Past Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar (past)

Tue, 07/05
15:45 		Jesse Wolfson (Northwestern) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Descent for n-Bundles
Given a Lie group $ G $, one can construct a principal $ G $-bundle on a manifold $ M $ by taking a cover $ U\to M $, specifying a transition cocycle on the cover, and then descending the trivialized bundle $ U \times G $ along the cocycle. We demonstrate the existence of an analogous construction for local $ n $-bundles for general $ n $. We establish analogues for simplicial Lie groupoids of Moore's results on simplicial groups; these imply that bundles for strict Lie $ n $-groupoids arise from local $ n $-bundles. We conclude by constructing a simple finite dimensional model of the Lie 2-group String($ n $) using cohomological data.
Thu, 02/05
14:00 		Ed Segal (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L2
Sheafy matrix factorizations and bundles of quadrics
A Landau-Ginzburg B-model is a smooth scheme $ X $, equipped with a global function $ W $. From $ (X,W) $ we can construct a category $ D(X,W) $, which is called by various names, including ‘the category of B-branes’. In the case $ W=0 $ it is exactly the derived category $ D(X) $, and in the case that $ X $ is affine it is the category of matrix factorizations of $ W $. There has been a lot of foundational work on this category in recent years, I’ll describe the most modern and flexible approach to its construction. I’ll then interpret Nick Addington’s thesis in this language. We’ll consider the case that $ W $ is a quadratic form on a vector bundle, and the corresponding global version of Knorrer periodicity. We’ll see that interesting gerbe structures arise, related to the bundle of isotropic Grassmannians.
Tue, 30/04
15:45 		Jonny Evans (University College London) Algebraic and Symplectic Geometry Seminar Add to calendar L2
Unlinking and unknottedness of monotone Lagrangian submanifolds
I will explain some recent joint work with Georgios Dimitroglou Rizell in which we use moduli spaces of holomorphic discs with boundary on a monotone Lagrangian torus in $ {\mathbb C}^n $ to prove that all such tori are smoothly isotopic when $ n $ is odd and at least 5
Tue, 23/04
15:45 		Richard Rimanyi (University of North Carolina) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Equivariant classes, COHA, and quantum dilogarithm identities for Dynkin quivers II
Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE. A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials. The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.
Tue, 23/04
14:00 		Richard Rimanyi (University of North Carolina) Algebraic and Symplectic Geometry Seminar Add to calendar L1
Equivariant classes, COHA, and quantum dilogarithm identities for Dynkin quivers I
Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE. A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials. The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.
Tue, 05/03
15:45 		Frances Brown (visiting Newton Institute) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Modular forms and multiple zeta values
Thu, 21/02
15:30 		Anatoly Preygel (UC Berkeley) Algebraic and Symplectic Geometry Seminar Add to calendar L2
Centers and traces of categorified affine Hecke algebras (or, some tricks with coherent complexes on the Steinberg variety)
The bounded coherent dg-category on (suitable versions of) the Steinberg stack of a reductive group G is a categorification of the affine Hecke algebra in representation theory.  We discuss how to describe the center and universal trace of this monoidal dg-category.  Many of the techniques involved are very general, and the description makes use of the notion of "odd micro-support" of coherent complexes.  This is joint work with Ben-Zvi and Nadler.
Thu, 14/02
14:00 		Stephane Guillermou (Grenoble) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Microlocal sheaf theory and symplectic geometry III
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $ M $ and the symplectic geometry of the cotangent bundle of $ M $ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $ \Lambda $ are deduced from the existence of a sheaf with microsupport $ \Lambda $, which we call a quantization of $ \Lambda $. In the third talk we will see that $ \Lambda $ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $ \Lambda $.
Wed, 13/02
14:00 		Stephane Guillermou (Grenoble) Algebraic and Symplectic Geometry Seminar Add to calendar L1
Microlocal sheaf theory and symplectic geometry II
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $ M $ and the symplectic geometry of the cotangent bundle of $ M $ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $ \Lambda $ are deduced from the existence of a sheaf with microsupport $ \Lambda $, which we call a quantization of $ \Lambda $. In the second talk we will introduce a stack on $ \Lambda $ by localization of the category of sheaves on $ M $. We deduce topological obstructions on $ \Lambda $ for the existence of a quantization.
Tue, 12/02
15:45 		Stephane Guillermou (Grenoble) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Microlocal sheaf theory and symplectic geometry I
Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $ M $ and the symplectic geometry of the cotangent bundle of $ M $ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $ \Lambda $ are deduced from the existence of a sheaf with microsupport $ \Lambda $, which we call a quantization of $ \Lambda $. In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.
Tue, 05/02
15:45 		Jake Solomon (Jerusalem) Algebraic and Symplectic Geometry Seminar Add to calendar L3
The space of positive Lagrangian submanifolds
A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. A Hamiltonian isotopy class of positive Lagrangian submanifolds admits a Riemannian metric with non-positive curvature. Its universal cover admits a functional, with critical points special Lagrangians, that is strictly convex with respect to the metric. If time permits, I'll explain how mirror symmetry relates the metric and functional to the infinite dimensional symplectic reduction picture of Atiyah, Bott, and Donaldson in the context of the Kobayashi-Hitchin correspondence.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

Syndicate content