# Graduate Student Seminar

## Fridays 1:10–2:00 pm in Surge 284

### Organizers

 Xavi Ramos-Olivé Kyle Castro olive@math.ucr.edu kcastro@math.ucr.edu Christina Knox Xander Henderson sargent@math.ucr.edu henderson@math.ucr.edu

### Spring 2017

9 June 2017
Annual Organizational Meeting
2 June 2017
Speaker: Xander Henderson
Title: The Complex Dimensions of Self-Similar Subsets of p-adic Product Spaces
Abstract: The higher dimensional theory of complex dimensions developed by Lapidus, Radunović, and Žubrinić provides a language for quantifying the oscillatory behaviour of the geometry of subsets of Rn. In this talk, we will describe how the theory can be extended to metric measure spaces that meet certain homogeneity conditions. We will provide examples from p-adic spaces and discuss the geometric information that can be recovered from the complex dimensions in these cases. This is an expanded version of the talk given at Math Connections 2017.
26 May 2017
Speaker: Charley Conley
Title: Curvature of Hypersurfaces in the Euclidean Ambient
Abstract: This talk will introduce curvature of surfaces in a friendly way. It will then describe how to find curvatures of hypersurfaces in n-dimensional Euclidean space without having to buy n-dimensional graph paper. Finally, it will discuss a relatively (2013) new curvature invariant from the joint work of Charley T. R. Conley, Rebecca Etnyre, Brady Gardener, Lucy H. Odom, and Dr. Bogdan Suceava, with some interpretation.
19 May 2017
Speaker: Daniel Cicala
Title: Universal algebra and functorial semantics
Abstract: In his landmark thesis, William Lawvere introduced "functorial sematics" to the study of universal algebra. This method turns certain algebraic definitions into an actual mathematic objects. This talk gives an introduction to functorial semantics and shows that everything you know is just a functor.
12 May 2017
Speaker: Christina Knox
Title: Recovery of Both Sound and Source in Photo-Acoustic Tomography
Abstract: Photo-acoustic tomography is an imaging method that attempts to combine the high resolution of ultrasound and the high contrast capabilities of electromagnetic waves. In this talk we will first introduce the mathematical problems photo-acoustic tomography presents. Uniqueness results will briefly be discussed for the situation when sound speed is known and the source term is to be recovered. Then the case when both sound speed and source term are unknown will be considered. Partial uniqueness results in this case proved by Liu and Uhlmann will be presented along with an outline of the proof which relies on the temporal Fourier transform.
5 May 2017
Speaker: Joe Moeller
Title: The Grothendieck Construction
Abstract: Alexander Grothendieck is inarguably one of the greatest mathematicians of the 20th century. His influence in algebra, geometry, and category theory is unavoidable. In this talk, we will see an intuitive development of the Grothendieck construction, one of many things named after him. We will also see a simple application in algebra.
28 April 2017
Speaker: Edward Voskanian
Title: Mathematical Quasicrystals and the Complex Roots of a Nonlattice Dirichlet Polynomial
Abstract: The discovery of quasicrystals in 1982 by Dan Shechtman, which are nonperiodic solids with a diffraction pattern consisting of bright spots, brought forth a surge of new mathematics with which to model the new geometry involved. Beyond the physical questions raised by the discovery of quasicrystals, there is at least one mathematical question: How can one construct a 'well ordered' pattern that is 'aperiodic'. An example of such a mathematical structure is the Penrose tiling, which was developed by Sir Rodger Penrose even before the discovery of quasicrystals. The complex roots of a nonlattice Dirichlet polynomial have an interesting quasiperiodic structure too. However, to actually see this pattern, one must approximate the roots of a polynomial with very large degree. In this talk, we will see where this quasiperiodic pattern comes from and discuss the numerical attempts to see more of it. We ask if there is a natural way in which the quasiperiodic pattern of the complex roots can be understood in terms of a suitable generalized quasicrystal.
21 April 2017
Speaker: Brandon Coya
Title: Graphs and Circuits
Abstract: Circuits have been widely studied in electrical engineering and physics. In this talk we will view circuits as special types of graphs. First we will look at which graph property determines the behavior of a circuit. Additionally we look at circuits in a compositional way," meaning that we can stick circuits together to form larger ones. This comes from allowing our graphs to have distinguished nodes be inputs or outputs. We can then understand any circuit by breaking it up into smaller pieces and first understanding the behavior of the smaller pieces.
14 April 2017
Speaker: Jonathan Wolfram Siegel (UCLA)
Title: Information Theory and Communication
Abstract: Data compression and reliable communication in the presence of noise make modern information technology possible. The goal of this talk is to introduce Shannon's source coding theorem and noisy channel coding theorem. These remarkable theorems put precise limits on lossless compression and communication over a noisy channel. In the process, I will introduce fundamental concepts in information theory, for instance the notion of entropy, which find applications in statistics and physics, as well as in the study of PDEs.
7 April 2017
Speaker: Nicholas Newsome (CSU Fresno)
Title: An investigation of power sums of integers
Abstract: Sums of powers of integers have been studied extensively for many centuries. The Pythagoreans, Archimedes, Fermat, Pascal, Bernoulli, Faulhaber, and other mathematicians have discovered formulas for sums of powers of the first n natural numbers. Among these is Faulhaber's well-known formula which expresses the power sums as polynomials whose coefficients involve Bernoulli numbers.

In this talk, we sketch an elementary proof that for each natural number p, the sum of pth powers of the first n natural numbers can be expressed as a polynomial in n of degree p + 1. We also present a novel identity involving Bernoulli numbers and use it to show symmetry of these polynomials. In addition, we make a few conjectures regarding the roots of these polynomials, and speculate on the asymptotic behavior of their graphs. Finally, we study the remainders of the power sums upon division by integers. In particular, we generalize a well-known result on congruence of Sp(n) from prime n to any power of prime and state periodicity properties of Sp(n) mod k.

### Winter 2017

17 March 2017
Speaker: John Simyani
Title: Pursuing the Poisson Cohomology of k-step Nilmanifolds
Abstract: Geometry comes in many flavors, and Nigel Hitchin and his students have, in a sense, added another: Generalized Geometry. In this talk, I will discuss the generalized geometry and Poisson cohomology of particular nilmanifolds, focusing on stabilization of their spectral sequences. Although much of the material will be from my completed oral exam, I will also try to present some common computational tools that are often needed when working with forms and vectors.
10 March 2017
Speaker: Kyle Castro
Title: Multiplicative character sums and their applications to problems in analytic number theory
Abstract: The main goal of this talk will be to present new and existing bounds on multiplicative character sums as well as to provide the reader with an understanding of the various applications. In particular, we will discuss how upper bounds on these character sums provide lower bounds on the size of a subgroup of a finite field containing the image of an interval of consecutive integers under a polynomial function.
3 March 2017
Speaker: Mike Pierce
Title: Functional programming in Mathematica: a crash course
Abstract: Do you ever program in Mathematica? Well, you might be doing it wrong. In this talk we'll start with the basics of low-level Mathematica programming, like expressions, the Head of an expression, and Set versus SetDelay. Then we'll cover List manipulation, how to program in a functional style using things like Map and MapThread. Then we'll cover some basics of pattern matching.
24 February 2017
Speaker: Ethan Kowalenko
Title: Root systems by example
Abstract: Math is hard without examples, so I'm gonna pilot a ship from the abstract sky to a more concrete jungle. Historically, Weyl groups arise in Lie Theory, and have been studied extensively in conjunction with so-called "root systems," which can seem quite abstract if you never look at one. In this talk, we'll consider Weyl groups as a special case of finite reflection groups, and examine some properties of root systems through small examples.
17 February 2017
Speaker: Edward Voskanian
Title: An introduction to mathematical quasicrystals
Abstract: The discovery of physical quasicrystals in 1982 led to a surge of new mathematics with which to model the new geometry involved. This talk will be a partial survey of a two part paper titled "Geometric Models for Quasicrystals" by Jeffery C. Lagarias, and our focus will be the first part titled "Delone Sets of Finite Type". A Delone set is a subset of n dimensional space whose points are, in some sense, evenly spread out. Because the property of being a Delone set of finite type is determined by "local rules", these sets form a natural class for modeling the long-range order of the atomic structure of physical quasicrystals.
10 February 2017
Speaker: Jesse Cohen
Title: Symbols and ellipticity
Abstract: Explicitly solving partial differential equations is difficult in general but, with a small shift in perspective, it is possible to make strong conclusions about existence and regularity of solutions by considering the equations themselves. We will discuss this shift via symbols of linear differential operators of a very general class, with examples and computations, the property of ellipticity, and provide a hint toward the theory of pseudodifferential operators.
3 February 2017
Speaker: Andrew Walker
Title: Finite free resolutions
Abstract: Linear algebra over a field is great. But, if you had to settle, linear algebra over a ring is not so bad. There are some complications though. For example, not every module is free anymore. In other words, we might have some nontrivial relations in a set of generators for our module. It gets worse: we could have relations among a set of generators for these relations... and so on.
27 January 2017
Speaker: Lawrence Mouillé
Title: Morse theory for distance functions
Abstract: Morse theory gives topological information about a manifold by studying the critical points of smooth real-valued functions defined on it. On a Riemannian manifold, one might try to apply Morse theory to distance functions because they are intrinsic to the manifold, as opposed to most "Morse functions" which are extrinsic (e.g. the height of a surface when embedded into an ambient space). Such functions, however, are almost never smooth, and thus don't have critical points in the usual sense. Luckily, Karsten Grove and Katsuhiro Shiohama developed a definition of critical points for distance functions using metric notions. They used this to prove the celebrated diameter sphere theorem, while establishing a useful tool along the way: the isotopy lemma. Because the isotopy lemma is a direct counterpart to a crucial result in classical Morse theory, it is natural to ask whether a full-fledged "Morse theory" can be developed for distance functions and this new critical point theory. I will present work of Barbara Herzog and Fred Wilhelm concerned with addressing this issue, comparing the results with those in classical Morse theory, and discuss future work for improving the scope of their results and finding new applications.
20 January 2017
Speaker: Dylan Noack
Title: The historical development of modern complex analysis
Abstract: Modern Complex Analysis was arguably founded by Augustine Cauchy in the early nineteenth century. In this talk we watch the history of complex analysis unfold, starting with the fundamentals of holomorphic functions discovered by Cauchy to the more recent notions of pseudoconvex domains introduced by Levi. Technical details will be kept to a minimum.
13 January 2017
Speaker: Christina Osborne (University of Virginia)
Title: The first step towards higher order chain rules for abelian calculus
Abstract: One of the most fundamental tools in calculus is the chain rule for functions. Huang, Marcantognini, and Young developed the notion of taking higher order directional derivatives, which has a corresponding higher order iterated directional derivative chain rule. When Johnson and McCarthy established abelian functor calculus, they constructed the chain rule for functors which is analogous to the directional derivative when n=1. In joint work with Bauer, Johnson, Riehl, and Tebbe, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the HMY chain rule. Our initial investigation of this result involved a concrete computation of the case when n=2, which will be presented in this talk.

### Fall 2016

18 November 2016
Speaker: Daniel Cicala
Title: Spans of cospans
Abstract: We introduce the notion of a span of cospans and define, for them, horizonal and vertical composition. When in a topos C, these compositions satisfy the interchange law. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting. Technical details will be kept to a minimum.
4 November 2016
Speaker: Joshua Buli
Title: The discontinuous Galerkin (DG) method
Abstract: This talk will be an introduction to the Discontinuous Galerkin (DG) method applied to conservation laws. The DG method is us a class of finite element methods that use discontinuous basis functions. The method will be introduced on the simple Burgers' equation, and then the DG method will be used to numerically solve the single BBM and coupled BBM system, which are used to model water waves moving through a channel. We then provide numerical tests to demonstrate the usefulness of the DG method.
28 October 2016
Speaker: Priyanka Rajan
Title: Cohomogeneity one manifolds
Abstract: Let G be a compact Lie group acting effectively on a compact Riemannian manifold M. We say that group action is by cohomogeneity k if the orbit space M/G has dimension k. And the manifold M is then called a cohomogeneity k manifold. In this talk, we will discuss some general results regarding the curvature properties of cohomogeneity 1 manifolds.
21 October 2016
Speaker: Xavier Ramos Olivé
Title: An introduction to geometric mechanics
Abstract: During the 19th century, J.L. Lagrange and W.R. Hamilton reformulated classical mechanics by introducing equations that were independent of the chosen coordinate system. Moreover, their formulation allowed the study of constrained systems: we can study the motion of a particle on the surface of a sphere without understanding the force that keeps the particle attached to it. This lead to the development, during the 20th century, of Geometric Mechanics, that studies Lagrangian and Hamiltonian mechanics using differential geometry. This talk will be an introduction to Geometric Mechanics, with the goal to motivate the study of analysis in manifolds, as well as geometric objects like symplectic manifolds, Poisson manifolds, or Lie groups and Lie algebras.
14 October 2016
Speaker: Sean Watson
Title: An introduction to the worm-ridden Laakso spaces
Abstract: Laakso Spaces were first introduced by Tomi Laakso in 2000 as examples of Q-Regular spaces admitting a (1,1)-weak Poincare inequality, for any Q>1. In other words, they are examples of geometrically strange spaces that are strong enough that most analysis can still be done. The Laakso spaces were the first general examples of such spaces, with the added bonus that they are relatively simple spaces to work within. Geometrically we can picture the Laakso spaces as Cantor sets crossed with the unit interval, along with a countably dense collection of wormholes throughout the space connecting it all together. This talk will focus on constructing a simple Laakso space and, if time permits, a related Laakso graph while keeping technical details to a minimum.
7 October 2016
Speaker: Lawrence Mouillé
Title: What is comparison geometry, and why does anyone care?
Abstract: From Euclid to Gauss to Riemann to Nash to current mathematicians, so-called Riemannian geometry has had an interesting and complicated development. In this talk I will outline this story, describe the goals of comparison geometry and global Riemannian geometry, and present important results in this area. Technical details will be kept to a minimum, and all who are interested are encouraged to attend.
30 September 2016
Speaker: Jesse Cohen
Title: Operator algebras and topological K-Theory
Abstract: Vector bundles—assignments, in an appropriate sense of a vector space to every point of a topological space—are beautiful objects that arise naturally in many contexts in mathematics and physics. In particular, in topological K-theory, these structures form the basic building blocks of homotopy invariants of pointed spaces called K groups which, roughly speaking, tell us something about how twisted a vector bundle over a given space can be. In this talk, we will examine the construction of K0 and K1 via operator algebras and discuss the K-theory exact sequence.
23 September 2016
Speaker: Xander Henderson
Title: A brief introduction to non-Archimedean Fields
Abstract: Non-archimedean fields are an often overlooked collection of mathematical objects that are nevertheless beautiful and compelling. In this talk, we will discuss an analytic and an algebraic approach to constructing the p-adic numbers—the standard examples of non-archimedean fields—and explore some of their basic algebraic and topological properties.