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T.W.I.G.S.
The "What Is ...?" Graduate Seminar

Room 1634, Lederle Graduate Research Tower
University of Massachusetts, Amherst 		ComplexCubicUnitAction
The Seminar meets Wed 1:30-2:30 in 1634 LGRT.    
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Overview of the Seminar
Maintained by Farshid Hajir.

NOTE Here are the schedules for Fall 2002, Spring 2003, Fall 2003, Spring 2004, Fall 2004, Spring 2005, and Fall 2005.
Wednesday February 15 Farshid Hajir, UMass Amherst. "What is the Shannon capacity of a graph?"
Abstract: In 1956, Claude Shannon (the inventor of "information theory") asked whether one could determine the maximum rate of transmission over a noisy channel at which the receiver can recover the original message without errors. I'll define these terms and discuss Shannon's and Lovasz's contributions to this question.
Wednesday March 1, Hao Wu, UMass Amherst. What is a Knot? What is the Khovanov-Rozansky cohomology of a knot?
Abstract: In 1999, Mikail Khovanov constructed a cohomology theory for knots. He and Lev Rozansky generalized this construction later, and gave a infinite sequence of knot cohomology theories. Their work lead to many new developments in low-dimensional topology. In this talk, I will quickly explain what a knot is, and then slowly explain how Khovanov and Rozansky constructed their knot cohomologies.
Wednesday March 8, Tom Weston, UMass Amherst. What is a Galois Representation?
Abstract: One of the most important concepts in number theory is that of a Galois group, which is the group of algebraic symmetries among roots of a polynomial. Usually the best way is to understand a group is to represent its elements as matrices. A Galois representation is simply a representation of a Galois group as a group of matrices. It is quite remarkable that algebraic geometry (via cohomology) and complex analysis (via modular forms) are rich sources of Galois representations. These Galois representations in fact serve to establish highly important links between modular forms and algebraic varieties. The proof of Fermat's Last Theorem, for example, relied on this circle of ideas.
Wednesday March 15, Jim Humphreys, UMass Amherst. What is Monstrous Moonshine?
Abstract: In the late 1970s John McKay observed a striking numerical coincidence, suggesting a relationship between two remote areas of mathematics: the classical theory of the modular j-function (coming from the action of the modular group SL(2,Z) on the complex upper half-plane) and the character degrees of the recently discovered Fischer-Griess finite simple group (dubbed the Monster). In their 1979 paper "Monstrous Moonshine" in the LMS Bulletin, John Conway and Simon Norton formulated systematic conjectures which fed into later work on affine Lie algebras and vertex operator algebras (Frenkel-Lepowsky-Meurman, Borcherds). We explain some of the ideas in this still unfinished story.
Wednesday March 29, Hans Johnston, UMass Amherst. What are Navier-Stokes Equations?
Abstract: Viscous incompressible fluid flow plays an important role in many scientific and industrial applications. The fundamental mathematical model of interest is the time dependent incompressible Navier-Stokes equations. Although these equations were derived more than a century ago our understanding remains limited, for the Navier-Stokes equations are a system of nonlinear partial differential equations, exactly solvable for only the simplest of configurations and initial conditions. Thus, in order to use them as a predictive model one must apply approximation techniques. It is in this context that numerical methods are playing an increasingly important role. In this talk we will discuss the history and derivation of the Navier-Stokes equations, some fundamental analytic results, the important distinction between 2D and 3D flows, and finally some issues concerning the design and implementation of numerical schemes for simulating such flows.
Wednesday April 5 Farshid Hajir, UMass Amherst. "What is an error-correcting code?"
Abstract: I will define codes and their basic parameters, and discuss some methods for producing "asymptotically good" error-correcting codes.
Wednesday April 12 No talk this week.
Wednesday April 19 Dan Yasaki, UMass Amherst. "What is a symmetric space?"
Abstract: Symmetric spaces and locally symmetric spaces show up in Algebraic Geometry, Mathematical Physics, Number Theory, and Representation Theory. They arise as parameter spaces for variations of geometric and arithmetic objects. We will define these spaces and look at some examples.
Wednesday April 26 Sukhendu Mehrotra, UMass Amherst. "What is a derived category?"
Abstract: Introduced by Alexander Grothendieck and his students, derived categories are important invariants of algebraic varieties that are finding striking applications in string theory, representation theory and the minimal-model program. In this talk, I shall provide a quick intoduction to derived categories and try to motivate why these are natural structures to study. I also hope to present some classical examples in this setting and will end with some interesting non-classical applications.
Wednesday May 3 Farshid Hajir and Rob Kusner, and ...?, UMass Amherst. "Stump-the-chump edition of TWIGS"
Abstract: Stump the Chump: You bring the questions, we'll botch the answers.


The picture above was created by Paul Gunnells. It visualizes the natural action of the group of units of a complex cubic field on 3-space. Consult Paul for more details.

Last modified: Feb 2006 by Farshid Hajir