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Spring Semester 2014

Date / Time Speaker Title Location
6 March 2014
15:15-16:15 		Grigory Mikhalkin
Université de Genève
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Talks in Mathematical Physics

Title Algebraic knots and links in the tropical limit
Speaker, Affiliation Grigory Mikhalkin, Université de Genève
Date, Time 6 March 2014, 15:15-16:15
Location HG G 43
Abstract Algebraic curves in RP3 (the real projective space) may be viewed as knots and links. Unlike their conventional (topological) counterparts they come naturally enhanced with additional structure, such as the canonical framing. We look at these algebraic knots after passing to the so-called tropical limit which collapses tropical curves onto metric graphs, but retains information of their topological type and framing as certain combinatorial data. Applications include computation of 3d-Welschinger invariants.
Algebraic knots and links in the tropical limitread_more
HG G 43
13 March 2014
15:15-16:15 		Antti Knowles
ETH Zurich
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Talks in Mathematical Physics

Title The Altshuler-Shklovskii formulas for random band matrices
Speaker, Affiliation Antti Knowles, ETH Zurich
Date, Time 13 March 2014, 15:15-16:15
Location HG G 43
Abstract We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the variance and the two-point correlation function are governed by a universal power law behaviour that differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii, and describes the eigenvalue density correlations in general metallic samples with weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an arithmetic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner.
The Altshuler-Shklovskii formulas for random band matricesread_more
HG G 43
27 March 2014
15:15-16:15 		Yaël Frégier
University of Zurich
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Talks in Mathematical Physics

Title Homotopy Lie algebras governing simultaneous deformations
Speaker, Affiliation Yaël Frégier, University of Zurich
Date, Time 27 March 2014, 15:15-16:15
Location HG G 43
Abstract In this talk we will present a way to explicitly construct L-infinity algebras governing deformation problems for which several structures are simultaneously deformed. The typical examples include simultaneous deformations of two algebras and a morphism between them, or, in a geometrical context, of Poisson manifolds and their coisotropic submanifolds. This is a joint work with Marco Zambon (Madrid), and is based on a derived brackets construction due to T. Voronov.
Homotopy Lie algebras governing simultaneous deformationsread_more
HG G 43
10 April 2014
15:15-16:15 		Pavel Mnev
Universität Zürich
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Talks in Mathematical Physics

Title Classical free boson on Lorentzian surfaces with boundary
Speaker, Affiliation Pavel Mnev, Universität Zürich
Date, Time 10 April 2014, 15:15-16:15
Location HG G 43
Abstract For a large class of classical field theories on manifolds with boundary, solutions of the Euler-Lagrange equations induce a Lagrangian submanifold in the phase space associated to the boundary. E.g. this is so for theories admitting a Hamiltonian formulation, considered on cylinders. We will consider a system which does not admit a reasonable Hamiltonian description - 2D free massless boson on a compact domain on Minkowski plane, and will sketch the proof that Lagrangianity of the evolution relation still holds (which provides grounds for the geometric quantization of this system). On more general Lorentzian surfaces however the evolution relation may fail to be Lagrangian, e.g. in case of the Misner cylinder. The exposition is based on a joint work with Alberto S. Cattaneo, arxiv:1308.5592.
Classical free boson on Lorentzian surfaces with boundaryread_more
HG G 43
24 April 2014
15:15-16:15 		Alexandr Buryak
ETH Zurich
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Talks in Mathematical Physics

Title Double ramification cycles and integrable hierarchies
Speaker, Affiliation Alexandr Buryak, ETH Zurich
Date, Time 24 April 2014, 15:15-16:15
Location HG G 43
Abstract We will present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. It is based on an integration over double ramification cycles and is motivated by the Symplectic Field Theory. We conjecture that in the semisimple case our hierarchy is related to the Dubrovin-Zhang hierarchy by a Miura transformation and check it in several examples.
Double ramification cycles and integrable hierarchiesread_more
HG G 43
8 May 2014
15:15-16:15 		Jörg Teschner
DESY Hamburg
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Talks in Mathematical Physics

Title Conformal blocks for non-rational CFT
Speaker, Affiliation Jörg Teschner, DESY Hamburg
Date, Time 8 May 2014, 15:15-16:15
Location HG G 43
Abstract We revisit the representation-theoretic definition of the Virasoro conformal blocks. This construction yields infinite-dimensional vector spaces in general. Our goal will be to explain how to define bases for interesting topological subspaces of the spaces of conformal blocks by means of the gluing construction. This yields a first example where the Friedan-Shenker program can be realized for a non-rational conformal field theory.
Conformal blocks for non-rational CFTread_more
HG G 43
15 May 2014
15:15-16:15 		Ben Webster
University of Virginia and Paris 7
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Talks in Mathematical Physics

Title On isomorphisms of categorified sl_n knot invariants
Speaker, Affiliation Ben Webster, University of Virginia and Paris 7
Date, Time 15 May 2014, 15:15-16:15
Location HG G 43
Abstract Abstract: the '00's were marked by the appearance of a remarkable number of homological knot invariants categorifying Reshetikhin-Turaev invariants for sl_n. Khovanov's original homologies for sl_2 and sl_3, Khovanov-Rozansky homology for sl_n and its generalization by Wu and Yonezawa, the foam invariants defined by Mackaay, Stosic and Vaz, the invariants of Cautis and Kamnitzer based on the geometry of the affine Grassmannian, Mazorchuk, Stroppel and Sussan using category O, colored Jones homology defined using the categorified Jones-Wenzl projector of Cooper and Krushkal and invariants defined by myself using categorifications of tensor products. Unfortunately, we got a bit ahead of ourselves, and for a long time didn't know which of these were the same, and which ones potentially different. Luckily, these issues are now resolved, based on observations of Cautis and Kamnitzer, developed further by Lauda, Queffelec and Rose. All of these homologies are the same, because all of them are controlled by a categorification of sl_\infty and the Chuang-Rouquier braid group inside it. I'll try to explain this general framework, and how one can check it applies in the case of categorifications of tensor products, requiring the use of a categorical skew Howe duality (work in progress, joint with Marco Mackaay).
On isomorphisms of categorified sl_n knot invariantsread_more
HG G 43
22 May 2014
15:15-16:15 		Ivan Contreras
Berkeley
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Talks in Mathematical Physics

Title Lagrangian correspondences and extensions for the Poisson sigma model
Speaker, Affiliation Ivan Contreras, Berkeley
Date, Time 22 May 2014, 15:15-16:15
Location HG G 43
Abstract In arXiv:1401.7319 we define relational symplectic groupoids and we show that they integrate Poisson manifolds, in a compatible way with the usual integration via symplectic groupoids, from the phase space construction in the Poisson sigma model (PSM). This new object is described in an extended symplectic categroy of weak symplectic manifolds and Lagrangian correspondences. In this talk, we will describe the construction and and show that this gives rise naturally to two different extensions for PSM: the case of higher genus and a description of PSM as an extended topological field theory (a la Lurie). This is based in work in progress with Alberto Cattaneo and with Claudia Scheimbauer. ---
Lagrangian correspondences and extensions for the Poisson sigma modelread_more
HG G 43
* 26 May 2014
17:15-18:15 		Dmitry Roytenberg
MPI Bonn
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Talks in Mathematical Physics

Title Equivalence of models of "up to homotopy" algebras of differentiable functions
Speaker, Affiliation Dmitry Roytenberg, MPI Bonn
Date, Time 26 May 2014, 17:15-18:15
Location Y27 H 28
Abstract Whereas coordinate rings of algebraic varieties are commutative algebras, coordinate rings of differentiable manifolds have an important extra structure: they are so-called C-infinity algebras. In the derived setting, coordinate rings additionally acquire a homotopy type; thus, coordinate rings of derived manifolds can be modeled either by simplicial C-infinity algebras, or "differential graded" ones (understood in an appropriate sense). We construct a Quillen equivalence between these two models in a conceptually elegant way, using finite Grassmann algebras as a "kernel". This generalizes and sheds new light on an old result of Quillen about commutative algebras and Lie algebras.
Equivalence of models of "up to homotopy" algebras of differentiable functions read_more
Y27 H 28
9 July 2014
11:15-12:15 		Alexander Goncharov
Yale University
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Talks in Mathematical Physics

Title Representation theory and mirror symmetry I
Speaker, Affiliation Alexander Goncharov, Yale University
Date, Time 9 July 2014, 11:15-12:15
Location HG G 43
Abstract We develop a uniform way to parametrise canonical bases in various vector spaces which appear in representation theory, for instance invariants in tensor products of representations of a reductive group G. In all cases there is an underlying dual pair of moduli spaces M(G) and M*(G^L) related to the group G and the Langlands dual group G^L. Each of the spaces has a natural positive structure. Furthermore there is a positive function W on one the space M(G), the potential. The pair (M(G), W) allows to define a set of positive integral tropical points. We show that it parametrise a canonical bases in the vector space of regular functions on the dual space. We suggest that this parametrisation is a manifestation of homological mirror symmetry between the Landay-Ginzburg model on (M(G), W) and the dual space. For example, in the case when the vector spaces are finite dimensional representations of G, we get the Mirkovic-Vilonen basis. The corresponding mirror picture for SL(n) turns out to be just the one developed by Givental (1994) describing quantum cohomology of flag varieties via Toda integrable systems. For other groups we get a similar description developed by Rietsch, Gerasimov-Lebedev-Oblezin, and others. This is a joint work with Linhui Shen (Northwestern University).
Representation theory and mirror symmetry Iread_more
HG G 43
9 July 2014
14:15-15:15 		Alexander Goncharov
Yale University
Details

Talks in Mathematical Physics

Title Representation theory and mirror symmetry II
Speaker, Affiliation Alexander Goncharov, Yale University
Date, Time 9 July 2014, 14:15-15:15
Location HG G 43
Abstract We develop a uniform way to parametrise canonical bases in various vector spaces which appear in representation theory, for instance invariants in tensor products of representations of a reductive group G. In all cases there is an underlying dual pair of moduli spaces M(G) and M*(G^L) related to the group G and the Langlands dual group G^L. Each of the spaces has a natural positive structure. Furthermore there is a positive function W on one the space M(G), the potential. The pair (M(G), W) allows to define a set of positive integral tropical points. We show that it parametrise a canonical bases in the vector space of regular functions on the dual space. We suggest that this parametrisation is a manifestation of homological mirror symmetry between the Landay-Ginzburg model on (M(G), W) and the dual space. For example, in the case when the vector spaces are finite dimensional representations of G, we get the Mirkovic-Vilonen basis. The corresponding mirror picture for SL(n) turns out to be just the one developed by Givental (1994) describing quantum cohomology of flag varieties via Toda integrable systems. For other groups we get a similar description developed by Rietsch, Gerasimov-Lebedev-Oblezin, and others. This is a joint work with Linhui Shen (Northwestern University).
Representation theory and mirror symmetry IIread_more
HG G 43

Notes: events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Organisers: Anna Beliakova, Alberto Cattaneo, Giovanni Felder, Matthias Gaberdiel, Gian Michele Graf, Horst Knörrer, Thomas Hans Willwacher

Archive: AS 24 SS 24 AS 23 SS 23 AS 22 SS 22 AS 21 SS 21 AS 20 SS 20 AS 19 SS 19 AS 18 SS 18 AS 17 SS 17 AS 16 SS 16 AS 15 SS 15 AS 14 SS 14 AS 13 SS 13 AS 12 SS 12 AS 11

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