T.W.I.G.S. The "What Is ...?" Graduate Seminar Room 1634, Lederle Graduate Research Tower University of Massachusetts, Amherst
The Seminar meets on most Fridays, 12:15-1:15 in 1634 LGRT.     Driving Directions   and   Campus Maps Overview of the Seminar

		25 February	Michael Bush, UMass Amherst
		12:15-1:15	What is a profinite group?
		4 March	Tom Weston, UMass Amherst
		12:15-1:15	What is a p-adic L-function?
		11 March	Farshid Hajir, UMass Amherst
		12:15-1:15	SPECIAL STUMP THE CHUMP EDITION OF TWIGS
		25 March	Farshid Hajir, UMass Amherst
		12:15-1:15	What is the Mahler measure of a polynomial?
		31 March	Rob Benedetto, Amherst College
		1:00-2:00	What is an affine scheme? [RESCHEDULED FOR APRIL 7 DUE TO AQAD]
		7 April	Rob Benedetto, Amherst College
		1:00-2:00	What is an affine scheme?
		15 April	Franz Pedit, UMass Amherst
		12:15-1:15	What is GANG?
		22 April	Tom Braden, UMass Amherst
		12:15-1:15	What is a characteristic class?
		29 April	John Staudenmayer, UMass Amherst
		12:15-1:15	What is a Support Vector Machine?
		6 May	Keith Conrad, UCONN
		12:15-1:15	What is infinite Galois theory?

Abstract
Profinite groups are topological groups that often arise in mathematics by providing a convenient way to package together infinite families of finite groups into a single object. We will give two definitions of such groups (one topological, the other algebraic) and explain why they are equivalent. We will then look at several important examples that arise in algebra and number theory. This talk should be accessible to anyone who knows the basic facts about finite groups and topological spaces.

Abstract
FUN WITH BERNOULLI NUMBERS! First we'll meet or become reaquainted with p-adic numbers, then we'll meet or become recquainted with L-functions, and finally we'll introduce the two of them to each other to make p-adic L-functions.

Abstract
This is the first installment of the Stump the Chump Edition of TWIGS, where you get to ask Farshid to define mathematical objects on the spot (with no outside sources). You can also ask about major theorems/conjectures if you wish. Here is your chance to put a professor through the ringer! Come find out how little Farshid knows! (It's bound to make you feel good). [Questions I was asked: "What is quantum ergodicity?" (the chump was stumped). "What is a moduli space?" (isomorphism classes of geometric objects often natually become points of another geometric objec; I described how upper-half plane modulo SL(2,Z) classifies complex tori).]
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Abstract
The Mahler measure of a monic one-variable polynomial with complex coefficients, so-called because it was first considered by Lehmer, is the product of the absolute values of its roots outside the unit circle. That takes care of the "What is...?" part. The rest of the talk will concentrate on the "What is ... good for?!" part. In particular, I'll discuss a theorem of Kronecker and an intriguing question of Lehmer about the Mahler measure of polynomials with integer coefficients and its connections to deep questions in number theory.
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Abstract
There is an important and very fruitful correspondence between objects in algebraic geometry and commutative algebra according to which the elements of a (commutative) ring "act as" functions on an associated geometric object. The geometric object in question is an affine scheme. [Now, if we could just take derivatives of those functions ... -- The Editors.]

Abstract
The Center for Geometry, Analysis, Numerics & Graphics (GANG) is an interdisciplinary Differential Geometry research team in the Dept of Mathematics & Statistics at the University of Massachusetts, Amherst. This talk will give an overview of some mathematical problems of interest to current members of GANG.

Abstract
Vector bundles are important tools in many areas of geometry. A vector bundle over a space X looks locally like a product of X with a vector space, but it need not be a product globally. Characteristic classes provide one way to measure the extent to which a bundle is "twisted". I will introduce these ideas using simple examples and not too much heavy machinery.

Abstract
Suppose one observes pairs (x_i,y_i) i=1,...,n, where x_i is a vector and y_i is a scalar that has m possible values (y_i is a categorical response). Those data can be used to develop a classification model that can be applied to a new x_j to predict the value of the unobserved associated y_j. A support vector machine is a bad name (in my opinion) for a general tool that can be used as a classification model. We'll describe support vector machines in some detail in this talk. Possible highlights: separating hyperplane, kernel function, quadratic program, discriminant analysis, logistic regression.

Abstract
When a finite extension of fields is a Galois extension, there is a one-to-one correspondence between intermediate fields and subgroups of the Galois group. The concept of Galois extension can be generalized to field extensions of infinite degree, but the usual Galois correspondence breaks down: different subgroups of the (infinite) Galois group can correspond to the same intermediate field. There is a way to restore the classical Galois correspondence for infinite extensions, by introducing a suitable topology on the Galois group and using topological notions.

Last modified: Feb 2004 by Farshid Hajir