Graduate Student Seminar

UC Riverside Department of Mathematics
Fridays 1–2pm in Skye 284


Charley Conley
Savanna Gee
Derek Lowenberg
Alex Pokorny

Scheduled Talks, Spring 2019

7 June 2019 An Introduction to Quantum Lie Algebras in the Example of $\mathfrak{sl}_2$
Jonathan Dugan
Lie Algebras have long been known to be rigid, in the sense that they cannot be deformed. However, Drinfeld and Jimbo discovered instead a quantization of an associated algebra called its universal enveloping algebra. Analogous to the embedding of a Lie algebra in its universal enveloping algebra, Delius, Huffmann, and Gould defined a quantum Lie algebra embedded inside the quantized universal enveloping algebra by using its structure as a Hopf algebra. Such a construction has yielded applications to the study of quantum integrable systems. In this talk, we will explain this construction of the quantum Lie algebra, by working through the example of $\mathfrak{sl}_2$.
31 May 2019 Some Aspects of Generalized Covering Space Theory
Jacob Garcia
Covering space theory is a classical tool used to characterize the geometry and topology of spaces. It seeks to separate the main geometric features from certain algebraic properties. For each conjugacy class of a subgroup of the fundamental group, it supplies a corresponding covering of the underlying space and encodes the interplay between algebra and geometry via group actions. The full applicability of this theory is limited to spaces that are, in some sense, locally simple. However, many modern areas of mathematics, such as fractal geometry, deal with spaces of high local complexity. This has stimulated much recent research into generalizing covering space theory by weakening the covering requirement while maintaining most of the classical utility. This talk will focus on the relationships between generalized covering projections, fibrations with unique path lifting, separation properties of the fibers, and continuity of the monodromy.
24 May 2019 Understanding Tissue Formation via Cell-Based Mathematical Models
Christian Michael
Emergent tissue structure is difficult to study experimentally; in particular, linking hypotheses about molecular or local behavior of biological systems to understanding macroscale phenomena proves prohibitively difficult in some experimental cases. In our model, we attempt to recover the tissue-scale behavior of the shoot apical meristem (SAM) of Arabidopsis thaliana by implementing biologically relevant simplifications of such hypotheses to two-dimensional off-lattice node based simulations of SAM growth. To demonstrate the methods of a potential future direction of this model, another example of a multicellular system's model will be presented with an overview of the coarse-graining the model into a PDE system of nonlinear diffusion probability density evolution equations.
17 May 2019 The Hartogs's Phenomenon
Rubelio Berganza
In this talk I will go over the standard first result of several complex variables, the Hartogs's extension theorem, a result that gives insight to the singular set of a holomorphic function of several complex variables. Once that is done, we will move to the next natural topic, domains of holomorphy.
10 May 2019 Lang's Lebesgue Integral
Joshua Meyers
In his book Real and Functional Analysis, Serge Lang constructs the Lebesgue integral in a more natural way than the standard approach. He first defines the integral on step mappings, and then uses the Linear Extension Theorem to extend it to a large class of functions. Rather than going through the positive, real, and complex cases separately, Lang's definition applies immediately to the general Banach space-valued case, demonstrating that the ordering of the codomain is not a relevant structure to integration.
3 May 2019 Structural Differences Between Cold Dark Matter and Self-Interacting Dark Matter Models Throughout Time
Renata Koontz
We investigate fundamental structural differences as a function of time between the Cold Dark Matter and Self-Interacting Dark Matter $\sigma_x = 1$ models of dark matter halos using $N$-body simulations at scales of $30$ Mpc. To examine differences in structural formation of dark matter halos for both models, we compare the time $t_\frac{1}{2}$ at which a dark matter halo achieves half of its mass for masses ranging from $10^8 \sim 10^{12}$ $M_\odot$. Furthermore, we also compare mass-concentration parameter $c$ with $t_\frac{1}{2}$ and $z_\frac{1}{2}$ for these same masses ranges to find statistically significant differences. Once these differences are statistically significant, we investigate dark matter halo density and velocity dispersion profiles closest to the median using the Navarro-Frenk-White Profile.
26 April 2019 What is geometry and how can it be complex?
Alec Martin
This talk will introduce the subjects of Riemannian geometry and complex geometry. We will see how linear algebra and topology combine to form geometry, then how complex analysis and some more linear algebra give the subject of complex geometry. We will also see how real analysis can play a role. Some examples will be given to compare and contrast the worlds of complex and real (Riemannian) geometry.
19 April 2019 Combinatorics of Polytopes as a Shadow of Algebraic Geometry
Ethan Kowalenko
A polytope is the convex hull of finitely many points in Euclidean space, and is familiar object to everyone: examples including a point, a line segment, a tetrahedron, a pentagon, any Platonic solid, etc. To an $n$-dimensional polytope $P$, we will consider its face vector, which has its $i$-th coordinate given by the number of faces of $P$ of dimension $i$. This can be written as a polynomial $f(x)$, and we define the $h$-polynomial of $P$ to be $h(x)=f(x-1)$. This $h$ vector has some surprising properties, including non-negative coefficients. We'll discuss some of these constructions and how properties of $h(x)$ come from cohomology of certain algebraic varieties with nice symmetries.
12 April 2019 Honeycomb Model of $GL_n(\mathbb{C}$) and Tensors of $GL_n$ Irreducible Representations
Maranda Smith
A long standing question about the representation theory of $GL_n(\mathbb{C})$ is " for which triples of dominant weights $\lambda, \mu$, and $\nu$ does the tensor product $V_{\lambda} \otimes V_{\mu} \otimes V_{\nu}$ contains a $GL_n(\mathbb{C})$ invariant vector?" With the help of honeycomb modeling on the hexagonal lattice of triples of weights, this problem becomes a matter of checking the boundary edges of a triangle laid over our lattice.This will be a brief tour of this process for general $GL_n(\mathbb{C})$ and a concrete example on $GL_3 (\mathbb{C})$.

Scheduled Talks, Winter 2019

15 March 2019 Flag Algebras
Gabriel Elvin
Flag algebras, first developed by Alexander Razborov in 2007, are a vital tool used to solve problems in extremal graph theory. In this talk, we will build up the necessary machinery to define flag algebras and explore through examples, as well as talk about some applications and results that are obtained through the use of flag algebras. Just to preview, our algebras are generated by the set of all finite graphs (or certain interesting subsets, e.g. triangle-free graphs). Recall that an algebra needs to have addition and multiplication of its elements, so we must somehow define these two operations on graphs in a useful way. We will accomplish this by using probability, specifically (but informally) the probability of seeing a certain graph inside of another graph if you look at vertices randomly.
8 March 2019 Your Daily Recommended Serving of Fiber Bundles
Jonathan Alcaraz
Fiber bundles are an analogue to short exact sequences in the category of topological spaces. In this talk, I will define the structure, give examples and explain some cases when this analogy actually has mathematical substance.
1 March 2019 Subfactors, Conformal Field Theory, and the Thompson group F
Lex Luthor
In the inaugural talk of his national collegiate speaking tour, Mr. Luthor will discuss annular subfactors and their connection to conformal field theory. This will be followed by a demonstration of Vaughan Jones' recent WYSIWYG representation of the Thompson group F, and whether it is changing opinions on a certain conjecture.
22 February 2019 Foliations and the Godbillon-Vey Class
Ben Russell
A foliation is a decomposition of a manifold into injectively immersed submanifolds in a way that is locally modeled on the decomposition of $\mathbb R^n$ by $(n-q)$-planes. As such foliations reside in the intersection of geometric topology and dynamical systems. In this talk we introduce the basic definitions and machinery of foliations with an eye towards constructing the so-called Godbillon-Vey class of a codimension $1$ foliation, which Thurston described as measuring the "helical wobble" of the foliation.
15 February 2019 The fundamental tone of a Riemannian manifold
Xavier Ramos Olive
The Laplacian is a differential operator that's key to model the evolution of sound waves. A general solution to the wave equation can usually be written as a superposition of simpler pure sound waves, with a well defined frequency of oscillation. The lowest frequency is called the fundamental tone. It is possible to define a notion of Laplacian on a general Riemannian manifold and thus to study it's fundamental tone. The geometry of the manifold will influence the fundamental tone: it's diameter has a clear influence (larger strings, drums or flutes produce lower pitch sounds). So, from a geometer's perspective, a natural question is: does the curvature of the manifold influence the fundamental tone? We'll discuss a few results in this direction, including recent research carried out together with Shoo Seto, Guofang Wei and Qi S. Zhang.
8 February 2019 TBA
Xander Henderson
1 February 2019 Distributional Taylor Series for Languid Distributions
Matthew Overduin
An ordinary fractal string is defined as a bounded open set of the real numbers. The string can be represented by its counting function: $N_{\eta}(x)=\# \{l_{j}^{-1} \leq x:j \in \mathbb{N} \}$, where $\{ l_{j} \}_{j \in \mathbb{N}}$ are the associated interval lengths of the string. Viewing $N_{\eta}$ as a distribution, we can use an explicit formula to recover $N_{\eta}$ from the poles of the string's associated geometric zeta function. Moreover, this formula can be used to calculate the tubular volume of a fractal string from its geometric zeta function. When the ordinary fractal string is self-similar, this explicit formula does not have an error term. In this talk, I will discuss the properties of ordinary fractal strings as well as properties of self-similar strings. I will then present the explicit formula in two ways, one as a sum of residues, and another way as a Taylor Series. I will then calculate $\eta$ and the tubular formula for a self-similar string using the explicit formula.
25 January 2019 The Calculus of Variations and the Variational Bicomplex
Michael McNulty
Throughout Mathematics and Physics, the Calculus of Variations allows us to precisely formulate some very simple questions. For example, given two points on some geometric object, how might someone find the shortest path between those two points? The Calculus of Variations provides us with the tools necessary to consider and answer such questions among many others in fields such as Geometry, Analysis, and Differential Equations. Even better, the Calculus of Variations fits beautifully into a cohomology theory called the Variational Bicomplex which illuminates the underlying structure of the problems one considers in the field. After reviewing some example problems from the Calculus of Variations, we will expose ourselves to and explore the Variational Bicomplex among some of its very striking properties.
18 January 2019 The Pi Calculus - Toward Global Computing
Christian Williams
Historically, code represents a sequence of instructions for a single machine. Each computer is its own world, and only interacts with others by sending and receiving data through external ports. As society becomes more interconnected, this paradigm becomes more inadequate - these virtually isolated nodes tend to form networks of great bottleneck and opacity. Communication is a fundamental and integral part of computing, and needs to be incorporated in the theory of computation. To describe systems of interacting agents with dynamic interconnection, in 1980 Robin Milner invented the pi calculus: a formal language in which a term represents an open, evolving system of processes (or agents) which communicate over names (or channels). Because a computer is itself such a system, the pi calculus can be seen as a generalization of traditional computing languages; there is an embedding of lambda into pi - but there is an important change in focus: programming is less like controlling a machine and more like designing an ecosystem of autonomous organisms. We review the basics of the pi calculus, and explore a variety of examples which demonstrate this new approach to programming. We will discuss some of the history of these ideas, called “process algebra”, and see exciting modern applications in blockchain and biology. “… as we seriously address the problem of modelling mobile communicating systems we get a sense of completing a model which was previously incomplete; for we can now begin to describe what goes on outside a computer in the same terms as what goes on inside - i.e. in terms of interaction. Turning this observation inside-out, we may say that we inhabit a global computer, an informatic world which demands to be understood just as fundamentally as physicists understand the material world.” — Robin Milner
11 January 2019 A Better Path to Math Careers
Tim McEldowney
To help advance women and minorities in mathematics, we first need to ensure individuals are aware of the different options and pathways available to them. Many women and underrepresented minorities graduate with a degree in mathematics without even knowing about the possible career paths in math, or are underprepared to pursue them. Last year, as a graduate student at University of California, Riverside, I decided to do something about this. created the Advanced Mathematics Program (AMP) with the support of Drs. Po-Ning Chen and Yat-Sun Poon, graduate students, and staff. AMP is a free Summer Program that prepares students for abstract algebra and real analysis, two topics which often prove to be barriers to reaching careers in math. In addition, we helped students learn about what future careers they can consider with talks from mathematicians in pure math, applied math, and math education. I will address how I initially designed the program and the lessons I learned in its implementation. I will also address how we have expanded the program in its second year to include events we had during the academic year, including attending conferences with the former AMP participants and the additional invited speakers.

Scheduled Talks, Fall 2018

7 December 2018 Translating DisCoCat
Jade Master
The two most popular ways of representing meaning in natural language are the formal algebraic grammars of Lambek and Chomsky and the distributional methods often used in computer science. DisCoCat (Distributional Compositional Categorical) models of meaning are a way of combining these techniques into a coherent model. A critique of DisCoCat models is that they can be inflexible because they don't allow updating or changing, a feature that is ubiquitous in natural language. In this talk I will give an introduction to DisCoCat models as well as present a mathematical model of translation between two DisCoCat models which constitutes a first step towards making DisCoCat models more flexible.
30 November 2018 The Monad
Christian Williams
Theoretical computer science has been blossoming for decades, through the dao of category theory. In my time here at UCR, I will be telling you all about these beautiful ideas and why you should care - I mean, if you let me. The first step is clear: the monad. Contemplated by philosophers throughout history, it is now loved and feared by modern programmers. But we, as learn-ed mathematicians, will grasp this concept as effortlessly as a childhood dream. What is it all about? Action by composition. The monad is the two-dimensional generalization of the monoid, and the core of higher algebra. We will explore provocative statements such as "a category is just a monad", "RMod is just a slice of a double category", and "a distributive law is just a monad in a bicategory of monads". At the end, I will give a brief demonstration of formal verification software, and talk about why we should learn to code our research. Thank you for reading, and I hope to see you there!
16 November 2018 Diffraction by Complex Dimensions and the Poisson Summation Formula
Edward Voskanian
Following a measure theoretic idealization of kinematic diffraction, in which the diffraction pattern of a structure is described by the Fourier transform of the autocorrelation of that structure, the unique autocorrelation measure for the set of complex roots of a regular lattice Dirichlet polynomial is computed. By deriving a suitable version of the Poisson summation formula for certain degenerate lattices in Euclidean space, the Fourier transform of the autocorrelation is also obtained. While this work addresses the open problem stated by Lapidus and Van Frankenhuijsen, which asks if the ``quasiperiodic'' set of complex dimensions of a nonlattice self-similar fractal form a (generalized) mathematical quasicrystal.
9 November 2018 Fractional Differentiation Part II
Matthew Overduin
In this talk, I will continue discussing fractional differentiation. I will present a general formula for finding the fractional derivative of a polynomial. After I discuss fractional differentiation of functions, I will discuss how to fractional differentiate distributions. One example that I will discuss will be the fractional derivative of the Dirac Delta distribution. I will then present a general form of a fractional derivative Taylor series expansion of a function and a distribution.
2 November 2018 Global Riemannian Geometry
Lawrence Mouillé
In global Riemannian geometry, the main goal is to understand the connections that exist between curvature and topology. In this talk, I will give a survey of active research topics in the field and showcase the types of tools that are used to study them.
26 October 2018 Spectral Triples, Quantum Compact Metric Spaces, and the Sierpinski Gasket
Therese Landry
One of the fundamental tools of noncommutative geometry is Connes' spectral triple. Michel Lapidus and his collaborators have developed spectral triples for the Sierpinski gasket that recover the Hausdorff dimension, the geodesic metric, and the $\log_2 3$-dimensional Hausdorff measure. The space of continuous, complex-valued functions on the Sierpinski gasket can be viewed as a quantum compact metric space. The Gromov-Hausdorff distance is an important tool of Riemannian geometry, and building on the earlier work of Rieffel, Latr\'emoli\`ere introduced a generalization of the Gromov-Hausdorff distance to the quantum compact metric space. Aspects of geometry that can be recovered via the Gromov-Hausdorff propinquity will be discussed and compared with the geometric information that can be obtained from spectral triples.
19 October 2018 Showing the Ropes of Subfactor Planar Algebras
Charley Conley
Watch me get tied up with string diagrams! Topics include:
  1. Subfactors
  2. Planar Algebras
  3. Subfactor Planar Algebras
12 October 2018 Fractional Differentiation, Distributional Derivatives, and Distributional Taylor Series
Matthew Overduin
Examples of fractional derivatives include taking the "1/2 ", "2/3", or "1 + i" derivative and so on of a function. In this talk, I will present how to compute the fractional derivatives of continuous functions, and I will do some examples. There are three main formulas that I will go over. I will also present how to compute fractional derivatives of distributions.
5 October 2018 Survey of Several Complex Variables
Dylan Noack
After any course in complex analysis with one variable, the next logical step is several variables. Adding a few more z's into the pot changes the theory drastically. One major difference is that the Riemann Mapping Theorem no longer holds true, so naturally one of our chief efforts has been in generalizing it. We use ideas from both differential and algebraic geometry, so even if you don't want to be a pure SCV specialist, geometers of all varieties are welcome!

Talks from Previous Years

2017–2018 2016–2017 2015–2016 2014–2015 2013–2014 2012–2013