau. In this talk, we will have a look at surgery on manifolds and we will see that the surgery theorem and prime decomposition theorem give a classification of positive scalar curvature 3-manifolds .
08 March 2024
Abelian extension and Second cohomology
This talk will go through some basic definitions of extensions, group actions, and cohomology. We will eventually classify abelian extensions by their second cohomology. This is something I recently read through for my research, since the proof is not complicated, I will try to make this talk more approachable for people that are new to the topic. Hopefully this process helps me to organize my learning results and helps others to learn a bit more about these topics.
01 March 2024
Think-and-say Loops and Stuff We Don't Know About Them
The overlooked younger sibling of Look-and-say sequences, think-and-say sequences are "digitally descriptive iterations" of numbers (and numberish things) which usually result in self-describing numbers like 21322314 (which has two 1's, three 2's, two 3's, and one 4). We cover silly results and open problems from the past 30 years like: the limited loop length lemma, transfinite trimming tricks, multiset multiplicity maps, pre-period predictions, growth spurt exceptions, and pea-pattern prime pickings.
23 February 2024
An application of (co)homology to covering spaces
The underlying idea behind homology is that you can count the 'holes' in a space. There is an interesting connection between the first homology group of a surface and the fundamental group. From this relation, it is possible to classify G-covering spaces of a surface using the first cohomology group with coefficients in G. In this talk we will outline the ideas behind this connection and spend some time working through examples.
16 February 2024
Besov-Valued Rough Heat Equations
In preparation for the speaker's oral exam, we'll have a look at Besov spaces, various embeddings, stochastic integration on Besov spaces and a formulation of the stochastic heat equation in the rough path setting. Inspired by very recent work by Peter Friz, Benjamin Seeger, and Pavel Zorin Kranich, this work spans many topics in analysis and PDE theory. This line of research is novel for (among other reasons) its application of Besov-valued rough differential equations to SPDE theory and the reframing of stochastic integration in terms of the language of increments
09 February 2024
A Very Brief Introduction to Magnetohydrodynamics
Magnetohydrodynamics (MHD) is the study of electrically conducting fluids with many applications in geophysics, astrophysics, and engineering processes. In this talk, we will derive the main governing equations of MHD. We will also talk about Alfvén waves, turbulence, and Landau damping (as time permits).
02 February 2024
Representation Theory of Finite Monoids
While most mathematicians only encounter the representation theory of finite groups and, in the realm of research, delve into representations of Lie algebras, the representation theory of monoids is a less-discussed but richly developed area with nearly a century of theory behind it. In this talk, we'll navigate this lesser-explored terrain, exploring monoid representations through the lens of cell theory. Time permitting, we'll also touch upon their generalization to algebras and their categorification.
26 January 2024
The Surreal World of Go
Being the oldest games still being played, Go has a lot of theory behind it. While studying the endgame theory of Go, John Conway got inspired and develop the idea of the surreal numbers. In this talk I'll give you some of the intuition behind them and the more general concept of "game". Finally, we will look at some scenarios and be able to compute the result.
19 January 2024
What are connections?
In this talk, I will start with a brief introduction to the geometric ideas needed to understand connections, and then attempt to give an intuitive understanding of what connections are, why they are useful and how they work.
Fall 2023
8 December 2023
Constructing Span 2-Categories From Categories Without Pullbacks, And Examples "In The Wild"
Span categories are used to give an abstract compositional model for certain types of systems. The category of smooth manifolds, a starting point for a compositional model for classical mechanical systems, does not have all finite pullbacks. Prior work by Weisbart and Yassine identifies two necessary conditions for constructing span categories to circumvent this obstacle: F-pullbacks, and span tightness, but these conditions are insufficient to generalizing to higher categories. We introduce a new condition, tentatively named cospan tightness, to construct span 2-categories from categories without pullbacks, and investigate examples of F-pullbacks, span tightness, and cospan tightness. Warning: This talk's gonna just be category theory BS. I hope for it to be interesting category theory BS.
1 December 2023
Practice Oral Exam
In this talk I will present my current published work on diffusion in two or more $p$-adic dimensions. Time permitting I will sketch future research directions.
17 November 2023
An Application of the Alternating Split Bregman Algorithm for Least Gradient Problems
Finding a minimizer of a least gradient problem is often a difficult task. In the particular problem from my current research, with the proper assumptions, we may use Fenchel-Rockafellar to consider the dual problem and an Alternating Split Bregman Algorithm which converges to weak solutions of the dual problem and the original (primal) problem. In this talk, we will derive the algorithm and discuss its convergence.
03 November 2023
Alysha's Practice Defense
Developmental biologists study processes such as pattern formation, morphogenesis, and cellular signaling in an attempt to unravel the complexities of how organisms develop. Providing insights into the fundamental biology of life often requires joint work by both theoretical and experimental biologists. In this talk I will be discussing our contributions to understanding these processes in two different systems: pavement cells in plant leaves and the Drosophila (fruit fly) wing disc.
27 October 2023
Symplectic Groups Actions on Manifolds
There is a class of group actions on symplectic manifolds that preserve the symplectic form. In this talk, we will give an introduction to symplectic manifolds and the interesting features of symplectic group actions, highlighting examples that motivated the study of symplectic manifolds in general.
20 October 2023
A Very Brief Introduction to Tensor Decomposition
In recent years, tensor decomposition has become a crucial tool in the areas of machine learning, signal processing, and statistics. In this talk, we will discuss the basics of tensor decomposition methods, namely the canonical polyadic decomposition (CPD/PARAFAC/CANDECOMP) and the higher-order singular value decomposition (HOSVD). We will also discuss some recent applications.
13 October 2023
Getting in Line: An Introduction to Queuing Theory
In an increasingly interconnected world, efficient data transmission and network performance are paramount. In this introductory talk, we will unravel the complexities and common challenges of queuing theory—a powerful tool for comprehending and optimizing network performance. Queuing theory plays a vital role in data traffic management, delay reduction, and overall network efficiency enhancement. We will explore standard examples like Deficit Round Robin (DRR) and Start-Time Fair Queuing (STFQ) as classic algorithms that serve as our stepping stones into the world of network queues. If time permits, I'll also share insights and applications from my summer internship.
06 October 2023
Lightning Talks
Come drop by to see various math grads give short introductory talks about the math they do here at UCR! We especially encourace newer grad students to attend and meet our department.
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
Chris Grossack, Vice President
Jialin, Jennifer Wang, Secretary
Rahul Rajkumar, Treasurer