Past GATA seminars
| 2008-03-18 | Tue | Alastair King (Bath) | |
| 15:00 | J11 | Dimers, quivers and Calabi-Yau algebras | |
| 2008-03-05 | Wed | Ezra Getzler (Northwestern) | |
| 14:00 | J11 | Operads revisited | |
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Abstract: I give a new formulation of the axioms for operads, which covers all of the variants of the theory, such as cyclic operads, modular operads, PROPs, wheeled PROPs, as well as simplicially enriched variants, such as topological field theories of different types. |
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| 2008-03-04 | Tue | Ezra Getzler (Northwestern) | |
| 14:00 | J11 | Lie theory for L-infinity algebras (Part II) | |
| 2008-02-20 | Wed | Ezra Getzler (Northwestern) | |
| 14:00 | J11 | Lie theory for L-infinity algebras | |
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Abstract: I show how to associate to a nilpotent differential graded Lie algebra (or, more generally, L-infinity algebra) concentrated in degrees >-n an n-groupoid. This construction generalizes the case of a Lie algebra (it gives the associated Lie group) and when the dg Lie algebra is abelian (i.e. a chain complex), it becomes the Eilenberg-MacLane space of the complex. |
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| 2008-02-19 | Tue | Ezra Getzler (Northwestern) | |
| 16:00 | J11 | Open-closed topological field theory in 2 dimensions | |
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Abstract: We use Deligne and Mumford's compactification of moduli spaces of Riemann surfaces, and its generalization to Riemann surfaces with boundary introduced by Liu, to analyse the structure of two-dimensional topological field theories. |
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| 2007-06-08 | Fri | Craig Huneke (University of Kansas) | |
| 16:00 | J11 | Absolute integral closures in mixed characteristic and characteristic p (GATA lecture III) | |
| 2007-06-07 | Thu | Craig Huneke (University of Kansas) | |
| 16:00 | J11 | Reduction to characteristic p, and further refinements of vanishing along algebraic subsets (GATA lecture II) | |
| 2007-06-06 | Wed | Craig Huneke (University of Kansas) | |
| 16:00 | J11 | How many times does a polynomial vanish at a point? (GATA lecture I) | |
| 2007-05-23 | Wed | Andrei Caldararu (Wisconsin) | |
| 16:00 | J11 | The Pfaffian-Grassmannian derived equivalence | |
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Abstract: We argue that there exists a derived equivalence between Calabi-Yau threefolds obtained by taking hyperplane sections (of the appropriate codimension) of the Grassmannian G(2,7) and the Pfaffian Pf(7). The existence of such an equivalence has been conjectured by physicists for almost ten years, as the two families of Calabi-Yau threefolds are believed to have the same mirror. It is the first example of a derived equivalence between Calabi-Yau threefolds which are provably non-birational. |
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| 2007-05-22 | Tue | Andrei Caldararu (Wisconsin) | |
| 15:45 | J11 | The Mukai pairing on Hochschild homology | |
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Abstract: Given a Calabi-Yau three-fold X, string theory constructs two so-called topological twists, the A-model and the B-model. A piece of the mathematical incarnation of the A-model is the singular cohomology ring of X (or its quantum deformation). The corresponding piece in the B-model is encoded by the Hochschild cohomology ring of X. Physics predicts both sets of data are Frobenius algebras, i.e., they are endowed with a non-degenerate pairing. In the A-model, this is given by the Poincare pairing on cohomology. In my talk I shall discuss the construction of the corresponding pairing on Hochschild (co)homology. I shall also discuss several important properties of this pairing, including the Cardy condition from open-closed topological string theory. |
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| 2006-11-21 | Tue | Mikhail Kapranov (Yale University) | |
| 14:00 | J11 | Talk 3: "Spaces of formal loops and gerbes of chiral differential operators." | |
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Abstract: For a complex manifold X physics considerations lead to a construction of certain sheaves of vertex algebras on X called chiral differential operators (CDO). While locally such a sheaf is unique, globally the situation is similar to that of spinor bundles on a Riemannian manifold. In the categorical terminology they form a gerbe. We relate this gerbe with the gerbe describing 'determinantal anomaly' for the space of free loops in X. The calculation of the class of the gerbe of CDO due to Gorbounov, Malikov and Schechtman turns out to be a particular case of a local Riemann-Roch-type theorem for determinantal gerbes. |
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| 2006-11-16 | Thu | Mikhail Kapranov (Yale University) | |
| 15:10 | J11 | Talk 2: "Spaces of formal loops and gerbes of chiral differential operators." | |
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Abstract: For a complex manifold X physics considerations lead to a construction of certain sheaves of vertex algebras on X called chiral differential operators (CDO). While locally such a sheaf is unique, globally the situation is similar to that of spinor bundles on a Riemannian manifold. In the categorical terminology they form a gerbe. We relate this gerbe with the gerbe describing 'determinantal anomaly' for the space of free loops in X. The calculation of the class of the gerbe of CDO due to Gorbounov, Malikov and Schechtman turns out to be a particular case of a local Riemann-Roch-type theorem for determinantal gerbes. |
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| 2006-11-13 | Mon | Mikhail Kapranov (Yale University) | |
| 14:00 | J11 | Talk 1: "Spaces of formal loops and gerbes of chiral differential operators" | |
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Abstract: For a complex manifold X physics considerations lead to a construction of certain sheaves of vertex algebras on X called chiral differential operators (CDO). While locally such a sheaf is unique, globally the situation is similar to that of spinor bundles on a Riemannian manifold. In the categorical terminology they form a gerbe. We relate this gerbe with the gerbe describing 'determinantal anomaly' for the space of free loops in X. The calculation of the class of the gerbe of CDO due to Gorbounov, Malikov and Schechtman turns out to be a particular case of a local Riemann-Roch-type theorem for determinantal gerbes. |
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| 2006-06-07 | Wed | Bertrand Toen (Toulouse) | |
| 15:00 | J11 | Stacks and derived categories II | |
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Abstract: The second talk is concerned with the problem of constructing a reasonable moduli space for triangulated (dg)-categories themselves. The main theorem of this second talk states that the (derived ∞-) stack of "saturated dg-categories" is algebraic. The infinitesimal theory of this moduli stack can be used to explain the relation between the deformation theory of dg-categories and Hochschild cohomology. Two other applications will be discussed. To start with I will describe, for any given rational number p/q, a circle action on the moduli stack of saturated dg-categories whose fixed points are "Calabi-Yau dg-categories of dimension p/q". This can be used to prove that the deformation theory of Calabi-Yau dg-categories is controlled by cyclic cohomology. Finally, I will explain how the "period map", from the stack of varieties to the stack of saturated dg-categories, can be used to study derived equivalence classes of algebraic varieties. |
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| 2006-06-06 | Tue | Bertrand Toen (Toulouse) | |
| 15:00 | J11 | Stacks and derived categories I | |
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Abstract: The purpose of these two talks is to report on recent works which use stack theory to study derived categories. In the first talk I will discuss the problem of constructing a reasonable moduli space for compact objects in a given triangulated category (or rather a triangulated "dg-category"). In a first part I will explain some motivations coming from algebraic geometry and representation theory (e.g. the contruction of moduli spaces of complexes of sheaves on an algebraic variety, the definition of "Hall algebras" for derived categories). The second part of the talk will be devoted to present a solution to this problem using a notion of "derived ∞-stack": the main theorem states that the (derived ∞-) stack of compact objects in a given "saturated" dg-category is algebraic. Some corollaries and possible future applications will be discussed. |
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