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# Mathematics Colloquium Schedule

**Spring 2016 **

**Date: 2-19-2016 (12-1pm, F03-200A), Dr. Bogdan Suceava, California State University, Fullerton**

**Title:** Geometry in the Dark Ages: Games of Shadows and Lights

**Abstract:** Isidore's Etymologies enjoyed a wide audience during the medieval period. We examine the structure of mathematics, as it is described in the Etymologies, and we discuss the sources on which Isidore relied when he collected his etymological definitions. We remark that for Isidore, mathematics is described as ``the science of learning'', and among his sources there have been the classical Greek authors, most likely available in Boethius' and Cassiodorus' Latin translations performed in the early 6th century. These translations are today lost. That's why the authors writing in the Middle Ages had to start from scratch in many of their investigations. We will illustrate this idea with one example, the discovery of curvature. In a paper published in 1952, J. L. Coolidge points out that ``the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicolas Oresme". Coolidge writes further: ``Oresme conceived the curvature of a circle as inversely proportional to the radius; how did he find this out?" Tractatus de configurationibus qualitatum et motuum, written by Orseme sometime between 1351 and 1355, contains the key. We discuss N. Orseme's original work in the scholarly environment of his time, and the moment when the first definition of curvature was given.

**Date: 3-4-2016 (12-1pm, F03-200A), Dr. Jim Stein, California State University, Long Beach**

**Title:** Liberal Arts Math

At least three different approaches have been tried in liberal arts math: reinforcing previously-taught material, emphasizing topics in finite mathematics, and making students aware of new developments in the subject (which gave birth to the textbook For All Practical Purposes). None of these approaches have proved to be spectacularly successful. This talk presents a different approach to liberal arts math by setting different goals (and BTW, what are the goals of current liberal arts math courses?) and trying to achieve those goals by methods which the students taking the course may find more palatable. This won’t be the most edifying math talk you’ll ever hear – but it will be right up there when it comes to entertainment value.

**Date: 4-8-2016 (2-3pm, F03-200A), Dr. Thomas Murphy, California State University, Fullerton**

**Title:** Hearing the Shapes of Surfaces of Revolution

**Abstract:** Most of the talk will involve concepts from multivariable calculus, and we will mostly focus on the two-dimensional sphere. I will explain what the Laplacian is on a smooth surface, why you would want to know what its eigenvalues are, and how you can compute them. There are surprising phenomena, even for surfaces obtained by revolving a smooth curve around an axis. Time permitting, I will try outline how mathematicians wish to generalize these ideas to higher dimensions.

**Date: 4-22-2016 (12-1pm, F03-200A), Dr. M. Andrew Moshier, Chapman University**

**Title:** A Relational Category of Formal Contexts

**Abstract:** Formal contexts (or polarities in Birkhof’s terminology) provide a convenient combinatorial way to present closure operators on sets. They have been studied extensively for their applications to concept analysis, particularly for finite contexts, and are used regularly as a technical device in general lattice theory (for example, to describe the Dedekind-MacNeille completion of a lattice). In the most common uses, morphisms of contexts do not play a role. Although various scholars (most thoroughly, Marcel Erne ́) have considered certain notions of context morphisms, these efforts have generally concentrated either on special kinds of contexts that closely match certain “nice” lattices or on special kinds of lattice morphisms.

Here we propose a category of formal contexts in which morphisms are relations that satisfy a certain natural combinatorial property. The idea is to take our cue from the fact that a formal context is simply a binary relation between two sets. So the identity morphism of such an object should be that binary relation itself. From this, we get the combinatorial properties of morphisms more or less automatically.

The first main result of the talk is that the category of contexts is dually equivalent to the category INF, of complete meet lattices with meet-preserving maps. To get a duality with complete lattices, the second result characterizes those context morphisms that correspond to complete lattice homomorphisms.

We also consider various constructions that are well-known in INF to illustrate that formal contexts yield remarkably simple, combinatorial descriptions of many common constructions.

**Date: 4-29-2016 (12-1pm, F03-200A), Dr. Wai Yan Pong, California State University, Dominguez Hills**

**Title:** Generalized Wronskians and linear dependence of formal power series

**Abstract:** In this talk, I will talk about a new proof of a generalization of the following result: A family of formal power series (in several variables) are linearly independent over the field of constants if and only if some of its generalized Wronskians does not vanish. Our proof also works for quotients of germs of analytic functions. It make an interesting use of arithmetic functions and formal Dirichlet series. This is a joint work with Keith Ball and Cynthia Parks supported by Project PUMP.

**Fall 2015 **

**Date: 9-11-2015 (12-1pm, F03-200A), Dr. Hui Sun, UCSD**

**Title:** Numerical Simulation of Solvent Stokes Flow and Solute-Solvent Interface Dynamics

**Abstract:** Fundamental biological molecular processes, such as protein folding, molecular recognition, and molecular assemblies, are mediated by surrounding aqueous solvent (water or salted water). Continuum description of solvent is an efficient approach to understanding such processes. In this work, we develop a solvent fluid model and computational methods for solvent dynamics and solute-solvent interface motion. The key components in our model include the Stokes equation for the incompressible solvent fluid which governs the motion of the solute-solvent interface, the ideal-gas law for solutes, and the balance on the interface of viscous force, surface tension, van der Waals type dispersive force, and electrostatic force. We use the ghost fluid method to discretize the flow equations that are reformulated into a set of Poisson equations, and design special numerical boundary conditions to solve such equations to allow the change of solute volume. We move the interface with the level-set method. To stabilize our schemes, we use the Schur complement and least-squares techniques. Numerical tests in both two and three-dimensional spaces will be shown to demonstrate the convergence of our method, and to demonstrate that this new approach can capture dry and wet hydration states as observed in experiment and molecular dynamics simulations.

**Date: 10-09-2015 (12-1pm, F03-200A), Dr. Andrew J. Bernoff, Harvey Mudd College**

**Title:** Energy driven pattern formation in thin fluid layers: The good, the bad and the beautiful

**Abstract:** A wide variety of physical and biological systems can be described as continuum limits of interacting particles. Their dynamics can often be described in terms of a monotonically decreasing interaction energy. We show how to exploit these energies numerically, analytically and asymptotically to characterize the observed behavior. Examples are drawn from the dynamics of thin fluid layers including ferrofluids.

**Date: 10-16-2015 (2-3pm, F03-200A), Dr. Peter Jipsen, Chapman University Center of Excellence in Computation, Algebra and Topology (CECAT)**

**Title:** From Residuated Lattices to Boolean Algebras with Operators

**Abstract:** This general audience talk introduces lattices and residuated operations on them, and explains how these algebraic structures are related to substructural logics. Adding some natural axioms defines Heyting algebras (corresponding to intuitionistic logic) and Boolean algebras (corresponding to classical propositional logic). No special background in abstract algebra or logic is assumed for this talk.

For many applications in logic and computer science additional operations are introduced, leading to the classical theory of Boolean algebras with operators (BAOs) as well as the still largely unexplored theory of Heyting algebras with operators (HAOs). As an example of BAOs, I will define Boolean semilattices and present some recent results about them. In the area of HAOs, generalized bunched implication algebras (GBI-algebras) are Heyting algebras expanded with a residuated monoid operation, and they have found interesting applications in the past decade in the form of separation logic for reasoning about pointers, data structures and parallel resources.

I will indicate why BAOs with a monoid operator generally lack decision procedures for their equational theories, whereas GBI-algebras, residuated lattices and several of their subclasses are equationally decidable. Some algorithms for enumerating finite algebras in these classes will be presented, as well as computational tools that are useful for exploring research questions in these areas.

**Date: 10-23-2015 (12-1pm, F03-200A), Dr. Christian Rose, Technische Universität Chemnitz**

**Title:** Compact manifolds with integral bounds on the negative part of Ricci curvature and the Kato class

**Abstract:** Bochner’s theorem states that a compact manifold with non-negative Ricci curvature and positive somewhere admits a trivial first cohomology group. Starting from a generalization by Elworthy and Rosenberg we show using Kato conditions for certain Schrödinger operators that L^p criteria for the part of curvature below a certain depth is sufficient that Bochner still holds.

**Date: 10-30-2015 (12-1pm, F03-200A), Dr. Jasbir Chahal, Brigham Young University**

**Title:** Two Applications of the Arithmetic of Elliptic Curves

**Abstract:** We will explain everything about elliptic curves needed to show how the arithmetic of elliptic curves can be used to solve two ancient problems. One is: what whole numbers are the areas of right triangles when the side lengths are allowed to be rational numbers and not just the whole numbers? The second problem is: for what triangles with all side lengths rational, an altitude, an angle bisector and the median are concurrent? No knowledge beyond high school math is required. However, the topics are very beautiful and lie at the frontier of research in number theory.

**Date: 11-13-2015 (12-1pm, F03-200A), Applied Math and Statistics Graduate Students, CSULB**

**Title:** Part I: Collaborative Filtering and the Yelp Dataset Part II: Python Introduction

**Abstract:** We will have a special colloquium featuring two presentations from our Applied Math and Statistics Master's students. Maike Scherer and Daniel Hallman will speak on collaborative filtering techniques for predicting user ratings of restaurants on Yelp. Juan Apitz and Truong Tran will then give an introduction to Python and the Jupyter Notebook project.