| 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" ## Abstracts
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.
This talk is the first in a mini-course with T. Schedler.
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.
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.
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.
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.
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.
I'll discuss quantum counterparts (in the context of the Berezin-Toeplitz quantization)
of some rigidity-type phenomena discovered within symplectic topology. 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.
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.
This talk is the second in a mini-course with P. Etingof.
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.
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).
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.
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.
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). Joint work with P.E. Paradan.
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.
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.
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).
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). |