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Thomas Schellhous
schellhous at math.ucr.edu
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Joshua Strong
strong at math.ucr.edu
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| Speaker: | Jesse Cohen | |
| Title: | Connections on Principal Bundles | |
| Abstract: | Of
particular importance in low-dimensional topology, geometry, and high
energy physics, principal bundles are natural generalizations of vector
bundles admitting a fiberwise free, transitive, and fiber-preserving
Lie group action. On any principal bundle, one may construct a
geometric object, called an Ehresmann connection, which allows us to
define familiar notions such as curvature in the context of principal
bundles. In this talk, we will define principal bundles and connections
and give examples to illustrate some of their properties. |
| Speaker: | Hyun-Seung Choi | |
| Title: | Multiplicative Lattices and Semistar Operation | |
| Abstract: | Some
well known results and concepts on commutative rings can be extended to
multiplicative lattices. Some of these transitions, for example,
defining the localization and semistar operation on certain types of
multiplicative lattices, will be discussed. |
| Speaker: | Nick Woods | |
| Title: | PROPs of Linear Systems | |
| Abstract: | A PROP is a symmetric monoidal category whose objects are the natural numbers and whose tensor product is given on objects by ordinary addition. They were devised as a tool for describing algebraic structures; since a morphism from m to n can be visualized as a string diagram from m to n, they are also useful in describing diagrams found in physics and engineering. We will investigate PROPs in all of these contexts by describing their relationship with finite-dimensional linear maps. |
| Speaker: | Xander Henderson | |
| Title: | An Introduction to Fractal Dimensions | |
| Abstract: | One of the most important tools in the study of fractal sets is the Hausdorff dimension. The definition of the Hausdorff dimension is somewhat opaque, and it has the somewhat disconcerting property that it can take non-integer values. In this talk, I will present an informal derivation of the Hausdorff dimension, and demonstrate that it is a natural generalization of the concept of dimension that is taught in elementary school. |
| Speaker: | Taylor Baldwin | |
| Title: | A Physical Derivation of the Incompressible Navier-Stokes Equations | |
| Abstract: | The Navier-Stokes equations are used widely in physics and engineering to describe the evolution of viscous fluids, and are the subject of intense and ongoing research in mathematics. In this talk, I will give an intuitive physical derivation of the incompressible Navier-Stokes Equations. If time allows, I will also talk about certain mathematical properties and applications of the system. |
| Speaker: | Lawrence Mouillé | |
| Title: | Nets and Filters | |
| Abstract: | Sequential convergence does not play the same central role in generalized topological spaces as it does in metric spaces. The purpose of the concept of a net (introduced by Moore-Smith in 1922) or a filter (introduced by Cartan in 1937 and publicized by Bourbaki) is to generalize the notion of a sequence in general topological spaces in a way that is useful. In this talk, I will introduce nets and filters, motivating the definitions, and show how they better encode information about a general topological space than do sequences. I will also give examples of how they can be applied to solve problems. |
| Speaker: | Tim Cobler | |
| Title: | Everything You Ever Wanted to Know About the Riemann Zeta Function! | |
| Abstract: | This talk will define the Riemann Zeta function, examine its convergence, look at alternate representations and then cover some of the main properties: functional equation, existence and location of poles/zeros, the prime number theorem, universality and anything else interesting that I can fit in. |
| Speaker: | Christina Osborne | |
| Title: | Introduction to Model Categories | |
| Abstract: | One of the purposes of model categories is to provide a natural context for homotopy theory. In this talk, we will discuss the definition of a model category as well as a few examples. Also, we will investigate what it means for two functions to be homotopic in the model category sense and why the homotopy category is important. This talk should be accessible to all grad students. |
| Speaker: | Matthew Lee | |
| Title: | Representations and the Decomposition of Lie Algebras | |
| Abstract: | Starting with the basic definitions, I will talk about some introductory theory of Lie Algebras. After introducing the adjoint representation, I will quickly discuss finite dimensional representations of sl2. Then we will use this information to decompose a Lie Algebra into the direct sum of ideals. This decomposition corresponds to a collection of simple reflections, which have a group structure, called the Weyl Group. |
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| Speaker: | Andrea Arauza | |
| Title: | The Commutative Motivation for Noncommutative Fractal Geometry | |
| Abstract: | First, we discuss the correspondence between compact Hausdorff spaces and commutative C* algebras. We then use the ideas behind this correspondence to formulate new ways of describing certain geometric properties of topological spaces. This can be especially helpful when dealing with fractal spaces. We will see how one can use algebraic tools to recover the (Minkowski) dimension, geodesic distance, and even a notion of integration, on the Sierpinski gasket. |
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| Speaker: | Lisa Schneider | |
| Title: | A Look at Combinatorial Representation Theory | |
| Abstract: | In this talk, we will discuss what combinatorial representation theory is. One of the common tools used in combinatorial representation theory is the Young diagram. We will introduce the Young diagram and see how it relates to the representation theory of symmetric groups as well as general linear groups. |
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| Speaker: | Thomas Schellhous | |
| Title: | An Introduction to Fluid Mechanics and Vorticity | |
| Abstract: | Fluid mechanics is a branch of PDE that studies the Navier-Stokes and Euler equations for fluid motion. After introducing some basic ideas, we will investigate some of the properties of these equations and their solutions, including the useful concept of vorticity. We will conclude by discussing some open problems, recent developments, and some current work in progress. This talk should be accessible to all grad students. |
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| Speaker: | Donna Blanton | |
| Title: | Origin and Basics of Representation Theory | |
| Abstract: | Representation theory was initiated by Frobenius in an attempt to solve a problem that arose in correspondence with Dedekind. We will discuss that problem and some basic ideas and questions in representation theory. Finally, we will see some uses of representations in other areas of mathematics. |
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| Speaker: | Xander Henderson | |
| Title: | On the Assouad Dimension of Self-Similar Sets with Overlaps | |
| Abstract: | The Assouad dimension of a set is of interest as it provides information about the local complexity of the set. Unfortunately, direct computation of the Assouad dimension is often quite difficult. In this talk, I will show that if a self-similar set satisfies the weak separation property (WSP), then the Assouad dimension coincides with the easily computed similarity dimension. Moreover, if the WSP is not satisfied, then the Assouad dimension is bounded below by 1, giving a precise dichotomy for subsets of the real line. |
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| Speaker: | Parker Williams | |
| Title: | Szemerédi's Regularity Lemma: A Friendly Introduction | |
| Abstract: | People will use words like "central," "essential," and "important" when speaking of the lemma. I will take a very long road to this theorem in order to illuminate why combinatorists care about it and why it is such a powerful idea. This lemma is thought of primarily as being a statement about extremal graph theory but in fact was developed to answer a famous conjecture of Erdös regarding arithmetic progressions. On the surface, this lemma can be understood as the notion that in some sense every graph can be approximated by a random graph. This notion will be expounded upon and I will start from very basic notions to give an overview leading to an appreciation of the lemma. |
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| Speaker: | Tim Cobler | |
| Title: | Basics of Algebraic Geometry and the Weil Conjectures | |
| Abstract: | This talk will cover some beginning definitions, ideas, and examples in Algebraic Geometry. They will then be used to define the zeta function of an algebraic variety in order to state the Weil Conjectures. If time permits, there will be some discussion of the methods used in the proofs of the Weil Conjectures. |
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| Speaker: | Franciscus Rebro | |
| Title: | Iterated Pullbacks and Span2(C) | |
| Abstract: | In this talk I'll go through some of the main steps in showing directly that given any category C with pullbacks and a terminal object, one can construct a bicategory called Span2(C); its objects are those of C, morphisms are spans in C, and 2-cells are isoclasses of spans of spans in C. The construction involves several computations with iterated pullbacks, and I will discuss a general conjecture that handles each of them as special cases. |
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| Speaker: | Sean Watson | |
| Title: | Self-Similar Fractals on the Heisenberg Group | |
| Abstract: | Analysis on the Heisenberg Group is motivated by its appearance in quantum mechanics and several complex variables. It can be viewed as a sub-Riemannian manifold with a natural horizontal distribution and induced metric. Sets (and in particular, fractals) in the standard Euclidean plane can be lifted to the Heisenberg group in a natural way through what is called horizontal fractals. I will show that any contractive iterated function system can be lifted, and that the horizontal lifts completely classify any affine Lipschitz self-map on the Heisenberg group. |
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| Speaker: | Taylor Baldwin | |
| Title: | An Introduction to Complex Fluids and Entropy Methods | |
| Abstract: | The
evolution of dispersed particles in a fluid may be modeled by a system
coupling the Navier-Stokes equations with the Vlasov-Fokker-Planck
equation. I will introduce this system, and discuss the behavior of a
particular Vlasov-Fokker-Planck/Navier-Stokes asymptotic regime. I will
also discuss entropy in the context of PDE and, using entropy methods,
will show that weak solutions of the asymptotic regime converge to
solutions of a multi-fluid system. |
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| Speaker: | Matt Barber | |
| Title: | An introduction to operads | |
| Abstract: | The term operad (or multicategory) was originally coined by Peter May to study infinite loop spaces. Consider a pointed space X, then the loop space of X is something you would want to think of as a topological monoid because you are able to compose paths. However, such composition is only associative up to homotopy. We want a good way to hide all of these homotopies. So an operad will be something like a collection of operations together with some way to compose them. Then many common algebraic structures will become algebras over some operad. |
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| Speaker: | Peri Shereen | |
| Title: | Voting Theory and Representation Theory | |
| Abstract: | Voting theory gives a mathematical framework to voting structures such as: presidential elections, bills, UN decision-making, etc. I will talk about three different contexts of voting theory that have interested me. The first will talk about 'dimensions' of voting systems. The second will talk about the paradoxes and geometry of voting theory. Lastly, I will talk about an algebraic framework of voting theory and how one can use representation theory to study voting systems. |
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| Speaker: | Priyanka Rajan | |
| Title: | What are exotic spheres? | |
| Abstract: | In 1956, Milnor revolutionized topology by discovering spaces homeomorphic to the standard spheres but not diffeomorphic to it, which are now known as exotic spheres. In this talk I will be explaining the construction of these spaces, which should be accessible to a broad range of audience. |
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| Speaker: | Scott Roby | |
| Title: | Self-Similar Strings and Multifractals | |
| Abstract: | Did you know that open sets in the real number line can have non-integer dimensions? In this talk I will show how to construct an important class of these sets called self-similar fractal strings and use associated zeta functions to calculate families of complex-valued dimensions. Furthermore, I will show that by imposing a probability measure on the construction of any such set we can analyze the multifractal structure. |
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| Speaker: | Josh Strong | |
| Title: | Gromov Hyperbolicity and the Kobayashi Metric in Convex Domains | |
| Abstract: | Have you ever stood on one side of a triangle and wished that you were close enough to at least on of the other sides? Well if you are in a Gromov hyperbolic space, your dreams have come true! In this talk, we shall discuss a necessary condition on the boundary of a convex domain to be Gromov hyperbolic under the Kobayashi metric. |
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