Symplectic Khovanov cohomology
Speaker: Ivan Smith
Department: Cambridge
Time: 2013-04-10 13:30 - 14:30
Location: Polhemsalen, Ångströmlaboratoriet
Symplectic Khovanov cohomology is a Floer-theoretic invariant of oriented links in the three-sphere, conjecturally isomorphic to its combinatorial sibling. I will outline a partial proof of that conjecture in characteristic zero. The key ingredient is a formality theorem for the Fukaya categories of certain symplectic manifolds arising in Lie theory; the proof of formality is motivated in part by ideas from homological mirror symmetry. This talk reports on joint work with Mohammed Abouzaid.
Advances in the theory of knot polynomials
Speaker: Alexei Morozov
Department: ITEP, Moscow
Time: 2013-04-10 15:00 - 16:00
Location: Polhemsalen, Ångströmlaboratoriet
Knot polynomials are Wilson-loop averages in topological Chern-Simons theory and its generalisations. They depend on a variety of variables and satisfy rich set of equations - only few of which are already known. Generating functions of knot polynomials belong to the class of Hurwitz partition functions with non-trivial integrability properties. We review technical approaches, which allow to study these relations and connect knot theory to other branches of theoretical physics.
McKay correspondence and the partition functions of 4d N=2 gauge theories
Speaker: Vasily Pestun
Department: IAS, Princeton
Time: 2012-10-03 13:30 - 14:30
Location: Polhemsalen, Ångströmlaboratoriet
To any discrete subgroup of SU(2), or an affine ADE quiver, equipped with a certain data, there is an associated 4d N=2 gauge theory partition function, describing equivariant cohomology of the moduli space of self-dual connections on a four-manifold and defined combinatorially on the space of multi-colored partitions. For all such theories, the asymptotics of this partition function is found explicitly in terms of the periods of certain algebraic integrable system associated to the moduli of holomorphic ADE bundles on elliptic curves.
The geometric construction of the Reshetikhin-Turaev Topological Quantum Field Theory
Speaker: Jørgen Ellegaard Andersen
Department: Aarhus University
Time: 2012-10-03 15:00 - 16:00
Location: Polhemsalen, Ångströmlaboratoriet
In this talk, we will discuss the geometric construction of the Reshetikhin-Turaev Topological Quantum Field Theory using the geometric quantization of the moduli spaces of flat connections on two dimensional surfaces. We will then discuss various results on the large level asymptotics of these theories.
Recent progress on four-dimensional symplectic embedding problems
Speaker: Michael Hutchings, UC Berkeley
Time: 2012-03-26 13:30 - 14:30
Location: Häggsalen, Ångströmlaboratoriet
We discuss recent results on the problem of when one four-dimensional symplectic manifold (usually with boundary) can be symplectically embedded into another. For example, Dusa McDuff proved a number-theoretic criterion for when one four-dimensional ellipsoid can be symplectically embedded into another. Numerical invariants called "ECH capacities" give general obstructions to four-dimensional symplectic embeddings which turn out to be sharp in the case of ellipsoids.
String-theory applications of integrable systems
Speaker: Konstantin Zarembo, Nordita
Time: 2012-03-26 15:00 - 16:00
Location: Häggsalen, Ångströmlaboratoriet
Integrability plays an important role in many areas of physics, yielding exact results for systems where interactions may be arbitrary strong and difficult to handle by any other means. One example is string theory, where integrability has led to important insights into the nature of gauge/string dualities.
Rigidity and Flexibility in Symplectic Geometry
Speaker: Yakov Eliashberg
Department: Stanford
Time: 2011-09-06 12:30 - 14:30
Location: Å80101, Ångströmlaboratoriet
Symplectic topology was born in the 1980s on the borderline between the worlds of rigid and flexible mathematics. In the talk I will describe some recent advances on both sides of the border. On the flexible side, I will discuss a surprising h-principle for Legendrian knots of dimension >1, proven by my student Max Murphy, as well as its consequences for symplectic topology of Stein manifolds. On the rigid side, I will discuss effective techniques for computing symplectic invariants of Stein manifolds via Legendrian surgery. The flexible side of the story is joint work with K.
A piece of 21st century mathematics that didn't make it into 20th century physics
Speaker: Sergei Gukov
Department: Caltech
Time: 2011-09-06 15:00 - 17:00
Location: Å80101, Ångströmlaboratoriet
This lecture will be about the physics of new knot invariants invented by Khovanov, Rozansky, Ozsvath, Szabo, Rasmussen, and other mathematicians circa 2000. The key ingredients of the proposed physical framework involve standard tools from gauge theory and string theory. It leads to a wealth of generalizations and comes with a few surprising features.
December 16, 2010
13:30-14:30, Polhemsalen
Marcos Mariño (University of Geneva)
Quantum theory and enumerative problems
In quantum theory we are very often interested in counting objects. For example, to understand perturbation theory in quantum field theory we have to count graphs by using combinatorial techniques, while in string theory we want to count curves in manifolds, by using algebraic geometry. In this talk I will explore some of these enumerative problems, starting with simple examples in quantum theory and ending with string theory. I will also use Chern-Simons theory and its mathematical incarnation (the LMO invariant) to discuss the interplay between counting problems and the 1/N expansion. I will be particularly interested in the connection between non-perturbative effects in quantum theory and the asymptotic behavior of these enumerative problems. This will lead to some interesting applications of instanton techniques in string theory and large N theory to enumerative mathematics.
December 16, 2010
15:00-16:00, Polhemsalen
Ezra Getzler (Northwestern Univ, Chicago)
Derived brackets in generalized geometry
The derived bracket of Koszul is an algebraic construction which associates to a differential on a graded Lie algebra a "derived" Lie algebra. This turns out to generalize the way in which the Poisson bracket is associated to a Poisson tensor on a manifold.
In their important paper, Roytenberg and Weinstein generalized this construction so that it works for Courant algebroids. (These are related to Lie algebroids and arise, for example, in generalized complex geometry.) Whereas in the Poisson case, the graded Lie algebra is concentrated in degrees -1,0,1,..., in the new setting, the graded Lie algebra is concentrated in degrees -2,-1,0,1,..., and the construction of Roytenberg and Weinstein yields a Lie 2-algebra, that is, an L-infinity algebra concentrated in degrees 0 and -1.
In this talk, I generalize the construction of Roytenberg and Weinstein. I show that there is a sequence of operations on the negatively graded part of a differential graded algebra making it into an L-infinity algebra. The formulas for the higher brackets involve Bernoulli numbers (see arXiv:1010.5859)
May 6, 2010, Häggsalen
13:30 - 14:30
Generalized holomorphic bundles
Nigel Hitchin, (Oxford)
Location:
Häggsalen, Ångströmlaboratoriet
Type :
Theoretical Physics
On a manifold with a generalized complex structure there is a natural notion of generalized holomorphic bundle introduced by Gualtieri. In the case of a symplectic structure this is just a bundle with flat connection, but for an ordinary complex structure, viewed from the generalized point of view, it becomes an interesting holomorphic object. We shall discuss this case, and also the case for a holomorphic Poisson structure, considering in particular the role of B-field transformations.
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Geometry and string duality
Speaker:
Chris Hull
Department:
Imperial College
Time:
2010-05-06 15:00 - 16:00
Location:
Häggsalen, Ångströmlaboratoriet
Type :
Theoretical Physics
Generalized geometry provides a natural framework for studying d-dimensional manifolds equipped with a metric and B-field. In this approach the tangent bundle is "doubled" to T+T* and there is a natural action of O(d,d) on the geometry. The group O(d,d) also arises naturally in string theory for backgrounds that are d-torus bundles. Dimensional reduction on the torus fibres gives a truncation to an effective theory with an O(d,d) symmetry.
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