Department of Mathematics

Organizers:

 Christina Osborne osborne at math.ucr.edu Kayla Murray murray at math.ucr.edu Andrea Arauza Matthew Lee arauza at math.ucr.edu mlee at math.ucr.edu

Spring 2016: Fridays 1:10-2:00pm in Surge 284

June 3, 2016

 Speaker: Matthew Barber Title: Presheaves, $\Delta$, and other tales Abstract: Presheaf categories are very common in Algebraic Topology and Homotopy theory.  In this talk we will discuss $\Delta$, its properties, and some generalizations.

May 27, 2016

 Speaker: Xavier Ramos Olive Title: Cheeger-Colding- Naber Theory Abstract: Cheeger-Colding- Naber Theory (CCN) provides us with tools to study limit spaces of Riemannian Manifolds, and tries to answer the question: how degenerate can the limit space be? In this talk, rather than studying CCN Theory itself, we will present the tools needed to understand the results that follow from this theory. For instance, given a sequence of Riemannian Manifolds, in which sense do they converge to anything? How do we know if a point in the limit space is in a regular or in a singular region? How can we measure the amount of smoothness of the limit space? After answering some of these questions, we will see how recent results on the Codimension 4 Conjecture answer the question above under certain curvature assumptions. If time permits, we will discuss some possiblegeneralizations to other curvature assumptions.

May 20, 2016

 Speaker: James Ogaja Title: Non-compact complete positively curved manifolds and Busemann function Abstract: A study of function theory (especially of geometrically interesting functions) on noncompact complete manifolds with positive curvature is essential for understanding their geometry and topology. For example, existence of a C^∞ strictly convex exhaustion function on a non-compact complete Riemannian manifold of positive sectional curvature implies that it is diffeomorphic to Euclidean space. In this talk I’ll present a simple deduction that a modified version of Busemann function on an open complete Riemannian manifold with positive sectional curvature is exhaustion. I will also talk about other properties of this function on the above named manifold.

May 13, 2016

 Speaker: Mikahl Banwarth-Kuhn Title: The dynamics of love affairs Abstract: In high-dimensional phase spaces, trajectories have a lot of room to maneuver, and a wide range of dynamical behavior ensues.  This talk will cover the simplest class of higher-dimensional systems, linear systems in two dimensions.  These systems are interesting in their own right, but they also play an important role in the classification of fixed points of nonlinear systems.  To arouse your interest in the classification of linear systems, we will focus our attention to a simple model for the dynamics of love affairs (pun provided by Steven Strogatz).

May 6, 2016

 Speaker: Edward Voskanian Title: Quasicrystals and the Complex Dimensions of a Nonlattice Fractal String. Abstract: The discovery of crystals with 'forbidden' symmetry (quasicrystals) posed fascinating and challenging problems in many fields of mathematics. In this talk, we give the basic definitions and results required to understand the following open problem: Is there a natural way in which the quasiperiodic pattern of the set of complex  dimensions of a nonlattice self-similar string can be understood in terms of a suitable (generalized) quasicrystal?

April 29, 2016

 Speaker: Kyle Castro Title: The (Eventually) All Knowing Oracle Abstract: I will be giving an introductory talk on the problem of recovering a hidden monic polynomial using an Oracle function.  You should be wondering what else an Oracle function can do; can it tell me about my future? Can it read my mind? What is at the bottom of the ocean? How will Game of Thrones end? The questions are endless.  This talk is sure to include at least one joke regarding “The Matrix” and potentially an appearance from the character, “Neo;” at the very least, Keanu Reeves.

April 22, 2016

 Speaker: Frank Kloster Title: Introduction to K-Theory for C*-Algebras Abstract: In this talk, I will introduce the concept of K-Theory. Specifically, I will look at K-Theory for C*-Algebras. The study of K-Theory consists of the study of two functors,K_0  and K_1, which are functors from the category of C*-algebras to the category of abelian groups. I will be introducing the K_0 functor

April 15, 2016

 Speaker: Brandon Coya Title: The Category of Circuits and the Blackbox Functor. Abstract: In this talk we will first construct a category which models circuits made of ideal wire. We then show that the process of considering only the relationship between inputs and outputs of a circuit, called "blackboxing," is functorial. Finally we extend this category and functor to circuits that include passive linear components, batteries, and current sources.

April 8, 2016

 Speaker: Christina Osborne Title:  Introduction to Calculus on Functors:  Goodwillie and Beyond Abstract: Thomas Goodwillie brought ideas from calculus into specific categories.  In this talk, we will discuss what it means for a functor to be of "degree n".  Analogous to the Taylor series, Goodwillie constructed a "Taylor tower" of functors.  Georg Biedermann and Oliver Röndigs were able to extend Goodwillie's work into model categories.  The possibility of extending this work further into complete Segal spaces will also be discussed.

April 1, 2016

 Speaker: Josh Buli Title: Numerical Methods for Stochastic Ordinary Differential Equations (SODEs) Abstract: Modeling of physical phenomenon involves some sort of uncertainty, whether in measurement of quantities of interest, or the presence of random forcing. In application, differential equations can include stochastic terms, either in initial/boundary conditions, coefficients, or forcing terms to attempt to capture these random effects. Explicitly solvable SODEs are rare in applications and thus require numerical methods in order to determine approximate solutions to such problems. This talk will focus on SODEs in particular, and will include an introduction to Brownian motion and stochastic calculus, and a variety of numerical methods that can be used to solve SODEs.

Winter 2016: Fridays 1:10-2:00pm in Surge 284

March 4, 2016

 Speaker: Donna Blanton Title: The Current Algebra of sl_2 and its Representations Abstract: The Lie algebra sl_2 is the algebra that is made up of 2x2 matrices with trace 0. We completely understand the representation theory of this algebra because we can decompose any module for sl2 into a sum of irreducible modules.  We get the current algebra by tensoring sl_2 with polynomials over the complex numbers. This algebra is more complicated and we can’t understand its modules in the same way.  So we will discuss this algebra and some things that we do know about its modules and other things that we would like to know.

February 26, 2016

 Speaker: Edward Voskanian Title: Euclidean crystallographic groups and the crystallographic restriction. Abstract: Quasicrystals were first observed in 1982. These are crystals with a diffraction pattern exhibiting symmetries forbidden by the crystallographic restriction. The purpose of this talk is to define the 2-dimensional Euclidean crystallographic group and then prove the crystallographic restriction.

February 19, 2016

 Speaker: Scott Roby Title: An Introduction to Hypergeometric Functions Abstract: The hypergeometric functions have been studied by many well-known mathematicians including Gauss, Euler, and Riemann. This talk will consist of a brief historical overview of the study of these functions, some fundamental properties including analytic continuation, and a list of topics and functions studied by means of the hypergeometric functions. This talk will be accessible to anyone familiar with the residue theorem for complex integration covered in 210.

February 12, 2016

 Speaker: Mikahl Banwarth-Kuhn Title: Random walk models in biology Abstract: Mathematical modeling of the movement of animals, micro-organisms, and cells is of great relevance in the fields of biology, ecology, and medicine. These models can take many different forms, and we will look at models based on the extensions of simple random walk processes. The aim is to introduce the mathematics behind random walks in a straightforward manner and explain how such models can be used to aid our understanding of biological processes. This talk is intended to be accessible to all graduate students.

February 5, 2016

 Speaker: Josh Strong Title: Bounded domains of finite type Abstract: A conjecture of Greene and Krantz states that if a smoothly bounded domain of several complex variables admits an automorphism orbit accumulation point in its boundary, then that point is of finite type. Though the conjecture is as yet unsolved, will discuss several results in support.

January 29, 2016

 Speaker: Jesse Cohen Title: A Brief Survey of Ergodic Theory Abstract: Roughly speaking, ergodic theory is the study of the behavior of measure preserving dynamical systems which exhibit a sort of maximally homogenizing behavior. In this talk, we will review the basic notions of mixing and ergodic measure preserving transformations, providing examples along the way, and state several important results of ergodic theory. Time permitting, we will also discuss applications of ergodic theory to measure theory and probability.

January 22, 2016

 Speaker: Andrea Arauza Title: Harmonic Functions on the Sierpinski Gasket Abstract: The theory of analysis on fractals stemmed from the desire to study differential equations and Laplacians on fractals. We will look at the construction of "harmonic" functions on the Sierpinski gasket and see results analogous to those known for the familiar harmonic functions of complex analysis. The majority of this talk will be accessible to all graduate students and in fact to a great many undergraduates.

January 15, 2016

 Speaker: Xavier Ramos Olive Title:  The Geometry of Relativity Abstract: This talk is an introduction to the Theory of General Relativity. After an overview of Special Relativity, we will present the Equivalence Principle and how it leads to a geometric theory of gravity, generalizing elegantly Newtonian mechanics. To do so, we will introduce the notion of a semi-Riemannian manifold and its curvature, and we will present Einstein's equation. If time permits, we will study some solutions to this equation and its consequences.No previous knowledge on advanced physics or geometry is needed; the talk is intended to be accessible to all graduate students.

January 8, 2016

 Speaker: None Title: Cancelled due to JMM Abstract:

Fall 2015: Fridays 1:10-2:00pm in Surge 284

November 13, 2015

 Speaker: Thomas Schellhous Title: Vortex Patches Abstract: Vortex patches are solutions to the 2D Euler Equtations for fluid motion whose vorticities (curl of the fluid velocity) are constant inside of a bounded region and zero everywhere else. The manner in which the region's boundary can deform over time has been studied since the late 1970s, whith current research ongoing. This talk will introduce the relevant ideas and discuss the interesting history of the vortex patch problem along with some of the insights that have driven past progress. It will conclude with a description of current work being done in this area. This talk should be accessible to all graduate students.

October 30, 2015

 Speaker: Brandon Coya Title: Categories, Cospans, Correlations, and Circuits Abstract: this talk I will discuss the relationship between categories and circuits by using the language of string diagrams. The dual notion to a relation between sets called a "corelation" will provide the properties of parallel and series junctions in circuit diagrams made of perfectly conductive wires.

October 23, 2015

 Speaker: Jesse Cohen Title: Introduction to Holonomy Abstract: Given a Riemannian manifold $(M,g)$, one can use the metric $g$ to define a way of translating tangent vectors along curves so that some geometric relationship between the curve and the vector is preserved; this is the notion of parallel transport. We will give a sketch of covariant derivatives and parallel transport and use the latter to define the holonomy group of $(M,g)$ and discuss its applications in mathematics and physics.

October 16, 2015

 Speaker: Priyanka Rajan Title: Introduction to Alexandrov Geometry Abstract: In this talk I will iintroduce the basics of Alexandrov geometry through examples, how it generalizes the notion of Riemannian manifolds and will give a short proof of "why submetries preserve lower curvature bound" using Alexandrov geometry tools.

October 9, 2015

 Speaker: Frank Kloster Title: C*-algebras Abstract: Operator and $C*$ algebras are one of the main branches of analysis, with several applications, including in Fourier analysis, noncommutative geometry, and quantum mechanics. In this talk, I will be introducing operator and $C*$ algebras. The talk will only assume knowledge out of the 209 sequence and some linear algebra.

October 2, 2015

 Speaker: Andrew Walker Title: Local Cohomology Abstract: Given an ideal in a commutative ring with identity, we have a torsion functor associated to this ideal. Its right-derived functors are called the local cohomology functors. We will discuss how to construct local cohomology modules and where they appear.