2009-10 Colloquia talks

Abstract: The Tutte polynomial is one of the most famous in combinatorics. This is a polynomial in two variables defined for a graph. It generalizes the chromatic polynomial. Many other graph invariants are specializations of the Tutte polynomial. However it became famous because its applications in statistical physics. It is a partition function of the Potts model and its close relative, random-cluster model describing the theory of phase transitions and critical phenomena. Also the Tutte polynomial and its generalizations have very important applications in knot theory.

In my talk I will define the Tutte polynomial and discuss its various properties. Then I am going to describe its applications and generalizations mentioned above.

Abstract: The concept of a minimal surface seems very intuitive, yet the subject has intrigued mathematicians from Euler to present day. In this talk I will provide an overview of minimal surfaces, including a family of surfaces that I study. I will also discuss a nice connection between minimal surfaces and complex analysis, but the talk will be accessible to a general mathematics audience.

Abstract: In this talk I will describe a system for composition, sound design, pedagogy, and sound analysis using FMS (Feature Modulation Synthesis). FMS employs a synthesis by analysis approach for sound manipulation focusing on salient acoustic features that capture timbral dimensions corresponding to relevant characteristics of sound perception. These feature vectors are then analyzed and modulated for re-synthesis. I will discuss core concepts of this new synthesis paradigm and introduce the current salient feature extraction algorithms and their respective modulation modules that have been developed and implemented. I will also present a new software system called ìFMS Toolboxî which is a GUI-based MATLAB application that allows for the analysis, modulation, and re- synthesis of the sound objects.

Abstract: All the statistical tests used in practice and the designs proposed for conducting clinical trials are abound with statistical assumptions. The underlying assumptions are often violated but their impact on the test procedures or on the design are seldom evaluated. Although, in testing framework these assumptions are often checked and alternative methods for testing are adopted. However, this practice often remains an academic exercise and does not get easily translated into routine practice. In this talk, two specific examples, one pertaining to the conducting of an analysis and the other pertaining to the designing of a clinical trial, will be discussed. With these two examples the role of assumptions and their impact on clinical trials will be highlighted and some alternative approaches will be discussed.

Abstract: Cancer is a dreaded diagnosis made in over 1.4 million individuals yearly in the United States. The most common reaction to this diagnosis is fear of death and pain. Coping with the diagnosis and distress in cancer patients in many individuals involves spiritual beliefs of the influence of a ìhigher powerî in order to successfully navigate the path of treatment and recovery. This discussion will focus on spirituality in coping with cancer Ö a story told by the patients. Also, with the institution of complementary therapy in the adjunctive treatment of cancer, mind-body therapies are becoming a part of this realm and allows researchers in the field the opportunity to study the process of spirituality and its impact of the survival and well being of the patients.

Abstract: Modular categories appear as the algebraic framework in a number of mathematical and physical settings. I will introduce modular categories and some of their connections to condensed matter physics, low-dimensional topology, Hopf algebras and quantum computation. With these applications in mind, we have been working on the classification of modular categories. This has led to some conjectures - both structural and enumerative - which I will describe. This is based on joint work with Zhenghan Wang, Michael Larsen, Sarah Witherspoon, Deepak Naidu and others.

Abstract: Studying cohomology is hard, but the problem can sometimes be made easier through the introduction of a geometric tool called support varieties. In this talk we'll discuss the support varieties that arise while studying the cohomology of restricted Lie algebras and quantized enveloping algebras (also called quantum groups). It is a somewhat amazing fact that the geometric objects that arise in these contexts admit explicit descriptions in terms of nilpotent matrices. We will look at some easy examples, discuss some recent progress on computing cohomology and support varieties, and mention some open questions. This talk is designed to be accessible to a wide mathematical audience.

Abstract: Quite often, and quite unexpectedly, questions in Mathematics and the Mathematical Sciences have concealed within them connexions to discrete structures that are key to their solution, and it is important to understand them. For example, Feynman diagrams are an instance of such structures in Quantum Field Theory, and a combinatorial Legendre Transform based on these appears to circumvent some analytic awkwardnesses in the theory.

Since discrete structures may not be familiar to all members of the Audience, I shall begin with an unusual solution of the familiar Derangement Problem (How many permutations have no fixed points?), then touch upon the relation between Cayley's result for trees and the Lagrange-Burmann Theorem from complex analysis as a concrete instance of an Unexpected Connexion alluded to above. The main topic of the Talk is a combinatorial study of the 1990's model of string theory for meson-meson interaction (associated with the quarks of the title), and what it suggests about combinatorial structures embedded in surfaces.

My intention is that this talk should be appealing to a general mathematical Audience. Where algebra that may be unfamiliar is used, I shall concentrate on the intuition behind it and what it signifies rather than on the details.

(In the sequel in the Algebra Seminar, tomorrow, I shall use techniques from algebraic combinatorics to study genus g ramified covers of the sphere, the impact of this material on the very short proof of Witten's Conjecture (Kontsevich's Theorem) by Alex Kazarian, and a very short proof of the lambda_g-Conjecture/Theorem of Getzler and Pandharipande.)