Spring 2021

Place: Zoom Meeting Room (Virtual)
Time: Central Time (CT)
Organizers: Ezra Getzler, Dogancan Karabas (dogancan.karabas[at]northwestern dot edu), and Eric Zaslow

Thursday, April 15, 2021 at 3:00pm

  • Speaker: Filip Živanović (University of Edinburgh)
  • Title: Symplectic Geometry of Conical Symplectic Resolutions
  • Abstract: Conical symplectic resolutions (CSRs) are a big family of spaces that are of central importance in Geometric Representation Theory. Quiver varieties, hypertoric varieties, Slodowy nilpotent slices are all examples of CSRs.
    All the existing examples of CSRs are known to be complete hyperkähler. In this talk, I will focus on their symplectic geometry. I will explain that each CSR has a natural Liouville structure and that one can obtain a family of compact exact Lagrangian submanifolds in it, whose Floer-theoretic invariants are of purely-topological nature, and which yields lower bounds on the symplectic cohomology.
    Time permitting, I will talk about the highly non-exact symplectic structure on CSRs, whose symplectic cohomology can be constructed (joint work with Alex Ritter) and give some consequences of this construction.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Recording: Zoom

Thursday, April 22, 2021 at 3:00pm

  • Speaker: Laura Starkston (UC Davis)
  • Title: Unexpected fillings, singularities, and plane curve arrangements
  • Abstract: I will discuss joint work with Olga Plamenevskaya studying symplectic fillings of links of certain complex surface singularities, and comparing symplectic fillings with complex smoothings. We develop characterizations of the symplectic fillings using planar Lefschetz fibrations and singular braided surfaces. This provides an analogue of de Jong and van Straten’s work which characterizes the complex smoothings in terms of decorated complex plane curves. We find differences between symplectic fillings and complex smoothings that had not previously been found in rational complex surface singularities.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Notes: PDF
  • Recording: Zoom

Thursday, April 29, 2021 at 1:00pm

  • Speaker: Chris Hull (Imperial College London)
  • Title: Special Holonomy Metrics, Degenerate Limits and Nilmanifold Fibrations
  • Abstract: This talk explores a construction of metrics for K3 and generalisations to higher dimensional special holonomy manifolds. The starting point is a degenerate limit of K3 recently constructed by Hein, Sun, Viaclovsky and Zhang, with a long neck that can be thought of as a 3-dimensional nilmanifold fibred over a line, with gravitational instanton insertions. Special holonomy generalisations of this neck region that are given by higher dimensional nilmanifolds fibred over a line will be discussed. They are dual to intersecting brane solutions, and this relation leads to more general solutions. The possibility of these arising as part of a degenerate limit of a compact special holonomy space will be considered, and applications will be explored.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Notes: Slides
  • Recording: Zoom

Thursday, May 6, 2021 at 3:00pm

  • Speaker: Maxim Jeffs (Harvard University)
  • Title: Mirror symmetry and Fukaya categories of singular varieties
  • Abstract: One problem that arises when studying mirror symmetry is that even mirrors of smooth varieties are often singular, and it is difficult to make sense of the A-model for these mirrors. In this talk I will explain Auroux’ definition of the Fukaya category of a singular hypersurface and two results about this definition, illustrated with some fundamental examples. The first result is analogous to Orlov’s derived Knorrer periodicity theorem, relating Auroux’ category to a Landau-Ginzburg model on a higher-dimensional variety; the second result is a homological mirror symmetry equivalence at certain large complex structure limits.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Notes: Slides

Thursday, May 13, 2021 at 3:00pm

  • Speaker: Bogdan Stoica (Northwestern University)
  • Title: A tale of two geometries: Riemann zeta from the p2-brane
  • Abstract: Ordinary bosonic string theory is a tale of two geometries, with the 26-dimensional gravitational target space emerging from the geometry of the two-dimensional string worldsheet. I will present a p-adic analogue of the bosonic string, where the worldsheet has been replaced by the “geometry” of a p-adic two-brane constructed from the Bruhat-Tits tree, building on earlier work with An Huang and Shing-Tung Yau. I will discuss restrictions arising from the time-evolution, as well as the canonical quantization procedure and connections to the Riemann zeta function. I will also explain the p-adic implications of requiring that the target is a gravitational space, which in the usual bosonic string case is the famous condition leading to the target space dimension of D = 26. Based on joint work with An Huang and Xiao Zhong.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Recording: Zoom

Thursday, May 20, 2021 at 3:00pm

  • Speaker: Orsola Capovilla-Searle (Duke University)
  • Title: Infinitely many Lagrangian Tori in Milnor fibers constructed via Lagrangian Fillings of Legendrian links
  • Abstract: One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular, one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings.
    We show that family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. We provide some of the first examples of a Legendrian link that admits infinitely many planar exact Lagrangian fillings. Weinstein domains are examples of symplectic manifolds with contact boundary that have a handle decomposition compatible with the symplectic structure of the manifold.  Weinstein handlebody diagrams are given by projections of Legendrian submanifolds. We provide Weinstein handlebody diagrams of the $4$-dimensional Milnor fibers of $T_{p,q,r}$ singularities, which we then use to construct infinitely many Lagrangian tori and spheres in these spaces.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Recording: Zoom

Thursday, May 27, 2021 at 1:00pm

  • Speaker: Nicolò Sibilla (SISSA)
  • Title: Fukaya category of surfaces and pants decomposition
  • Abstract: In this talk I will explain some results joint with James Pascaleff on the Fukaya category of Riemann surfaces. I will explain a local-to-global principle which allows us to reduce the calculation of the Fukaya category of surfaces of genus g greater than one to the case of the pair-of-pants, and which holds both in the punctured and in the compact case. The starting point are the sheaf-theoretic methods which are available in the exact setting, and which I will review at the beginning of the talk. This result has several interesting consequences for HMS and geometrization of objects in the Fukaya category. The talk is based on 1604.06448 and 2103.03366.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Recording: Zoom

Thursday, June 3, 2021 at 3:00pm

  • Speaker: Sylvain Courte (Université Grenoble Alpes)
  • Title: Twisted generating functions and the nearby Lagrangian conjecture
  • Abstract: I will explain the notion of twisted generating function and show that a closed exact Lagrangian submanifold L in the cotangent bundle of M admits such a thing. The type of function arising in our construction is related to Waldhausen’s tube space from his manifold approach to algebraic K-theory of spaces. Using the rational equivalence of this space with BO, as proved by Bökstedt, we conclude that the stable Lagrangian Gauss map of L vanishes on all homotopy groups. In particular when M is a homotopy sphere, we obtain the triviality of the stable Lagrangian Gauss map and a genuine generating function for L. This is a joint work with M. Abouzaid, S. Guillermou and T. Kragh.
  • Zoom Meeting ID: 916 7431 9341 (email Dogancan Karabas to access the meeting password.)
  • Recording: Zoom

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