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| John Dusel
jmd at math.ucr.edu |
Mathew Lunde
matlunde at math.ucr.edu |
Jacob West
west at math.ucr.edu |
| Speaker: | Lisa Schneider | |
| Title: | Cluster Algebras | |
| Abstract: | Since their discovery in 2002 by Fomin and Zelevinsky, cluster algebras had been applied to a variety of topics in commutative and non-commutative algebra, geometry, and topology. In this talk, we will define cluster algebras using quivers and quiver mutations. Then, we will show some examples of cluster algebras for simply-laced Dynkin diagrams and discuss properties of their cluster variables. |
| Speaker: | Oliver Thistlethwaite | |
| Title: | Clifford Algebras | |
| Abstract: | A Clifford algebra is an algebra that is naturally associated to a vector space equipped with a quadratic form. These algebras and their representations play a fundamental role in differential geometry. The goal of this talk is to present various examples of Clifford algebras and discuss their properties. This talk should be accessible to anyone who has taken undergraduate linear algebra. |
| Speaker: | Mathew A. Lunde | |
| Title: | Algebras, bialgebras, and Hopf algebras | |
| Abstract: | We will go through the definitions of these gadgets via commutative diagrams, and go through lots of examples. |
| Speaker: | John Dusel | |
| Title: | Mobius inversion in a poset | |
| Abstract: | We will review the construction of the incidence algebra for a locally finite poset and discuss its unit group. Next we will discuss the inversion problem and see how the Movius function provides a uniform solution. |
| Speaker: | Thomas Schellhous | |
| Title: | A Gentle Introduction to Fluid Mechanics | |
| Abstract: | Fluid mechanics is a branch of PDE that studies the motion of fluids both in all space and in bounded domains. In this talk, we will investigate the Navier-Stokes and Euler equations which describe this motion and look at some of the tools used in the field. We will conclude with a discussion of some open problems as well as some known results. The talk should be accessible to any human being who knows a little about derivatives and vector calculus, and its meant to convince you that Fluids is an awesome field. |
| Speaker: | John Dusel | |
| Title: | Quantum Groups and Crystal Bases | |
| Abstract: | I will describe the quantized universal enveloping algebra U_q(sl_2) and present the basics of its finite-dimensional representation theory. Next I will introduce the so-called crystal bases of Kashiwara and Lusztig, motivated by an example of the irreducible decomposition of the tensor product of two simple modules. A digraph with colored edges can be associated to a crystal base. The talk will conclude with an example of how algebraic information is encoded in this graph. |
| Speaker: | Dennis Gumaer | |
| Title: | Introduction to LaTeX | |
| Abstract: | This talk will cover the very basics of LaTeX usage. We will start with a download site of the necessary files. By the end of the talk, you should be ready to type homework and quizzes for a more professional touch. |
| Speaker: | Oliver Thistlethwaite | |
| Title: | The Seiberg-Witten Invariantes of 4-manifolds | |
| Abstract: | In the fall of 1996, the physicists Nathan Seiberg and Ed Witten introduced the world to a new set of invariants for compact smooth 4-manifolds. These Seiberg-Witten invariants are based on their work in gauge theory, and have since become one of the main tools in 4-manifold theory. In this talk, we will outline the information necessary to set up the invariants, as well as provide some of the basic results obtained by using them. |
| Speaker: | Oliver Thistlethwaite | |
| Title: | The Seiberg-Witten Invariants of 4-manifold (continued) |
| Speaker: | Mathew Lunde | |
| Title: | Representation theory of sl_2 and the quantum look algebra U_q(L(sl_2)) | |
| Abstract: | In this talk, we will discuss the Lie algebra sl_2 and a highest weight classification of its finite dimensional irreducible representations. We will also discuss the quantum loop algebra U_q(L(sl_2)) and a classification of its irreducible finite dimensional representations in terms of Drinfeld polynomials. |
| Speaker: | Adam Navas | |
| Title: | Applying Moser's Iteration to the 3D Axially Symmetric Navier_Stokes Equations (ASNSE) | |
| Abstract: | Ennio De Giorgi and John Forbes Nash independently solved Hilbert's 19th problem in 1957 and 1958, respectively, using what are now called the De Giorgi and Nash-Moser iteration schemes. In this talk, the ASNSE and corresponding vorticity equations will be derived, then the Nash-Moser iteration technique will be used to derive a local bound on an easier-to-deal-with elliptic partial differential equation. |
| Speaker: | Matthew Highfield | |
| Title: | Introduction to Sage | |
| Abstract: | Sage is free, open source math software available for download or use online at www.sagemath.org. We will explore a few features from its very large feature set, and see some reasons besides cose why one might (or might not) prefer it to Mathematica or a chalkboard and a piece of chalk. We will take a brief look at: the sage interface with LaTeX and the Python programming language, a computational problem from my own research on twisted graded Hecke algebras, applications in Lie theory, and more. |
| Speaker: | John Dusel | |
| Title: | Cones and Semigroups | |
| Abstract: | An integer sequence satisfying a set of linear inequalities gives a lattice point in a polynedral convex cone. Associated to such a cone is an affine semigroup; through this correspondence geometric and algebraic information are connected. We will discuss strongly convex rational polyhedgral cones, associated affine semigroups, and their respective generators. The next step in this program is to introduce affine toric varieties. We will go into this if time permits. |
| Speaker: | Jonas Hartwig | |
| Title: | Yangians and Their Applications | |
| Abstract: | Yangians are certain algebras closely related to Lie algebras and have many applications to representation theory and mathematical physics. In this talk, I will give a brief introduction to these algebraic structures, explain their historic origin, discuss their kinship with other quantum groups, and show some applications. No previous knowledge of Lie algebras or quantum groups will be assumed. |