# Spring 2012

### An Introduction to KLR-Algebras

#### Alexander Lang

#### Department of Mathematics, UC Davis

#### Thursday, June 21, 2012, DMSC 104, 4:15 - 5:30 pm

**Abstract:** Khovanov-Lauda-Rouquier algebras, or KLR-algebras, were independently discovered by Khovanov-Lauda and Rouquier as a means of studying quantum groups. They are conjectured to be the appropriate objects to categorify the quantum groups. We will define these algebras and discuss their basic properties. In particular we will describe the related diagrammatic calculus, as well as an important quotient of this algebra, known as the cyclotomic quotient. We will briefly sketch their significance to quantum group theory.

### Lagrangian fillings and caps in C^2

#### Emmy Murphy Department of Mathematics, Stanford University

#### Thursday, June 21, 2012, DMSC 104, 3:00-4:00 pm

**Abstract:** Thinking of S^3 as the unit sphere in C^2, a Legendrian knot K in S^3 is a smooth knot so that i*dK/dt is tangent to S^3 everywhere. Legendrian knots live on the border between geometry and topology: on one hand any curve in S^3 can be C^0 perturbed to be Legendrian, thus any smooth knot is smoothly isotopic to a Legendrian knot. On the other hand they are very geometrically rich, reflecting the fact that Legendrian knots can be used to construct any holomorphic/symplectic 4-manifold. In the talk we'll focus on the concept of Lagrangian caps and fillings. An exact Lagranagian in C^2 is a surface L satisfying a geometric condition with respect to the symplectic structure on C^2. A Lagrangian filling of a Legendrian knot K is an exact Lagrangian L in B^4 so that the boundary of L is K, similarly a Lagrangian cap is an exact Lagrangian in C^2 \ B^4 whose boundary is K. Since there is no holomorphic/symplectic diffeomorphism of C^2 \ (0,0) which exchanges infinity and the origin, Lagrangian fillings and caps appear in very different situations, and in fact no Legendrian knot admits both a filling and a cap. In some sense Lagrangian fillings are geometrically subtle and rich, whereas Lagrangian caps are constructed by topological methods which destroy any interesting geometry of a Legendrian. The main goal of the talk will be proving a recent (2012) theorem, which states that any smooth knot is isotopic to a Legendrian knot which admits a Lagrangian cap (of any sufficiently large genus). The talk will be accessible to an audience with no prior knowledge of contact/symplectic geometry, the only prerequisite is a basic understanding of smooth topology.

### Asymptotic Distribution of the Estimated CDF of the Bivariate Distribution with Truncated Logistic and Geometric marginals

#### Heidi Tan

#### Masters of Science in Mathematics Thesis Defense

#### Department of Mathematics and Statistics, University of Nevada, Reno

#### Friday, May 11, 2012, 11am-1pm, DMS 102

**Abstract:** In this work, we derive the limiting distribution of the cumulative distribution function for the bivariate model with truncated logistic and geometric marginals. We discuss the limiting behavior of the asymptotic variance and illustrate the convergence of the cumulative distribution function to the limiting normal law using Monte Carlo simulations. Last, we present an example of applying these results to estimate error probabilities for hydrological and financial data.

### The Mathematical Signature of a Composer: A Preliminary Analysis

#### Jessica Reynolds

#### Honors Thesis Defense

#### Department of Mathematics and Statistics, University of Nevada, Reno

#### Tuesday, May 1, 2012, 2:30-3:45pm, DMS 102

**Abstract:** The possibility of analyzing and comparing works of musical composers using statistical methods is explored. Statistical methods are already used to analyze literary authors, examining word lengths, frequencies, and idiosyncratic tendencies. These methods are studied and related to the field of music by analyzing musical patterns of different composers. In this research, information from individual notes from solo-instrument musical compositions is coded into five variables. This new coding system allows for observation of note frequencies, average note durations, note repetitions, sequences, and the composer's tendency to adhere to a specific key. Several works by Bach and Stravinsky are analyzed and compared based on these criteria. If these composers are successfully shown to be statistically different, it could be the starting point to the development of an identification system for works of unknown or uncertain authorship.

### Automated Solution of Equations with Uncertain Parameters

#### Dr. Andrzej Pownuk

#### The University of Texas at El Paso, Department of Mathematical Sciences & Computational Science Ph. D. Program

#### Thursday, April 12, 2012, 2:30-3:45pm, DMS 102

**Abstract:** Mathematical modeling allows prediction of the future characteristics of engineering structures without performing expensive experiments. In many cases it is hard to get exact values of the parameters necessary to specify mathematical model. If only limited information is available sometimes it is possible to obtain upper and lower bound of the parameter (i.e. ). In this talk an efficient method for solution of equations with uncertain parameters (interval, random, and fuzzy parameters) will be presented. The method allows adaptive error estimation.

Many numerical results, visualizations as well as mathematical theorems which are related to this presentation were developed automatically by the SelfNet system. The system is capable to perform many typical scientific tasks (e.g. prove theorems, prepare visualizations, get numerical/symbolic solution) automatically. SelfNet is capable to develop selected scientific ideas automatically and improve it itself. The system engenders not only the final results of the calculations, but also shows all intermediate steps which lead to its solution. New knowledge generated by the system is saved and can be used automatically in the future problems calculations. Once the new idea is added to the system it will never be forgotten and the system is capable to apply it for processing of future problems. SelfNet was developed by the author of this presentation.

### Localized Modes and Energy Trapping Phenomena in Coupled Nonlinear Oscillators

#### Dr. Valery Pilipchuk

#### Wayne State University

#### Thursday, April 5, 2012, 2:30-3:45pm, DMS 102

**Abstract:** High energy dynamics or resonance interactions between system subcomponents are usually accompanied by most interesting physical effects. At the same time, conventional analyses of such effects face challenging mathematical problems due to strong nonlinearities or dimension increase. The main idea of the present talk is to show that an adequate basis for understanding the essentially nonlinear phenomena must also be essentially nonlinear however still simple enough to play the role of basis. Such a stand point will be illustrated on practically reasonable mechanical models, whose dynamics may include both resonance interaction and nonlinear localization effects. The nonlinear mode localization has been known for a long time as both micro- and macro-level phenomenon. However, in the area of nonlinear dynamics, the mode localization recently became of a growing interest due to the idea of dynamic energy absorption. In contrast to stochastic localization in disordered linear systems discovered in quantum physics about one half of a century ago, nonlinear local modes may occur even in perfectly symmetric systems with symmetry braking. In general terms, localization means that one or few interacting particles become dynamically isolated from the rest of the system due to specific initial conditions and/or variation of physical parameters. As a result, the energy, while oscillating between two oscillators on low energy levels, does not oscillate any more and becomes trapped on one of the two oscillators.

### Recent advanced in time-domain simulation of electromagnetic wave propagation in metamaterials

#### Dr. Jichun Li

#### University of Nevada, Las Vegas, Department of Mathematical Sciences

#### Thursday, March 1, 2012, 2:30-3:45pm, DMS 102/h4>

**Abstract:** Since 2000, there is a growing interest in the study of metamaterials due to their potential applications in areas such as design of invisibility cloak and sub-wavelength imaging. In this talk, I'll first give a brief introduction to the short history of metamaterials. Then I'll focus on mathematical modeling of metamaterials, and discuss some numerical schemes we developed in recent years. Finally, I'll conclude the talk with our cloak simulation and some open issues for further exploration.

A short bio: Dr. Jichun Li is Professor of Mathematics and Director of Center for Applied Mathematics and Statistics (CAMS). He has worked at the Institute for Computational Engineering and Sciences (ICES) in University of Texas at Austin, the Institute for Pure and Applied Mathematics (IPAM) at UCLA, and U.S. Air Force Research Laboratory (AFRL). His research interest is in scientific computing, imaging processing, inverse problems, computational electromagnetics. He has published over 60 refereed journal papers and two books. He currently serves on editorial boards of three international journals.

### Sequential Confidence Limits for the Ratio of Two Binomial Proportions with Unequal Sample Sizes

#### Dr. Hokwon Anthony Cho

#### University of Nevada, Las Vegas

#### Department of Mathematical Sciences

#### Thursday, February 23, 2012, 2:30-3:45pm, DMS 102

**Abstract:** We propose the approximate confidence limits for the ratio of two independent binomial variates. Due to the nonexistence of an unbiased estimator for the ratio, we develop the procedure based on a modified maximum likelihood estimator. When sample sizes are not equal, by defining the sample-ratio we can generalize results of Cho and Govindarajulu (2008). We investigate the large-sample properties of the proposed estimator and its finite sample behavior through numerical studies. In addition, we make comparisons from the sample information view points.

### Moduli Spaces and Orbifolds: What is Half a Point?

#### Dr. Dorette Pronk

#### Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia

#### Monday, January 23 2012, 11am-12pm, DMS 102

**Abstract:** A moduli space is a geometric object whose points parametrize (encode) all structures of a particular algebraic or geometric type. For instance, there is a moduli space of annuli: it is 2-dimensional if you want to vary the diameter and the width independently and if you only keep track of the proportion of the two, you would obtain a 1-dimensional parameter space.

Moduli spaces are used in the study of elliptic curves and Riemann surfaces of higher genus, and other areas of mathematics that are related to mathematical physics.When the family of objects we are parametrizing comes with a symmetry structure this needs to be reflected in the moduli space. This leads us to moduli spaces for which the local structure consists of an open subset of Euclidean space together with the action of a finite group. Such gadgets are called orbifolds and form a natural generalization of manifolds. In this talk we will see how orbifolds arise in a natural way from a moduli problem and then we will discuss some techniques for modeling and studying them and in particular we will look at the question what smooth maps between orbifolds should be.

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