Poisson 2016  |  Geneva and Zurich School June 27 - July 2  |  Conference July 4 - 8

### Speakers and talks

Ana Cannas da Silva (ETH Zurich), "A Chiang-type lagrangian in CP^2"
Pavel Etingof (MIT), "D-modules on Poisson varieties, Poisson homology, and symplectic resolutions"
Matias del Hoyo (IMPA), "Categorifying Riemannian manifolds"
Ezra Getzler (Northwestern University), "The BV formalism for the spinning particle"
Dominic Joyce (Oxford), "'Fukaya categories' of complex Lagrangians in complex symplectic manifolds"
Ioan Marcut (Radboud University), "Rigidity of solutions to PDE's with symmetries"
Pavel Mnev (MPIM Bonn), "BF theory on cobordisms endowed with cellular decomposition"
Leonid Polterovich (Tel Aviv University), "Quantum footprints of symplectic rigidity"
Silvia Sabatini (University of Cologne), "On the geography of symplectic manifolds with group actions and applications"
Pavel Safronov (Oxford), "Poisson geometry of groups and shifted Poisson structures"
Travis Schedler (University of Texas), "Poisson-de Rham homology and special polynomials"
Pavol Ševera (University of Geneva), "Integration of Courant algebroids"
Geoffrey Scott (University of Toronto), "Lie Algebroids on Pinched S^1 Bundles"
Thomas Strobl (University of Lyon), "Curved Yang-Mills-Higgs Gauge Theories: Geometry and Physics"
Bertrand Toën (University of Toulouse), "Symplectic and Poisson structures in the derived setting"
Duco van Straten (Mainz), "Persistentce of invariant Lagrangian varieties"
Michèle Vergne (IMJ Paris), "Dirac operators and Geometric Invariant Theory in the differentiable setting"
Alan Weinstein (UC Berkeley), "Classification problems in Poisson vector spaces"
Harold Williams (University of Texas), "Cluster Algebra as the Moduli Theory of A-branes in 4-manifolds"
Milen Yakimov (Louisiana State University), "Noncommutative discriminants and Poisson geometry"
Chenchang Zhu (Göttingen), "String principal bundles and Courant algebroids"

### Abstracts

Ana Cannas da Silva (ETH Zurich), A Chiang-type lagrangian in CP^2

We analyse a lagrangian in $\mathbb{CP}^2$ which is an analogue of the Chiang lagrangian in $\mathbb{CP}^3$. Our lagrangian is topologically an $\mathbb{RP}^2$ but exhibits a distinguishing behavior under reduction by one of the toric circle actions (in particular it may be viewed as a "one-to-one transverse lifting" of a great circle in $\mathbb{CP}^1$). This behavior is relevant in connection with Weinstein's lagrangian composition and work of Wehrheim and Woodward in Floer theory.
VIDEO

Pavel Etingof (MIT), D-modules on Poisson varieties, Poisson homology, and symplectic resolutions

This talk is the first in a mini-course with T. Schedler.
In this talk we explain how to prove that the zeroth Poisson homology of varieties with finitely many symplectic leaves is finite, by defining a D-module whose solutions are invariants of Hamiltonian flow. More generally the technique applies to coinvariants of Lie algebra actions on varieties. In the case of varieties admitting symplectic resolutions, we explain conjectures identifying the result with the cohomology of the resolution and their status. In the case of complete intersection surfaces with isolated singularities, we compute the D-module, in which the Milnor numbers and genera of the singularities are encoded.
VIDEO

Matias del Hoyo (IMPA), Categorifying Riemannian manifolds

Lie groupoids constitute a unifying framework to perform differential geometry, with examples arising from actions, foliations, fibrations and Poisson manifolds, among others. Jointly with R. Fernandes we introduced metrics on Lie groupoids, extending and correcting previous attemps, and allowing arguments from Riemannian geometry within this context. For instance, the Weinstein-Zung Linearization Theorem, which provides normal forms for classic geometries, can be easily achieved via exponential maps. In this talk I will overview our construction of metrics on groupoids, describe variants for fibred groupoids and smooth stacks, present its applications to linearization and rigidity problems, and discuss some future lines of research.
VIDEO

Ezra Getzler (Northwestern University), The BV formalism for the spinning particle

The BV formalism is a generally covariant framework for field theories which allows the discussion of very general gauge symmetries in the Lagrangian framework. In this talk, I discuss the BV formalism for the spinning particle, which is a toy model for the superstring, and whose quantization is the Dirac operator. The main result is that the cohomology of the BV complex is nonzero in every negative degree. This seems to indicate that there are gaps in our understanding of the BV formalism in the presence of supersymmetries.
VIDEO

Dominic Joyce (Oxford), "Fukaya categories" of complex Lagrangians in complex symplectic manifolds

We outline a programme to define a "Fukaya category" of complex Lagrangians (and "derived Lagrangians") in a complex symplectic manifold, using perverse sheaves. The programme originates in the shifted symplectic derived algebraic geometry of Pantev-Toen-Vaquie-Vezzosi, but can be explained without using Derived Algebraic Geometry. Different parts of this programme are joint work with subsets of Lino Amorim, Oren Ben-Bassat, Chris Brav, Vittoria Bussi, Delphine Dupont, Pavel Safronov, and Balazs Szendroi.
VIDEO

Ioan Marcut (Radboud University), Rigidity of solutions to PDE's with symmetries

Local normal form theorems in differential geometry are often the manifestation of rigidity of the structure in normal form. For example, the existence of local Darboux coordinates in symplectic geometry follows from the fact that, locally, the standard symplectic structure has no deformations.
I will discuss a general local rigidity result for solutions to PDE's under the action of a closed pseudo-group of symmetries. The result is of the form: "infinitesimal tame rigidity" implies "tame rigidity"; it is in the smooth setting, and the proof uses the Nash-Moser fast convergence method.
Several classical theorems fit in our framework: the Newlander-Nirenberg theorem in complex geometry, Conn's theorem in Poisson geometry, Eliasson's theorem for integrable systems (around elliptic singularities), etc.
This is joint work with Roy Wang (PhD student at Utrecht University).
VIDEO

Pavel Mnev (MPIM Bonn), BF theory on cobordisms endowed with cellular decomposition

We will present an example of a topological field theory living on cobordisms endowed with CW decomposition (this example corresponds to the so-called BF theory in its abelian and non-abelian variants), which satisfies the Batalin-Vilkovisky master equation, satisfies (a version of) Segal's gluing axiom w.r.t. concatenation of cobordisms and is compatible with cellular aggregations. In non-abelian case, the action functional of the theory is constructed out of local unimodular L-infinity algebras on cells; the partition function carries the information about the Reidemeister torsion, together with certain information pertaining to formal geometry of the moduli space of local systems. This theory provides an example of the BV-BFV programme for quantization of field theories on manifolds with boundary in cohomological formalism. This is a joint work with Alberto S. Cattaneo and Nicolai Reshetikhin.
VIDEO

Leonid Polterovich (Tel Aviv University), Quantum footprints of symplectic rigidity

I'll discuss quantum counterparts (in the context of the Berezin-Toeplitz quantization) of some rigidity-type phenomena discovered within symplectic topology.
VIDEO

Silvia Sabatini (University of Cologne), On the geography of symplectic manifolds with group actions and applications

In this talk I will discuss some problems related to the geography of compact symplectic manifolds endowed with a symplectic group action. In particular I will present some recent results about rigidity properties of their Chern numbers and their dependence on the minimal Chern number of the manifold. These results can be used to give conditions for a symplectic manifold to support only Hamiltonian or only non-Hamiltonian circle actions with isolated fixed points, the existence of the latter being recently discovered by Tolman.
As a second application, I will present a generalisation of the famous "12" and "24" formulas for reflexive polytopes of dimension 2 and 3 to Delzant reflexive polytopes of any dimension. Finally we give bounds on the Betti numbers of certain monotone symplectic manifolds admitting Hamiltonian circle actions, these bounds depending on the minimal Chern number.
The talk is based on the preprints:
"On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action", arXiv:1411.6458v2 [math.AT]
"12, 24 and Beyond" (with L. Godinho and F. von Heymann), arXiv:1604.00277 [math.CO]
VIDEO

Pavel Safronov (Oxford), Poisson geometry of groups and shifted Poisson structures

I will discuss how the recently introduced theory of shifted Poisson and coisotropic structures can serve as an organizing principle for Poisson structures on groups which underline different versions of quantum groups. In the end I will mention a geometric way to think about some elliptic quantum groups.
VIDEO

Travis Schedler (University of Texas), Poisson-de Rham homology and special polynomials

This talk is the second in a mini-course with P. Etingof.
I will explain how to recover Tutte and Kostka polynomials from the Poisson-de Rham homology of hypertoric varieties and nilpotent cones (and conjectures about more general cones admitting symplectic resolutions), and also a connection, via D-modules, between a generalization of Poisson-de Rham homology and the Bernstein-Sato polynomial in the case of hypersurfaces with isolated singularities. In the latter case, this leads to a conjecture expressing the length of the D-module D.(1/f) as two more than the sum of the genera of the singularity, which we prove when f is quasihomogeneous. This includes joint work with Bellamy, Bitoun, Etingof, and Proudfoot.
VIDEO

Pavol Ševera (University of Geneva), Integration of Courant algebroids

In the same way as Poisson manifolds integrate to (local) symplectic groupoids, Courant algebroids integrate to (local) symplectic 2-groupoids. While this idea is old (it stems from Sullivan's Rational homotopy theory), to make it rigorous we would need to know that the set of solution of the corresponding generalized Maurer-Cartan equation is a well-behaved manifold. I will show that it is indeed true, explain how to construct the local symplectic 2- groupoids, and in what sense this construction is functorial. More generally, we shall integrate differential non-negatively graded manifolds and their (pre)symplectic forms. Based on a joint work with Michal Siran.
VIDEO

Geoffrey Scott (University of Toronto), Lie Algebroids on Pinched S^1 Bundles

In gauge theory, an abelian monopole is described as a singular connection on a 3-manifold M. In this talk, I will describe how these singular connections can be realized as smooth lie algebroid connections on the real oriented blowup of M. In addition, certain $S^1$ bundles that arise during this construction can answer the question, "What is a 'section' of a gerbe?" This is joint work with Marco Gualtieri (University of Toronto).
VIDEO

Thomas Strobl (University of Lyon), Curved Yang-Mills-Higgs Gauge Theories: Geometry and Physics

Yang-Mills-Higgs (YMH) gauge theories are a key element in our today's understanding of the interaction forces between elementary particles and their masses. The data needed to construct such a theory contain an action Lie algebroid, over the values of the Higgs fields, equipped with an invariant metric on the base and fibers. We show how the YMH theory can be generalized such that this action Lie algebroid is replaced by a more general Lie algebroid equipped with appropriately compatible connection $\nabla$ and metrics. The compatibilities are dictated by gauge invariance and imply in particular that for flat connections $\nabla$ one returns to the standard YMH gauge theory setting. We explain some of the geometry and physics of this new type of gauge theory.
VIDEO

Bertrand Toën (University of Toulouse), Symplectic and Poisson structures in the derived setting

The purpose of this talk is to give an overview of the notions of shifted symplectic and shifted Poisson structures. We will start by some quick reminders on derived algebraic geometry. We will present the existence results of shifted symplectic and shifted Poisson structures, as well as their quantizations. In the last part of the talk we will mention some perspectives and open problems.
VIDEO

Duco van Straten (Mainz), Persistentce of invariant Lagrangian varieties

The invariant torus theorem of Kolmogorov is a basic result on the perturbation of completely integrable systems, which marks the beginning of KAM-theory but which requires rather sophisticated analytic tools for its proof. In the talk I will present a general approach based on the notion of 'fixed point theorem in Kolmogorov spaces' that was introduced by M. Garay. This gives a transparent framework that can be applied in many other situations as well. (Work in progress with M. Garay).
VIDEO

Michèle Vergne (IMJ Paris), Dirac operators and Geometric Invariant Theory in the differentiable setting

Let $K$ be a compact Lie group, acting on a compact manifold $M$. If $D$ is a $K$-invariant elliptic operator on $M$, its space $Index_K(M, D)$ of (virtual) solutions is a virtual representation of $K$. We wish to understand the space of $K$-invariant solutions of $D$ as the space $Index(M_0, D_0)$ of solutions of an elliptic operator $D_0$ on a "smaller" space $M_0$. A familiar setting is when $M$ is a projective manifold, and $D$ the Dolbeaut operator acting on sections of the corresponding ample line bundle. In this case $M_0$ is again a projective manifold, the geometric quotient of $M$. Here we consider any Dirac operator on an oriented even dimensional compact $K$-manifold $M$ and we produce such manifold $M_0$ via a moment map construction analog to Mumford's construction of the geometric quotient. An important example is the case where $M$ is a compact complex manifold, $L$ an holomorphic line bundle, not necessarily ample, and $D = \bar\partial_L$ is the Dolbeaut operator with coefficients in $L$.
VIDEO

Alan Weinstein (UC Berkeley), Classification problems in Poisson vector spaces

This talk will be concerned with the classification of objects in Poisson vector spaces such as linear relations and arrangements of linear subspaces. An eventual goal is to find extensions of the usual linear representation theory of quivers to representations by relations compatible with Poisson structures.
This is joint work with Jonathan Lorand (University of Zurich).
VIDEO

Harold Williams (University of Texas), Cluster Algebra as the Moduli Theory of A-branes in 4-manifolds

We explain how the theory of cluster (Poisson) varieties can be organized into moduli theory in 4-dimensional symplectic geometry. Abstractly, a cluster variety is a space built out of infinitely many algebraic tori, and is defined by the initial data of a quiver. On the other hand, the quiver (together with some additional choices) can be used to specify a Weinstein 4-manifold, and we show that the cluster Poisson variety or a variant of it is recovered as a space of A-branes in this manifold. We study A-branes in the guise of microlocal sheaves on a Lagrangian skeleton, and explain how the combinatorial operation of quiver mutation lifts to a geometric operation on skeleta. This is joint work with Vivek Shende and David Treumann.
VIDEO

Milen Yakimov (Louisiana State University), Noncommutative discriminants and Poisson geometry

Discriminants play a key role in various settings in algebraic number theory, algebraic geometry, combinatorics, and noncommutative algebra. In the last case, they have been computed for very few algebras. We will present a general method for computing discriminants of noncommutative algebras which is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. This is a joint work with Bach Nguyen and Kurt Trampel (LSU).
VIDEO

Chenchang Zhu (Göttingen), String principal bundles and Courant algebroids

To a usual principal bundle, one can associate an Atiyah algebroid. For an $S^1$ gerbe, the higher version of an Atiyah algebroid is an exact Courant algebroid whose Severa class is the Dixmier-Douady class of the gerbe. In this talk, we'll explain the stack of transitive Courant algebroids built from local data. Then, in the case of the string principal bundle, the higher/noncommutative Atiyah algebroid turns out to be a transitive Courant algebroid. This explains why the obstruction to lift a principal G-bundle to a principal String(G)- bundle (controlled by one half of the Pontryagin class) coincides with the one for a twisted Courant algebroid to be Courant. (Joint work in progress with Yunhe Sheng and Xiaomeng Xu).
VIDEO