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

### Organizers

 Joe Moeller (email hidden, enable JavaScript) Dylan Noack (email hidden, enable JavaScript) Mike Pierce (email hidden, enable JavaScript)

### Scheduled Talks, Spring 2018

 1 June 2018 Constructing Arithmetic Hyperbolic Surfaces Jonathan Alcaraz In first-year Topology, we construct the so-called “Flat Torus” as the quotient of $2$-dimensional Euclidean space by integer linear combinations of the standard basis. This is used as an example for other topics in topology. In this talk, we will look at the abstract properties of this construction and apply them to hyperbolic space.
 25 May 2018 Kähler-Einstein metrics on compact cohomogeneity one Fano manifolds via effective approximations Pilar Orellana Kähler-Einstein metrics emerge when a complex, topological manifold, under additional conditions, admits a metric that is both Einstein and Kähler. They are beautiful objects which arise naturally in many facets of mathematics—and moreover, are of great importance in the study of string theory. We want to determine under what conditions a compact Fano manifold of Type I cohomogeneity one admits Kähler-Einstein metrics; for which is done by verifying the classes of the manifolds being Fano manifolds and their stability; however, by using the standard methods currently available to us, this proves to be quite a cumbersome task which yields very limited results. In order to overcome this obstacle, we have developed new specialized methods which are effective at retrieving large-scale information of classes of these compact Fano manifolds and their corresponding Kähler-Einstein properties.
 18 May 2018 The Eckmann-Hilton Argument and Some Applications Alex Pokorny There is a standard Munkres exercise assigned in 205A which asks to show that the fundamental group of a topological group is abelian. If you venture deeper into algebraic topology, you will stumble across a seemingly unrelated statement: that the higher homotopy groups of a topological space are all abelian. In this talk, I will prove the above statements and explore this idea of proving that a given operation is abelian using the Eckmann-Hilton argument. This argument is simple to prove, yet yields deep results. If time permits, I will generalize the definition of the center of a group and talk about $2$-categories.
 11 May 2018 Schur-Weyl duality and twisted commutative algebras Derek Lowenberg Schur-Weyl duality describes the link between the representation theories of the symmetric groups and the general linear groups. In this talk, I’ll tell you what it is and how it gives useful equivalences of certain symmetric monoidal categories. Following Sam and Snowden, one can define algebras (and their modules) as objects in such categories, which they call twisted commutative algebras. These in turn are used to study behaviors of families of symmetric and general linear group representations, and so the game continues.
 4 May 2018 Categorical Computation — Form and Content Christian Williams There is a duality of syntax and semantics – the form of a theory and the content of a model. This is a fundamental idea in category theory, which was introduced by William Lawvere in his 1963 PhD thesis. The notion of Lawvere theory provides an understanding of algebraic structures independent of presentation, improving upon the set-theoretic universal algebra. Soon after, these theories were proven equivalent to monads, the categorical manifestation of duality, through which the algebras of the monad correspond to models of the theory. Theories and monads provide complementary perspectives of algebraic structures, and both are becoming important to theoretical and practical computer science. We discuss the application to distributed computation, where enriched Lawvere theories can be used to create languages, programs, and data structures which have their operational semantics—the ways they can operate in context—integrated into their definition, effecting sound design of software.
 27 April 2018 Fractals and Finite Approximations with Respect to Noncommutative Metrics Therese-Marie Landry How can fractals be understood from the perspective of noncommutative geometry? Noncommutative geometry analyzes a space by studying the algebra of functions on that space. One of the fundamental tools of noncommutative geometry is Connes’ spectral triple. Via the efforts of Lapidus and his collaborators, there exist spectral triples for the Sierpinski gasket that recover the geodesic metric and encode some of its fractal qualities. Building on the work of Rieffel, Latrémolière introduced a generalization of the Gromov-Hausdorff distance to noncommutative, or quantum, compact metric spaces. Together with Aguilar, Latrémolière applied this new technique in noncommutative geometry—the Gromov-Hausdorff propinquity—to the space of continuous complex valued functions on the Cantor set. I am currently working on using the Gromov-Hausdorff propinquity to write the function space for the Sierpinski gasket as a limit of finite-dimensional $C^*$-algebras. In the process, I hope to understand which other fractals can be finitely approximated by noncommutative means.
 20 April 2017 What is condensed matter and why does it matter? Amir M-Aghaei The physics of a strongly interacting system—condensed matter—is usually drastically different than that of its building blocks; this is known as emergence. In this talk, I introduce the physics of condensed matter starting with a brief survey of how methods of statistical physics can explain some familiar but complicated phenomena around us. In particular, I will describe the physics of liquid-gas transition and discuss different aspects of an old question: why some materials conduct? Finally, I will mention the recent efforts of manipulating emergent physics to build quantum computers.
 13 April 2018 Open Petri Nets and the Reachability Problem Jade Master In computer science Petri nets are diagrams which are used to represent the transfer of resources in complex interacting systems of agents. These systems don’t usually exist in isolation and instead have inputs and outputs corresponding to external or environmental factors. To model this interconnectedness we define open Petri nets; Petri nets which can be glued together along specified inputs and outputs. We form a category of open Petri nets with open Petri nets as morphisms between their sets of inputs and outputs. Computer scientists are often interested in which states of a Petri net are reachable from a given initial state. We will put the category of open Petri nets to use by constructing reachability as a pseudo functor from the category of open Petri nets to the category of relations.
 6 April 2018 Model Theory and the Ax-Grothendieck Theorem Mike Pierce Model theory, from the perspective that I'll be talking about today, is the study of algebraic structures using ideas of pure logic. Or as logician Wilfrid Hodges said, model theory is algebraic geometry minus the fields. In this talk I'll start with a brief introduction to model theory, talk about completeness and compactness, and develop some facts about the theory of algebraically closed fields. Then if all goes well, this will culminate in a fantastic proof of the Ax-Grothendieck theorem, that every injective polynomial function $\boldsymbol{C}^n \to \boldsymbol{C}^n$ is surjective.

### Winter 2018

 16 March 2018 On the structure of complete open Kähler manifolds of positive curvature James Ogaja A central problem in complex geometry is to generalize the classical uniformization theorems on Riemann surfaces to higher dimension. In Kähler geometry, attention has been centered on how curvature affects the holomorphic structure of a Kähler manifold. In this talk I’ll discuss results related to Yau’s uniformization conjecture.
 9 March 2018 Some Combinatorial Representation Theory Justin Davis Combinatorics is an interesting topic on its own, but is also a very useful tool throughout mathematics. Due to the nature of the subject, combinatorics is extremely prevalent in representation theory, whether it’s classifying all finite dimensional irreducible representations, or decomposing representations into irreducible pieces. I will discuss a combinatorial rule, originally called the “Littlewood-Richardson Rule” for decomposing tensor products of two irreducible representations for the Lie algebra $\mathfrak{sl}_{n+1}$. This uses some interesting combinatorics of partitions of natural numbers. Lastly, I will discuss how I am using this rule to find the decomposition into irreducible representations of certain representations of $\mathfrak{sl}_{n+1}$ coming from a family of prime representations of quantum affine $\mathfrak{sl}_{n+1}$ recently defined by Brito and Chari.
 2 March 2018 Toric geometry Ethan Kowalenko A torus in normal everyday life is a product of circles, but in algebraic geometry a torus is a variety isomorphic to a product of $\mathbb{C}^\ast$’s. A toric variety $V$ is a variety with a dense open subset isomorphic to torus, such that multiplication in the torus extends to a group action on $V$. Recently, I’ve been looking at toric varieties with singularities, and blowing these singular points up (unrelated to Dylan’s talk) to get an overall smooth variety. The theory of toric varieties is actually very nice, with the ability to get almost any information you want about them via lattices and cones. In this talk, I’ll compute some examples of toric varieties, show how to glue affine pieces together, and maybe also compute how to resolve a singular point.
 23 February 2018 12:30 – 1:30pm Surge 268 Conference Travel Grants and You: Getting the Money You Need Jose Manuel Madrano, Conference Grant Coordinator We all want to travel and make the connections, both to further our studies and to land that job after graduating. Being a grad student is certainly does not make that easy, but the GSA has money to help. In this talk you can ask the GSA officer in charge of these funds any questions you might have about how to qualify for this money, and how to apply!
 16 February 2018 Towards quantifying fractality Xander Henderson While the term fractal is not well defined in mathematics, we generally understand the term to refer to a set that possess “roughness” or “complexity” at all scales. This complexity can be detected and quantified by studying zeta functions associated to the set. In this talk, we will introduce the distance zeta function associated to a bounded subset of a metric space, then discuss several examples of fractal and non-fractal sets.
 9 February 2018 Integrability, the singular manifold method and Darboux transformations: an algorithmic procedure to determine solutions. Paz Albares The Painlevé property has been proved to be a powerful test for identifying the integrability as well as a good basis for the determination of many properties of a given (nonlinear) PDE. The singular manifold method, based on the Painlevé analysis, provides the Lax pair and the Bäcklund transformation for the PDE. Furthermore, by employing the Darboux transformation approach, an iterative algorithmic method to obtain recursive solutions from a basic seed solution can be constructed. It will be illustrated by means of some examples, related to Nonlinear Schrödinger equations, in which solutions such as solitons, lumps and rogue waves will be thoroughly discussed.
 2 February 2018 Gauge Invariance and Charge Conservation Michael McNulty It is of no doubt to us mathematicians that mathematical abstraction is indispensable in our field of study. Yet when we consider mathematical applications to understanding the physical world, to what extent is it useful to separate from the seemingly concrete? In this talk, we will explore the concepts of gauge invariance and the conservation of electric charge through an abstracted lens; the former being a concept whose rich structure is familiar to those undergraduate students of physics who dug deeper than the typical classroom while the latter is an assertion familiar to most high school students. We will see how, given a simple mathematical framework, the phenomena of electromagnetism emerges nearly out of thin air and is completely self-contained within the initial framework. Our level of generality will lend itself toward viewing mathematical abstraction in applications to physics as not just useful but of utmost importance and as a crucial tool for the serious practitioner.
 26 January 2018 Blowing Things Up with Pinchuk and Frankel Dylan Noack In the complex plane there are a grand total of two simply connected domains: the plane itself and the ball (up to biholomorphism). This amazing result, known as the Riemann Mapping Theorem, has unfortunately proven not to be true in higher dimensions. Thus began the century-long journey to classify simply connected domains in higher dimensional complex space. A plethora of techniques have been developed in that time, and one such technique is the method of rescaling. There are two classic methods, Frankel rescaling and Pinchuk rescaling, each with its own strengths and weaknesses.
 19 January 2018 The Philosophical Science of Logic Christian Williams This week, I will try to explain the wild idea which led me to pursue mathematics. The introductory talk will serve to foster discussion, and hopefully some real interest. The topic cannot be summarized in an hour, let alone a paragraph. I will not defend a theory, but rather encourage a different way of thinking. Even in the best conditions the subject is extremely subtle and difficult, so I ask that you please come with an open mind. I will challenge basic assumptions, make provocative claims, and speak about something that is frankly still out of my cognitive league - so, a foundation of mutual respect is essential. If the hour can be free of pretense, prejudice, and preconception, we will be, as far as I know, the only people on earth thinking about this fascinating idea. This is a dream to which I am devoting my whole life, and I am excited to share it with you. Thank you for reading, and I hope to see you there.

### Scheduled Talks, Fall 2017

 8 December 2017 The Decay Lemma and Applications Matthew Overduin In the paper titled Decay Properties of Axially Symmetric D-Solutions to the Steady Navier-Stokes Equations, it is claimed that if $$\int\limits_{\boldsymbol{R}^3} r^{e_1} \left|f(r,z)\right|^2 \,\mathrm{d}x \leq C \quad\quad \int\limits_{\boldsymbol{R}^3} r^{e_2} \left|\nabla f(r,z)\right|^2 \,\mathrm{d}x \leq C \quad\quad \int\limits_{\boldsymbol{R}^3} r^{e_3} \left|\nabla \partial_z f(r,z)\right|^2 \,\mathrm{d}x \leq C$$ with nonnegative constants $e_1$ , $e_2$ , $e_3$, Then for any $r$ greater than zero we have, $$\int\limits_{-\infty}^{\infty} \left|f(r,z)\right|^2 \,\mathrm{d}z \leq Cr^{-\frac{1}{2}(e_1+e_2)-1} \quad \int\limits_{-\infty}^{\infty} \left|\partial_z f(r,z)\right|^2 \,\mathrm{d}z \leq Cr^{-\frac{1}{2}(e_2+e_3)-1} \quad \left|f(r,z)\right|^2 \leq Cr^{-\frac{1}{4}(e_1+2e_2+e_3)-1} \,.$$ While the paper outlines a proof for this lemma, the purpose of this talk is to fill in the gaps of this proof and to show how these estimates are obtained. We will also discuss how this lemma is relevant to solving the Axially Symmetric Navier-Stokes equation as a whole, and other types of equations.
 1 December 2017 Deformations and nonnegative curvature Lawrence Mouillé In Riemannian geometry, a natural question to ask is “what manifolds admit nonnegative or positive curvature?” This question has lead to interest in deformations of Riemannian metrics and collapse (convergence to a lower dimensional space) of Riemannian manifolds. I will discuss some general results in this area and describe a particular deformation due to Jeff Cheeger in the context of manifolds with isometric group actions.
 17 November 2017 Analysis on Manifolds via Li-Yau Gradient Estimates Xavier Ramos Olivé We are always told that the motivation for defining a smooth structure on a manifold is to be able to do calculus and analysis on manifolds. But how exactly is this done, and why? Will analysis give us information about our manifold? In this talk we will see how to define some natural differential equations on Riemannian manifolds, and how studying their solutions we can get topological information of the underlying manifold. We will do this via an example: by studying the so called Li-Yau gradient estimates of the heat kernel, with a particular focus to their relationship to the Ricci curvature. These estimates can be used to derive some bounds on the Betti numbers of the manifold. If time permits, we will explore some different strategies to derive the gradient estimate under different curvature assumptions, although to protect our sanity, we will skip all the messy computations. No previous knowledge about the concept of curvature will be required for the talk.
 3 November 2017 Covariance Computations for the Active Subspace Method Applied to a Wind Model Jolene Britton The method of active subspaces is an effective means for reducing the dimensions of a multivariate function $f$. This method enables experiments and simulations that would otherwise be too computationally expensive due to the high-dimensionality of $f$. By using a covariance matrix composed of the gradients of $f,$ one can find the directions in which $f$ varies most strongly, i.e. the active subspace. The current standard for estimating these covariance matrices is the Monte Carlo estimator. Due to the slow convergence of Monte Carlo methods, we propose alternative algorithmic approaches. The first utilizes a separated representation of $f,$ while the second uses polynomial chaos expansions. Such representations have well-defined sampling strategies and allow for the analytic computation of entries of the covariance matrix. Experimental results demonstrate how the Monte Carlo methods compare to our proposed alternative approaches as applied to a function representing power output of a wind turbine.
 27 October 2017 Better Faster Strongly Jacobson Modules Tim McEldowney Inventing new math is hard. However, there is a nice work around. Take old math and add an adjective. In this talk, I will build up to my most recent result which looks suspiciously like another theorem. I will start by talking about the base structures I study called ‘$G$-domains’ which are integral domains which are close to being their field of fractions. Next, I will define ‘$G$-ideals’ and ‘Hilbert rings’ which are made from these $G$-domains with some clear examples of these structures from common rings. Afterwards, I talk about the ‘strongly’ adjective and what that does to these objects. Lastly, I close with a game I like to call pin the adjective on your adviser’s theorem.
 20 October 2017 Network Models Joe Moeller A network is a complex of interacting systems which can often be represented as a graph equipped with extra structure. Networks can be combined in many ways, including by overlaying one on top of the other or sitting one next to another. We introduce network models — which are formally a simple kind of lax symmetric monoidal functor — to encode these ways of combining networks. By applying a general construction to network models, we obtain operads for the design of complex networked systems.
 13 October 2017 Can a nice variety of variety exist? Ethan Kowalenko Algebraic geometry is notorious for being difficult, as it is broadly the study of the zero sets of polynomials through some assigned rings. I will attempt to describe a very computable class of such zero sets, called toric varieties, by showing literal computations. Like, explicitly.
 6 October 2017 An Introduction to Hopf Algebras Dane Lawhorne What happens when you take the commutative diagrams that define an algebra and reverse all the arrows? The result is called a coalgebra, and with a few more axioms, you get a Hopf Algebra. In this talk, we will examine the role of Hopf algebras in representation theory. In particular, we will see that the category of left modules over a Hopf algebra has both tensor products and dual modules.

### Talks from Previous Years

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