Lisa Schneider
schneiderl
at math.ucr.edu | Peri Shereen
shereen
at math.ucr.edu | Jeffrey Wand
wand
at math.ucr.edu |

Speaker: | Matt Lee | |

Title: | Which Finitely Generated Abelian Groups Admit Isomorphic Cayley Graphs? | |

Abstract: | After introducing some graph theory terms, we will work through this relatively short and accessible paper by Clara Loh. A simple example will reveal why the result cannot be strengthened. |

Speaker: | Edward Burkard (University of Notre Dame) | |

Title: | A Primer on Pseudoholomorphic Curves | |

Abstract: | I will begin with an overview of almost complex manifolds and complex maniforlds. Next, I will define pseudoholomorphic curves (no one actually calls them that anymore...we usually just call them holomorphic curves these days, or at least J-holomorphic) and I will try to convince you that these are just the usual notion of a holomorphic function in a special case. After that, I will give some basic definitions in symplectic geometry and relations with almost complex structures. Finally, I will state some results in symplectic geometry which are applications of holomorphic curves, one of which can be thought of as a generalization of the Riemann Mapping Theorem. |

Speaker: | John Dusel | |

Title: | Balance Quotients or Normal Forms in the D-Weyl Group | |

Abstract: | After reviewing the word problem for finitely generated groups, we exposit a solution to this problem for a special subset of the D-Weyl group. The normal forms afforded by this solution are enumerated by the vertices in an infinite sequence of forest graphs. The only background required for this talk is a working knowledge of group presentations. |

Speaker: | Donna Blanton | |

Title: | Root Systems | |

Abstract: | A root system is a subset of Euclidean space satisfying certain axioms. Their applications range from classifying simple Lie algebras to the classification of certain types of graphs. In this talk, we will look at examples of root systems, and study the relationship between a simple Lie algebra and its corresponding root system. |

Speaker: | Sean Watson | |

Title: | Fractal Geometry and Complex Dimensions in Metric Measure Spaces | |

Abstract: | While classical analysis dealt primarily with smooth spaces, much research has been done in the last half century on expanding the theory to the nonsmooth case. Metric Measure (MM) spaces are the natural setting for such analysis, and it is thus important to understand the geometry of subsets of such spaces. This talk will be an introductory survey, first of MM spaces that arise naturally in varying fields, and second an overview of the current theory of complex dimensions in both the one dimensional case and the more recent higher dimensional theory. This recent theory should naturally generalize to MM spaces, and we will show preliminary results in that direction. |

Speaker: | Mathew Lunde | |

Title: | Ext^1 and Prime Factorization of Representations of Quantum Loop Algebras | |

Abstract: | The category of finite dimensional representations of a quantum loop algebra U_q(Lg) is not semi-simple. Moreover, the tensor product of irreducible representations remains irreducible generically. This leads naturally to the definition of prime objects: the factorization of irreducible objects into irreducible primes. We show that there is an interesting connection between the notion of primes and the homological properties of the category, namely for g=sl_2, an irreducible representation V is a tensor product of r prime representations if and only if the dimension of the space of self extensions of V is r. |

Speaker: | Joshua Strong | |

Title: | A Geometric Perspective of Complex Analysis | |

Abstract: | Differential geometric techniques, such as Ahlfors's generalization of the Schwarz lemma, can be used to provide new proofs of classical complex function theory. In this talk, we will discuss a differential geometric side of complex analysis and use it to prove the Ahlfors-Schwarz lemma as well as Picard's little theorem. |

Speaker: | Parker Williams | |

Title: | First and Second Moment Methods | |

Abstract: | In this talk, I will review a fundamental concept in probabilistic combinatorics where the first and later second moments may be used to prove the existence of desired configurations. This can range from sets with given properties to graphs to any other structure you might like to enumerate. I will be presenting this from a no prior knowledge standpoint and the goal will be to demonstrate these methods in a context that you might see how to use them in your own work. For those keeping count, lets see how many different ways I spell Chebyshev. |

Speaker: | Franciscus Rebro | |

Title: | Why Categories? | |

Abstract: | Category theory has been described as the linguistics of modern mathematics. This talk will give some brief historical background and motivation for categories, show some examples of common constructions in categories, and end with an introduction to my ongoing project with John Baez and Tu Pham, on showing that electric circuit diagrams form a bicategory. No Previous knowledge of category theory will be assumed from the audience. |

Speaker: | Jason Erbele | |

Title: | Categories in Control | |

Abstract: | Control
theory is a branch of engineering that studies dynamical systems with
inputs and outputs, and how to optimize their behavior. Diagrams called
signal-flow graphs are frequently used to describe these systems. These
signal-flow graphs exhibit features of a symmetric monoidal category,
bearing a striking resemblance to the depiction of morphisms as string
diagrams. This notion is made more precise by considering the symmetric
monoidal categories FinRel_{k} and its subcategory, FinVect_{k}.
These categories have morphisms that correspond to signal-flow graphs,
and are equivalent to symmetric monoidal categories generated by the
object k, a generating set of morphisms, and a set of relations between
morphisms. |

Speaker: | Jason Park | |

Title: | Brownian Motion | |

Abstract: | Brownian motion is a simple continuous stochastic process that widely used in physics and finance for modeling random behavior that evolves over the time. In this talk, I will introduce the definition of Brownian motion and the construction of the Wiener space and Wiener measure. |

Speaker: | Jacob West | |

Title: | Auslander-Reiten Theory in stable (∞,1)-categories | |

Abstract: | Auslander-Reiten theory was introduced by M. Auslander and I. Reiten in the early 1970's as a tool for understanding representations of Artin algebras (and in particular, finite dimensional algebras). Of central interest are the so-called Auslander-Reiten sequences, which are (roughly speaking) minimal non-split short exact sequences. In this talk, we introduce an analogue of Auslander-Reiten theory in stable (∞,1)-categories. |

Speaker: | Tim Cobler | |

Title: | Quantizing the Riemann Zeta Function |

Speaker: | Oliver Thistlethwaite | |

Title: | Alexander Polynomials of 3-manifolds | |

Abstract: | The Alexander polynomial originally arose as a knot invariant but was later extended by Turaev to an invariant for 3-manifolds. We will define this invariant as well as discuss some of the results obtained by using it. |

Speaker: | Thomas Schellhous | |

Title: | Weak Derivatives of Distributions and Applications to PDEs | |

Abstract: | We will begin by defining distributions, also known as generalized functions, and the notion of a weak derivative of a distribution. We will then look at some illustrative examples to investigate their behavior and then discuss how we can use these ideas to weaken differentiability requirements on solutions to PDEs, allowing for a much larger (and more practical) class of weak solutions. We will conclude with some applications to the Navier-Stokes and Euler equations for fluid motion. It should be accessible to all graduate students (and most 10B students as well). |

Speaker: | Adam Navas | |

Title: | Singular Integral Operators and Some Applications | |

Abstract: | In this talk, I will review facts about singular integral operators and then apply them to different partial differential equations. |

Speaker: | Mathew Lunde | |

Title: | Extensions of Abelian Groups | |

Abstract: | We will define and construct the Yoneda Ext group for abelian groups. We will see the explicit construction, examples, and some basic properties. Time permitting, we will also generalize the concept of Ext groups to more general module categories. |

Speaker: | Soheil Safii | |

Title: | Equivariant and Isovariant Function Spaces | |

Abstract: | When working with topological spaces, we have equivalence given by homeomorphisms but also a weaker notion of equivalence given by homotopy. When considering G-Spaces (topological spaces with an added group action), in addition to equivariant homotopy equivalences, which are analogous to homotopy equivalences in topological spaces, we also have a slightly stronger condition of isovariant homotopy equivalence. In this talk, we hope to discuss the two different notions and in particular, the conditions under which they are the same. |