Colloquia
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Fri, 19/01/2007 16:30 |
Professor Dennis Sullivan (City University, New York) |
Colloquia  |
L2 |
| Imagine a finite collection of directed closed curves in a manifold M which varies with respect to finitely many parameters...like multi dimensional time. At certain times various strands of these curves may cross. At these moments we can reconnect the strands, often in several combinatorially distinct ways. All of these operations on all such families can be organized using algebraic topology into an elaborate "closed string algebra", more or less like a concept introduced and promoted by Segal ,Getzler, Kontsevich, and Costello. There is a relative version using paths which start and end on a submanifold K in M. One again finds an elaborate "open string algebra" which in some ways is a noncommutative version of the closed string algebra. These two open and closed string algebras interact in a way that extends a well known description due to Graeme Segal and the physicist Greg Moore based somewhat on a relation due the physicist, John Cardy. Recently, motivated by a structure in symplectic topology called symplectic field theory and a combinatorial construction of Lenny Ng, in joint work with Mike Sullivan, we have learned the relative open string algebra of a knot K in 3-space provides rather good invariants - even after passing to homology in degree zero! | |||
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Fri, 02/03/2007 16:30 |
Prof. Angus MacIntyre (Queen Mary University, London) |
Colloquia |
L2 |
| Model theory typically looks at classical mathematical structures in novel ways. The guiding principle is to understand what relations are definable, and there are usually related questions of effectivity. In the case of Lie theory, there are two current lines of research, both of which I will describe, but with more emphasis on the first. The most advanced work concerns exponentials and logarithms, in both real and complex situations. To understand the definable relations, and to show various natural problems are decidable, one uses a mixture of analytic geometry with number-theoretic conjectures related to Schanuel's Conjecture. More recent work, not yet closely connected to the preceding, concerns the limit behaviour (model-theoretically), of finite -dimensional modules over semisimple Lie algebras, and here again, for decidability, one seems obliged to consider number-theoretic decision problems, around Siegel's Theorem. | |||
