Past Algebraic and Symplectic Geometry Seminar

18 November 2014
14:00
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.

  • Algebraic and Symplectic Geometry Seminar
13 November 2014
14:00
Kai Cieliebak
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.

  • Algebraic and Symplectic Geometry Seminar
4 November 2014
15:45
Akram Alishahi
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.

  • Algebraic and Symplectic Geometry Seminar
28 October 2014
15:45
Renato Vianna
Abstract
In previous work, we constructed an exotic monotone Lagrangian torus in $\mathbb{CP}^2$ (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it $T(1,4,25)$ because, when following a degeneration of $\mathbb{CP}^2$ to the weighted projective space $\mathbb{CP}(1,4,25)$, it degenerates to the central fibre of the moment map for the standard torus action on $\mathbb{CP}(1,4,25)$. Related to each degeneration from $\mathbb{CP}^2$ to $\mathbb{CP}(a^2,b^2,c^2)$, for $(a,b,c)$ a Markov triple -- $a^2 + b^2 + c^2 = 3abc$ -- there is a monotone Lagrangian torus, which we call $T(a^2,b^2,c^2)$.  We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.
  • Algebraic and Symplectic Geometry Seminar
21 October 2014
15:45
Pavel Safronov
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.

  • Algebraic and Symplectic Geometry Seminar
14 October 2014
15:45
Georgios Rizell
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.

 

  • Algebraic and Symplectic Geometry Seminar

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