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Forthcoming events in this series
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Wick Rotation in Quantum Field Theory
Abstract
Physical space-time is a manifold with a Lorentzianmetric, but the more mathematical treatments of the theory usually prefer toreplace the metric with a positive - i.e. Riemannian - one. The passage fromLorentzian to Riemannian metrics is called 'Wick rotation'. In my talk I shallgive a precise description of what is involved, and shall explain some of itsimplications for physics.
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Cohomology jump loci, sigma-invariants, and fundamental groups of alge-
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Upper bounds onReidemeistermoves
Abstract
Given any two diagrams of the same knot or link, we
provide an explicit upper bound on the number of Reidemeister moves required to
pass between them in terms of the number of crossings in each diagram. This
provides a new and conceptually simple solution to the equivalence problem for
knot and links. This is joint work with Marc Lackenby.
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Characters and pushforward for differential K-theory with the Index theorem interpretation
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The virtual fibering conjecture and related questions
Abstract
Thurston asked a bold question of whether finite volume hyperbolic 3-manifolds might always admit a finite-sheeted cover which fibers over the circle. This talk will review some of the progress on this question, and discuss its relation to other questions including residual finiteness and subgroup separability, the behavior of Heegaard genus in finite-sheeted covers, CAT(0) cubings, the RFRS condition, and subgroups of right-angled Artin groups. In particular, hyperbolic 3-manifolds with LERF fundamental group are virtually fibered. Some applications of the techniques will also be mentioned.
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The Blob Complex
Abstract
We define a chain complex B_*(C, M) (the "blob complex") associated to an n-category C and an n-manifold M. This is in some sense the derived category version of a TQFT. Various special cases of the blob complex are
familiar: (a) if M = S^1, then the blob complex is homotopy equivalent to the Hochschild complex of the 1-category C; (b) for * = 0, H_0 of the blob complex is the Hilbert space of the TQFT based on C; (c) if C is a commutative polynomial ring (viewed as an n-category), then the blob complex is homotopy equivalent to singular chains on the configuration (Dold-Thom) space of M. The blob complex enjoys various nice formal properties, including a higher dimensional generalization of the Deligne conjecture for Hochschild cohomology.
If time allows I will discuss applications to contact structures on 3-manifolds and Khovanov homology for links in the boundaries of 4-manifolds. This is joint work with Scott Morrison.
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Decomposition complexity of metric spaces
Abstract
I shall describe the notion of finite decomposition complexity (FDC), introduced in joint work with Romain Tessera and Guoliang Yu on the Novikov and related conjectures. The talk will focus on the definition of FDC and examples of groups having FDC.
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The asymptotic geometry of mapping class groups and application
Abstract
I shall describe the asymptotic geometry of the mapping class
group, in particular its tree-graded structure and
its equivariant embedding in a product of trees.
This can be applied to study homomorphisms into mapping class
groups defined on groups with property (T) and on lattices in semisimple groups.
The talk is based upon two joint works with J. Behrstock, Sh. Mozes and M. Sapir.
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moduli of flat bundles on Riemann surfaces
Abstract
Let G be a compact semisimple Lie group. A classical paper of Atiyah and Bott (from 1982) studies the moduli space of flat G-bundles on a fixed Riemann surface S. Their approach completely determines the integral homology of this moduli space, using Morse theoretic methods. In the case where G is U(n), this moduli space is homotopy equivalent to the moduli space of holomorphic vector bundles on S which are "semi-stable". Previous work of Harder and Narasimhan determined the Betti numbers of this moduli space using the Weil conjectures. 20 years later, a Madsen and Weiss determined the homology of the moduli space of Riemann surfaces, in the limit where the genus of the surface goes to infinity.
My talk will combine these two spaces: I will describe the homology of the moduli space of Riemann surfaces S, equipped with a flat G-bundle E -> S, where we allow both the flat bundle and the surface to vary. I will start by reviewing parts of the Atiyah-Bott and Madsen-Weiss papers. Our main theorem will then be a rather easy consequence. This is joint work with Nitu Kitchloo and Ralph Cohen.
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Representation of Quantum Groups and new invariants of links
Abstract
The colored HOMFLY polynomial is a quantum invariant of oriented links in S³ associated with a collection of irreducible representations of each quantum group U_q(sl_N) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure, which is the part of Labastida-Marino-Ooguri-Vafa conjecture. The new integer invariants are also predicted by the LMOV conjecture and recently has been proved. LMOV also give the application of Licherish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial could also be developed in a same fashion by using more complicated algebra method.
In a joint work with Lin Chen and Nicolai Reshetikhin, we rigorously formulate the orthogonal quantum group version of LMOV conjecture in mathematics by using the representation of Brauer centralizer algebra. We also obtain formulae of Lichorish-Millet type which could be viewed as the application in knot theory and topology. By using the cabling technique, we obtain a uniform formula of colored Kauffman polynomial for all torus links with all partitions. Combined these together, we are able to prove many interesting cases of this orthogonal LMOV conjecture.
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Applications of the Cobordism Hypothesis
Abstract
In this lecture, I will illustrate the cobordism hypothesis by presenting some examples. Exact content to be determined, depending on the interests of the audience.