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

### Organizers

 Charley Conley (email hidden, enable JavaScript) Savanna Gee (email hidden, enable JavaScript) Derek Lowenberg (email hidden, enable JavaScript) Alex Pokorny (email hidden, enable JavaScript)

### 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 TBA
 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

### 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.
 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.
 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.