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 |

14 April 2014 | |
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Speaker: | Jonathan Wolfram Siegel (UCLA) |

Title: | TBA |

Abstract: | TBA |

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. |

17 March 2017 | |
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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. |

18 November 2016 | |
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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 K_{0} and K_{1} 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. |