Past Seminars › Seminar Series

Algebraic and Symplectic Geometry Seminar (past)

Tue, 08/01
15:45 		Jinwon Choi (University of Illinois at Urbana Champaign) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Refined stable pair invariants on local Calabi-Yau threefolds
A refinement of the Pandharipande-Thomas stable pair invariants for local toric Calabi-Yau threefolds is defined by what we call the virtual Bialynicki-Birula decomposition. We propose a product formula for the generating function for the refined stable pair invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local $ {\bf P}^1 $. I will also describe how the proposed product formula is related to the wall crossing in my first talk. This is joint work with Sheldon Katz and Albrecht Klemm.
Tue, 08/01
14:00 		Jinwon Choi (University of Illinois at Urbana Champaign) Algebraic and Symplectic Geometry Seminar Add to calendar L3
On the moduli spaces of stable pairs on the projective plane
We study the birational relationship between the moduli spaces of $ \alpha $-stable pairs and the moduli space $ M(d,1) $ of stable sheaves on $ {\bf P}^2 $ with Hilbert polynomial $ dm+1 $. We explicitly relate them by birational morphisms when $ d=4 $ and $ 5 $, and we describe the blow-up centers geometrically. As a byproduct, we obtain the Poincare polynomials of the moduli space of stable sheaves, or equivalently the refined BPS index. This is joint work with Kiryong Chung.
Tue, 27/11/2012
15:45 		Andrei Caldararu (University of Wisconsin) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Formality of ordinary and twisted de Rham complex from derived algebraic geometry
Beautiful results of Deligne-Illusie, Sabbah, and Ogus-Vologodsky show that certain modifications of the de Rham complex (either the usual one, or twisted versions of it that appear in the study of the cyclic homology of categories of matrix factorizations) are formal in positive characteristic. These are the crucial steps in proving algebraic analogues of the Hodge theorem (again, either in the ordinary setting or in the presence of a twisting). I will present these results along with a new approach to understanding them using derived intersection theory. This is joint work with Dima Arinkin and Marton Hablicsek.
Tue, 20/11/2012
15:45 		Ed Segal (Imperial) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
SEMINAR CANCELLED
SEMINAR CANCELLED
Tue, 13/11/2012
15:45 		Dennis Borisov (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Derived manifolds and d-manifolds
Tue, 06/11/2012
15:45 		Yu-Jong Tzeng (Harvard) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Enumeration of singular curves with tangency conditions
How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my recent work on the generalization of the above problem to count singular curves with given tangency condition to a fixed smooth divisor on general surfaces. I will relate the enumeration to tautological integrals on Hilbert schemes of points and show the numbers of curves in question are given by universal polynomials. As a result, we can obtain infinitely many new formulas for nodal curves and understand the asymptotic behavior for all singular curves with any tangency conditions.
Tue, 30/10/2012
15:45 		Oren Ben-Bassat (Oxford and Haifa) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Sub-varieties and Descent
Let $ X $ be a variety and $ Z $ be a sub-variety. Can one glue vector bundles on $ X-Z $ with vector bundles on some small neighborhood of $ Z $? We survey two recent results on the process of gluing a vector bundle on the complement of a sub-variety with a vector bundle on some 'small' neighborhood of the sub-variety. This is joint work. The first with M. Temkin and is about gluing categories of coherent sheaves over the category of coherent sheaves on a Berkovich analytic space. The second with J. Block and is about gluing dg enhancements of the derived category of coherent sheaves.
Tue, 23/10/2012
15:45 		Arend Bayer (Edinburgh) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Birational geometry of moduli of sheaves on K3's via Bridgeland stability
I will explain recent results with Emanuele Macrì, in which we systematically study the birational geometry of moduli of sheaves on K3's via wall-crossing for Bridgeland stability conditions. In particular, we obtain descriptions of their nef cones via the Mukai lattice of the K3, their moveable cones, their divisorial contractions, and obtain counter-examples to various conjectures in the literature. We also give a proof of the Lagrangian fibration conjecture (due to Hassett-Tschinkel/Huybrechts/Sawon) via wall-crossing.
Tue, 16/10/2012
15:45 		Martijn Kool (Imperial College London) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Reduced classes and curve counting on surfaces
Counting nodal curves in linear systems $ |L| $ on smooth projective surfaces $ S $ is a problem with a long history. The Göttsche conjecture, now proved by several people, states that these counts are universal and only depend on $ c_1(L)^2 $, $ c_1(L)\cdot c_1(S) $, $ c_1(S)^2 $ and $ c_2(S) $. We present a quite general definition of reduced Gromov-Witten and stable pair invariants on S. The reduced stable pair theory is entirely computable. Moreover, we prove that certain reduced Gromov-Witten and stable pair invariants with many point insertions coincide and are both equal to the nodal curve counts appearing in the Göttsche conjecture. This can be seen as version of the MNOP conjecture for the canonical bundle $ K_S $. This is joint work with R. P. Thomas.
Thu, 11/10/2012
16:00 		Tom Bridgeland (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar Gibson Grd floor SR
Quadratic differentials as stability conditions
Thu, 11/10/2012
12:00 		Dmitry Kaledin (Moscow) Algebraic and Symplectic Geometry Seminar Add to calendar Gibson Grd floor SR
Hochschild-Witt complex
The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves $ W^*(X) $ on a smooth algebraic variety $ X $ over a finite field, computing the cristalline cohomology of $ X $. I am going to present a non-commutative generalization of this: even for a non-commutative ring $ A $, one can define a functorial "Hochschild-Witt complex" with homology $ WHH^*(A) $; if $ A $ is commutative, then $ WHH^i(A)=W^i(X) $, $ X = Spec A $ (this is analogous to the isomorphism $ HH^i(A)=H^i(X) $ discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.
Tue, 09/10/2012
14:00 		Zheng Hua (Kansas State) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
Donaldson-Thomas theory of toric CY 3-folds I
I will explain an approach to study DT theory of toric CY 3-folds using $ L_\infty $ algebras. Based on the construction of strong exceptional collection of line bundles on Fano toric stack of dimension two, we realize any bounded families of sheaves on local surfaces support on zero section as critical sets of the Chern-Simons functions. As a consequence of this construction, several interesting properties of DT invariants on local surfaces can be checked.
Tue, 02/10/2012
14:00 		Olivier Schiffmann (Jussieu) Algebraic and Symplectic Geometry Seminar Add to calendar SR1
$ W $-algebras and moduli spaces of sheaves on $ A^2 $ I
Motivated by a conjecture of Alday, Gaiotto and Tachikawa (AGT conjecture), we construct an action of a suitable $ W $-algebra on the equivariant cohomology of the moduli space $ M_r $ of rank r instantons on $ A^2 $ (i.e. on the moduli space of rank $ r $ torsion free sheaves on $ P^2 $, trivialized at the line at infinity). We show that the resulting $ W $-module is identified with a Verma module, and the characteristic class of $ M_r $ is the Whittaker vector of that Verma module. One of the main ingredients of our construction is the so-called cohomological Hall algebra of the commuting variety, which is a certain associative algebra structure on the direct sum of equivariant cohomology spaces of the commuting varieties of $ gl(r) $, for all $ r $. Joint work with E. Vasserot.
Tue, 05/06/2012
15:45 		Frank Gounelas (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Free curves on varieties
This talk will be about various ways in which a variety can be "connected by higher genus curves", mimicking the notion of rational connectedness. At least in characteristic zero, the existence of a curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.
Tue, 29/05/2012
15:45 		Gavin Brown (Loughborough) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Fano 3-folds in codimension 4
I show how to construct some Fano 3-folds that have the same Hilbert series but different Betti numbers, and so lie on different components of the Hilbert scheme. I would like to show where these fit in to a speculative (indeed fantastical) geography of Fano 3-folds, and how the projection methods I use may apply to other questions in the geography.
Tue, 22/05/2012
15:45 		Timo Schurg (Bonn) Algebraic and Symplectic Geometry Seminar Add to calendar L3
From perfect obstruction theories to commutative differential graded algebras
A perfect obstruction theory for a commutative ring is a morphism from a perfect complex to the cotangent complex of the ring satisfying some further conditions. In this talk I will present work in progress on how to associate in a functorial manner commutative differential graded algebras to such a perfect obstruction theory. The key property of the differential graded algebra is that its zeroth homology is the ring equipped with the perfect obstruction theory. I will also indicate how the method introduced can be globalized to work on schemes without encountering gluing issues.

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

Syndicate content