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Colloquium, Autumn 2005 Thursdays at 3:30 p.m. in South Hall, Room 6635
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| September 29, 2005 | Abstract: The B-field is a notion which occurs in many aspects of current theoretical physics, and we want to understand it in the language of differential geometry in the same way that the physicist's gauge theory was understood as the study of connections on principal bundles. There appears to be a formalism using some new tools, notably something called the Courant bracket, which incorporates the B-field in a natural way, and which sheds some light on familiar formulas in standard differential geometry. Contact person: Xianzhe Dai |
| October 13, 2005 | Abstract: We start with a brief introduction into knot homology theories and categorification of polynomial knot invariants. Of particular interest are homology theories of Ozsvath-Szabo-Rasmussen and Khovanov-Rozansky which provide a homological lift of the Alexander polynomial and the quantum sl(N) invariant, respectively. Motivated by the ideas from physics, we then present a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. We also describe the geometric meaning of the new knot invariants in terms of the enumerative geometry of Riemann surfaces with boundaries in a certain Calabi-Yau three-fold. Contact person: James Mckernan |
| October 20, 2005 | Abstract: Is circular reasoning necessary in mathematics? Many mainstream mathematicians might be surprised to learn that the standard axiomatizations of set theory involve fundamental circularities. Mathematical conceptualism is an alternative foundational philosophy, originating in views of Poincare and Russell, which strictly forbids all circularity. This sort of approach was originally thought to be far too weak to support ordinary core mathematics, and later was felt to be subject to severe limitations of a more abstract nature, but we now know that these limitations are not valid and in fact essentially all core mathematics is conceptualistically legitimate if interpreted properly. At the same time, conceptualism exorcises vast regions of set-theoretic pathology from the mathematical universe, so that it is in fact in better accord with actual mathematical practice than the Cantorian picture. I believe a strong case can be made for abandoning Cantorian set theory as a foundation for mathematics, and adopting conceptualism in its place. (Related papers are available at papers) Contact person: Chuck Akemann |
| November 1, 2005, ESB1001, Building 514 | Abstract: When Hamilton discovered quaternions he felt sure that this would be his greatest achievement and that it would transform mathematics and physics. This view was shared by few others. However, today 200 years after Hamilton’s birth, quaternions can be seen to be deeply embedded in modern physics and to have a signifiicant role in the associated geometry and analysis. I will explain how Roger Penrose’s theory of twistors goes a long way to justify Hamilton’s vision. Contact person: Xianzhe Dai |
| November 3, 2005 | Abstract: For a few examples of matrix valued orthogonal polynomials that satisfy a differential equation (with matrix coefficients) I study the noncommutative algebra of all these operators. In the scalar case this goes back to Burchnall and Chaundy and more recently Krichever. The situation here is considerably more complex. Joint work with Mirta Castro. Contact person: Milen Yakimov |
| November 10, 2005 | Abstract: Derived categories were first introduced in the 1960s by Grothendieck and Verdier as a technical and rather abstruse tool in homological algebra. More recently they have entered the mainstream of algebraic geometry, and have helped to forge links with non-commutative algebra, symplectic geometry and string theory. After explaining a little about what derived categories are, I shall try to give a flavour of the subject by describing a few examples where they give a useful point of view. Contact person: James McKernan |
