Forthcoming events in this series
15:45
The homological projective dual of Sym^2(P^n)
Abstract
In recent years, some powerful tools for computing semi-orthogonal decompositions of derived categories of algebraic varieties have been developed: Kuznetsov's theory of homological projective duality and the closely related technique of VGIT for LG models. In this talk I will explain how the latter works and how it can be used to understand the derived categories of complete intersections in Sym^2(P^n). As a consequence, we obtain a new proof of result of Hosono and Takagi, which says that a certain pair of non-birational Calabi-Yau 3-folds are derived equivalent.
15:45
Complex Geometry and the Hele-Shaw flow
Abstract
The goal of this talk is to discuss a link between the Homogeneous Monge Ampere Equation in complex geometry, and a certain flow in the plane motivated by some fluid mechanics. After discussing and motivating the Dirichlet problem for this equation I will focus to what is probably the first non-trivial case that one can consider, and prove that it is possible to understand regularity of the solution in terms of what is known as the Hele-Shaw flow in the plane. As such we get, essentially explicit, examples of boundary data for which there is no regular solution, contrary to previous expectation. All of this is joint work with David Witt Nystrom.
14:00
The Donaldson-Thomas theory of K3xE and the Igusa cusp form
Abstract
Donaldson-Thomas invariants are fundamental deformation invariants of Calabi-Yau threefolds. We describe a recent conjecture of Oberdieck and Pandharipande which predicts that the (three variable) generating function for the Donaldson-Thomas invariants of K3xE is given by the reciprocal of the Igusa cusp form of weight 10. For each fixed K3 surface of genus g, the conjecture predicts that the corresponding (two variable) generating function is given by a particular meromorphic Jacobi form. We prove the conjecture for K3 surfaces of genus 0 and genus 1. Our computation uses a new technique which mixes motivic and toric methods.
14:00
The topology of rationally and polynomially convex domains
Abstract
Rationally and polynomially convex domains in ${\mathbb C}^n$ are fundamental objects of study in the theory of functions of several complex variables. After defining and illustrating these notions, I will explain joint work with Y.Eliashberg giving a complete characterization of the possible topologies of such domains in complex dimension at least three. The proofs are based on recent progress in symplectic topology, most notably the h-principles for loose Legendrian knots and Lagrangian caps.
15:45
Cobordisms between tangles
Abstract
In a previous work, we introduced a refinement of Juhasz’s sutured Floer homology, and constructed a minus theory for sutured manifolds, called sutured Floer chain complex. In this talk, we introduce a new description of sutured manifolds as “tangles” and describe a notion of cobordism between them. Using this construction, we define a cobordism map between the corresponding sutured Floer chain complexes. We also discuss some possible applications. This is a joint work with Eaman Eftekhary.
Infinitely many monotone Lagrangian Tori in CP^2
Abstract
15:45
Hamiltonian and quasi-Hamiltonian reduction via derived symplectic geometry
Abstract
I will explain an approach to Hamiltonian reduction using derived
symplectic geometry. Roughly speaking, the reduced space can be
presented as an intersection of two Lagrangians in a shifted symplectic
space, which therefore carries a natural symplectic structure. A slight
modification of the construction gives rise to quasi-Hamiltonian
reduction. This talk will also serve as an introduction to the wonderful
world of derived symplectic geometry where statements that morally ought
to be true are indeed true.
15:45
Exotic spheres and the topology of the symplectomorphism group
Abstract
Using the fact that certain exotic spheres do not admit Lagrangian embeddings into $T^*{\mathcal S}^{n+1}$, as proven by Abouzaid and Ekholm-Smith, we produce non-trivial homotopy classes of the group of compactly supported symplectomorphisms of $T^*{\mathcal S}^n$. In particular, we show that the Hamiltonian isotopy class of the symplectic Dehn twist depends on the parametrisation used in the construction. Related results are also obtained for $T^*({\mathcal S}^n \times {\mathcal S}^1)$.
Joint work with Jonny Evans.
Torus action and Segre classes in the context of the Green-Griffiths conjecture
Abstract
The goal of this second talk is to study the existence of global jet differentials. Thanks to the algebraic Morse inequalities, the problem reduces to the computation of a certain Chern number on the Demailly tower of projectivized jet bundles. We will describe the significant simplification due to Berczi consisting in integrating along the fibers of this tower by mean of an iterated residue formula. Beside the original argument coming from equivariant geometry, we will explain our alternative proof of such a formula and we will particularly be interested in the interplay between the two approaches.
Jet techniques for hyperbolicity problems
Abstract
Hyperbolicity is the study of the geometry of holomorphic entire curves $f:\mathbb{C}\to X$, with values in a given complex manifold $X$. In this introductary first talk, we will give some definitions and provide historical examples motivating the study of the hyperbolicity of complements $\mathbb{P}^{n}\setminus X_{d}$ of projective hypersurfaces $X_{d}$ having sufficiently high degree $d\gg n$.
Then, we will introduce the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy due to Bloch, Demailly, Siu, that relies in an essential way on the relation between entire curves and jet differentials vanishing on an ample divisor.
Recent directions in derived geometry
Abstract
We will give an idea of derived algebraic geometry and sketch a number of more or less recent directions, including derived symplectic geometry, derived Poisson structures, quantizations of moduli spaces, derived analytic geometry, derived logarithmic geometry and derived quadratic structures.
The geometry of auctions and competitive equilibrium with indivisible goods
Abstract
Auctioneers may wish to sell related but different indivisible goods in
a single process. To develop such techniques, we study the geometry of
how an agent's demanded bundle changes as prices change. This object
is the convex-geometric object known as a `tropical hypersurface'.
Moreover, simple geometric properties translate directly to economic
properties, providing a new taxonomy for economic valuations. When
considering multiple agents, we study the unions and intersections of
the corresponding tropical hypersurfaces; in particular, properties of
the intersection are deeply related to whether competitive equilibrium
exists or fails. This leads us to new results and generalisations of
existing results on equilibrium existence. The talk will provide an
introductory tour to relevant economics to show the context of these
applications of tropical geometry. This is joint work with Paul
Klemperer.
Morse theory in representation theory and algebraic geometry
Abstract
Hamiltonian reduction arose as a mechanism for reducing complexity of systems in mechanics, but it also provides a tool for constructing complicated but interesting objects from simpler ones. I will illustrate how this works in representation theory and algebraic geometry via examples. I will describe a new structure theory, motivated by Hamiltonian reduction (and in particular the Morse theory that results), for some categories (of D-modules) of interest to representation theorists. I will then explain how this implies a modified form of "hyperkahler Kirwan surjectivity" for the cohomology of certain Hamiltonian reductions. The talk will not assume that members of the audience know the meaning of any of the above-mentioned terms. The talk is based on joint work with K. McGerty.
On the Gromov width of polygon spaces
Abstract
After Gromov's foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold
$(M, \omega)$ is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in it. I will discuss tecniques to compute the Gromov width of a special family of symplectic manifolds, the moduli spaces of polygons in real $3$-space. Under some genericity assumptions on the edge lengths, the polygon space is a symplectic manifold; in fact, it is a symplectic reduction of Grassmannian of 2-planes in complex $n$-space. After introducing this family of manifolds we will concentrate on the spaces of 5-gons and calculate for their Gromov width. This is joint work with Milena Pabiniak, IST Lisbon.
Mirror symmetry without localisation
Abstract
Mirror Symmetry predicts a surprising relationship between the virtual numbers of degree-d rational curves in a target space X and variations of Hodge structure on a different space X’, called the mirror to X. Concretely, it predicts that one can compute genus-zero Gromov–Witten invariants (which are the virtual numbers of rational curves) in terms of hypergeometric functions (which are the solutions to a differential equation that controls the variation of Hodge structure). Existing proofs of this rely on beautiful but fearsomely complicated localization calculations in equivariant cohomology. I will describe a new proof of the Mirror Theorem, for a broad range of target spaces X, which is much simpler and more conceptual. This is joint work with Cristina Manolache.
The Crepant Transformation Conjecture and Fourier--Mukai Transforms
Abstract
Comparing curve-counting invariants
Abstract
Counting curves with given topological properties in a variety is a very old question. Example questions are: How many conics pass through five points in a plane, how many lines are there on a Calabi-Yau 3-fold? There are by now several ways to count curves and the numbers coming from different curve counting theories may be different. We would then like to have methods to compare these numbers. I will present such a general method and show how it works in the case of stable maps and stable quasi-maps.