In a 1995 paper N. Hitchin, using twistor methods, has constructed new solutions of the Einstein equations. They were based on the results of K. Tod, who had (a year earlier) shown that the anti-selfdual Einstein $SU(2)$-invariant metrics on the $4$-manifolds generically are reduced to the Painleve-$6$ equation, satisfied by the independent variable, parametrizing the $SU(2)$-orbits. Hitchin expressed his solutions by rather cumbersome formulas it terms of theta-functions; their relation with the geometry of families of elliptic curves was not clarified in that paper.

However, in the case of algebraic solutions Hitchin has discovered this relation in a 2004 paper; it turned out to be expressible in terms of the Poncelet closure theorem. In the talk the corresponding isomonodromic families will be presented in several cases, including the simplest one where the twistor space is the projectivisation of the space of cubic polynomials.

A collection of these (and some others) ideas and constructions indicates the existence of a certain class of Einstein manifolds that have an arithmetic nature and are described completely by dessins d’enfants (the graphs, embedded into the oriented surfaces and related to arithmetic geometry by A. Grothendieck). Some related physical fantasies will possibly be mentioned.

We present a new method for classifying naturally reductive homogeneous spaces – i.e. homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion and curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such connections with the geometric structure on the given Riemannian manifold. We reproduce by much easier arguments the known classifications in dimensions $3$, $4$ and $5$, and obtain as a new result the complete classification in dimension $6$. In each dimension, we also exhibit a hierarchy of degeneracy for the torsion form, which we then treat case by case. For the complete degenerate cases, we prove results that are dimension independent. In some situations, we are able to show that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We prove that a generic parallel torsion form defines a quasi-Sasaki struture in dimension $5$ and an almost complex structure in dimension $6$.

References

I. Agricola, A. C. Ferreira, T. Friedrich. The classification of naturally reductive homogeneous spaces in dimensions $n\geq 6$. (To appear).

Lie groupoids serve as models for a certain kind of singular geometric spaces known as "smooth stacks". The classical orbifolds of Satake and Thurston provide examples of such spaces. Many more examples arise for instance in foliation theory, Poisson geometry or noncommutative geometry. Smooth stacks relate to Lie groupoids in much the same way smooth manifolds relate to differentiable atlases. Namely, there is a natural notion of equivalence between Lie groupoids, called "Morita equivalence", which has the property that two Lie groupoids are equivalent precisely when they model the same smooth stack. Motivated by the study of global geometric questions concerning the structure of differentiable stacks associated with proper Lie groupoids, we investigate the existence of multiplicative connections on such groupoids. We show that one can always deform a given connection which is only approximately multiplicative into a genuinely multiplicative connection. Our proof of this fact relies on a recursive averaging technique. We regard our results as an initial step towards the construction of an obstruction theory for multiplicative connections on proper Lie groupoids. Some applications of our results will be described.

In this talk, we will discuss the following conjecture.

Conjecture. Let $(M, \omega)$ be a compact symplectic manifold and $S^1$ be the circle group which acts on $(M, \omega)$ symplectically. If the fixed point set is non-empty and isolated, then the action is Hamiltonian.

The conjecture above originated from T. Frankel’s work (Ann. Math, 1959). He proved that any holomorphic circle action on a compact Kähler manifold preserving Kähler structure is Hamiltonian if and only if the fixed point is non-empty. A symplectic analogue of Frankel’s theorem was studied by K. Ono (1987) and D. McDuff (1988). In particular, McDuff proved that Frankel’s theorem does not hold in the symplectic category in general. But it remains still open when the fixed points are all isolated. The main aim of this talk is to describe several techniques to approach this conjecture (localization, Duistermaat-Heckman measure,...).

Also, if time permits, we will discuss the question of existence of symplectic circle action with only two fixed points, which is posed by Tolman and Weitsman (2000).

An arbitrary Lie groupoid gives rise to a groupoid of germs of local diffeomorphisms over its base manifold, known as its effect. The effect of any bundle of Lie groups is trivial. All quotients of a given Lie groupoid determine the same effect. It is natural to regard the effects of any two Morita equivalent Lie groupoids as being "equivalent". In this talk we will describe a systematic way of comparing the effects of different Lie groupoids. In particular, we will rigorously define what it means for two arbitrary Lie groupoids to give rise to "equivalent" effects. For effective orbifold groupoids, the new notion of equivalence turns out to coincide with the traditional notion of Morita equivalence. Our analysis is relevant to the presentation theory of proper smooth stacks.

Given a compact Kähler manifold $X$ polarized by some ample line bundle $L$, the Yau-Tian-Donaldson conjecture relates the existence of an extremal metric in the first Chern class of $L$ to a GIT stability notion of the pair $(X,L)$. This conjecture is valid for Fano varieties with the anti-canonical polarization but the general case is still open. In this talk, I will explain how the complex deformation theory of a projective manifold admitting an extremal metric gives an infinitesimal evidence for this conjecture. The deformation theory for extremal metrics will be illustrated with the examples of toric surfaces.

The $G$-character variety of a surface of genus $g$ is the moduli space parametrizing representations of the fundamental group of the surface into $G$. Twisted character varieties consist of representations of the fundamental group of a punctured surface with fixed holonomy around the puncture. Using a geometric technique based on stratifications and fibrations of the moduli space, we will show how to compute the E-polynomials of these spaces for $G = \operatorname{SL}(2,\mathbb{C})$, $\operatorname{PGL}(2,\mathbb{C})$ and arbitrary genus. (joint work with V. Muñoz)

I will prove a rigidity result for Lagrangian cobordisms using an $L^2$ version of Floer homology. The input from holomorphic curves will be compressed in an easily understandable black box (at the expense of generality) and the construction of cellular $L^2$ homology will be described in some detail; therefore the talk should be accessible also to non symplectic geometers. This is a joint work with Baptiste Chantraine, Georgios Dimitroglu Rizell and Roman Golovko.

I will describe examples of contact manifolds in every dimension that admit contactomorphisms that are smoothly isotopic to the identity but not isotopic through contactomorphisms. Joint with Patrick Massot.

It is well known that in any compact Kahler manifold the exterior multiplication by suitable powers of the symplectic form induces isomorphisms between the de Rham cohomology spaces of complementary degrees. This is the content of the celebrated Hard Lefschetz Theorem. In my talk I will present recent joint work with B. Cappelletti Montano and I. Yudin showing the existence of similar isomorphisms for compact Sasakian manifolds. We prove that such isomorphisms are independent of the choice of any compatible Sasakian metric on a given contact manifold. As a consequence, we find an obstruction for a contact manifold to admit compatible Sasakian structures.

Monopoles are solutions to the Bogomolnyi equation, which is a PDE for a connection and an Higgs field (a section of an certain bundle) on a $3$ dimensional Riemannian manifold. In this talk I plan to introduce these equations. Then I want to tell you some properties of its solutions on $\mathbb{R}^3$. Finally, I plan to speak about monopoles on a more general class of noncompact manifolds known as asymptotically conical. My main goal is to explain the geometric meaning of the parameters needed to give coordinates on an open set of the moduli space of monopoles.

In 1999 Florian Deloup and I were attempting to find closed formulae for all abelian quantum invariants. These invariants can be expressed in terms of generalized Gauss sums, which depend on a quadratic form obtained from the linking form of the 3-manifold. Toward this end we formulated a conjecture that was intended to refine a theorem by Kawauchi and Kojma that demonstrated all linking pairings on finite abelian groups (i.e. symmetric, non-degenerate bilinear forms into $\mathbb{Q}/\mathbb{Z}$) arise as a "linking form" of some 3-manifold. Their construction involves taking the connected sum of three different types of 3-manifolds. The basis of this theorem was Wall's work and the subsequent work of Kawauchi and Kojma that classified all linking pairings on finite abelian groups.

Our conjecture that was supposed to refine this theorem stated that any linking pairing on a finite abelian group arises from the linking form of a Seifert fibered rational homology sphere. We proved this result in the case when the abelian group has no 2-torsion by 2004. In 2010 Jonathan Hillman gave counterexamples in the 2-torsion case. The underlying reason for the failure of the linking form conjecture is that there are homology cobordism classes of 3-manifolds that do not contain any Seifert manifolds.

It is possible to reformulate the linking form conjecture so it fulfils its original purpose. A corollary of this "new" linking form "theorem" is that every homology cobordism class has a representative that arises from a "generalized Seifert presentation".

Furthermore, there are some interesting applications of these abelian quantum invariants to physics.

Polarizations (real or complex) play an important role in geometric quantization. The adapted Kähler structure is a natural complex polarization on the phase space $N$ of a Riemannian manifold. It turns out, that this Kähler structure is just one member of a family of (real or complex) polarizations parameterized by the complex plane, due to the symmetries of $N$. In fact to talk about these polarizations one doesn't need a Riemannian metric, a connection suffices. In the talk I shall discuss joint results with Lempert concerning these polarizations.

Diffeology, introduced around 1980 by Jean-Marie Souriau following earlier work of Kuo-Tsai Chen, gives a simple way to extend notions of differential topology beyond manifolds. A diffeology on a set specifies which maps from open subsets of Euclidean spaces to the set are “smooth”. Examples include (possibly non-Hausdorff) quotients of manifolds and (infinite dimensional) spaces of smooth mappings between manifolds.

I will present some examples and results that relate diffeology with more traditional aspects of Lie group actions.

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 fiber 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.

Based on the notion of translated points of contactomorphisms, in 2011 I proposed an analogue in contact topology of the Arnold conjecture on fixed points of Hamiltonian diffeomorphisms. In this talk I will present a proof, based on a variant of Floer homology that I am developing ad hoc to study this problem, in the case when there are no closed contractible Reeb orbits. In the exposition I will try to avoid technicalities and explain instead in details the variational principle that I am considering, the motivations behind it and, if time permits, some expected applications.

Given a reductive group $G$ and an affine $G$-scheme $X$, constellations are $G$-equivariant sheaves over $X$ such that their module of global sections has finite multiplicities. Prescribing these multiplicities by a function $h$, and imposing a stability condition $\theta$ there is a moduli space for $\theta$-stable constellations constructed by Becker and Terpereau, using Geometric Invariant Theory. This construction will depend on a finite subset $D$ of the set of irreducible representations of $G$. By reformulating the stability condition $\theta$ and the GIT stability condition, in terms of a slope condition (say $\mu_{\theta}$ and $\mu_D$) we are able to construct Harder-Narasimhan filtrations from both points of view, and prove a precise relation between the two filtrations. Finally, we show that the associated polygons to the $\mu_D$-filtrations converge to the one associated to the $\mu_{\theta}$-filtrations when $D$ grows.

These results are joint work with Ronan Terpereau.

In 1991, string theorists Candelas, de la Ossa, Green and Parkes made a startling prediction for the number of curves in each degree on a generic quintic threefold, in terms of periods of a holomorphic volume form on a “mirror manifold”. Givental and Lian, Liu and Yau gave a mathematical proof of this version of mirror symmetry for the quintic threefold (and many more examples) in 1996. In the meantime (1994), Kontsevich had introduced his “homological mirror symmetry” conjecture and stated that it would “unveil the mystery of mirror symmetry”. I will explain how to prove that the number of curves on the quintic threefold matches up with the periods of the mirror via homological mirror symmetry. I will also attempt to explain in what sense this is “less mysterious” than the previous proof.

In this talk I will explain how to use derived algebraic geometry and the idea of brane actions of Toën to obtain Gromov-Witten invariants from derived categories of sheaves. This is a joint work with Etienne Mann from the University of Montpellier 2.

Recent work of L. Polterovich exibits a link between rigidity properties in symplectic topology and some properties of quantum observables (positive operator valued measures). This link is established through Berezin-Toeplitz quantization. In view of this, it would be desirable to extend the asymptotic results on Berezin-Toeplitz quantization to symbols with lower regularity. In this talk, I will discuss some recent results in this direction as well as some open questions.

Let $X$ be a compact Kähler manifold polarized by an ample line bundle $L$. For $k$ big enough, $L^k$ is generated by holomorphic sections which can be used to embed $X$ holomorphically into complex projective space. The philosophy of quantisation is to canonically associate to each “classical” object defined on $X$ a sequence of “quantized” objects defined on the embedded submanifolds. These objects should converge back to the classical objects when $k$ tends to infinity (in the semi-classical limit). Building on earlier work of Wang, Fine, Ma and Marinescu I show how to work out this program for the Laplacian and construct a sequence of self-adjoint operators $D_k$ acting on some finite dimensional vector spaces. In the semi-classical limit the eigenvalues and eigenspaces of $D_k$ converge to those of the Laplacian in an appropriate sense. This opens the route to study of the spectral properties of the Laplacian purely in terms of projective geometry. If time permits we explain how to use these results to quantise the heat flow on $X$. This is joint work with Julien Keller and Reza Seyyedali.

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Rui Loja Fernandes, University of Illinois at Urbana-Champaign Symplectic gerbes

Gerbes were introduced by Giraud, following ideas of Grothendieck, to deal with non-abelian cohomology in degree 2. In their simplest form, they allow to give geometric representatives of degree 3 integer cohomology classes, in the same way as one can think of principal circle bundles as geometric representatives of degree 2 integer cohomology classes (via their Chern class). In this talk I will report on current joint work with M. Crainic (Utrecht) and D. Martinez-Torres (PUC-Rio) where we describe a symplectic version of gerbes and relate them to quasi-hamiltonian $T^n$-spaces.

Consider an algebraic variety defined by system of polynomial equations with integer coefficients. For each prime number $p$, we may reduce the system modulo $p$ to obtain an algebraic variety defined over the field of $p$ elements.

A standard problem in arithmetic geometry is to understand how the geometry of one of these varieties influences the geometry of the other.

One can take a statistical approach to this problem.

We will illustrate this with several examples, including: polynomials in one variable, algebraic curves and surfaces.

Let $\Omega =B(p,r)$ be a geodesic ball centered at $p$ and radius $r>0$ of a Riemannian manifold $(M,ds^2)$. Let $\sigma (\Omega)=\{\lambda_{k}\}_{k=1}^{\infty}\subset [0, \infty)$ be the spectrum of $L=-\Delta\vert_{W_{0}^{2}(\Omega)}$.

The radial spectrum of $\Omega$ is the set of the eigenvalues $\lambda_{k_i}\in \sigma (\Omega)\subset \sigma (\Omega)$ whose eigenfunctions are radial functions, i.e., functions that depend only on the distance $r_p(x)={\rm dist}_{M}(p,x)$.

Recall that a $m$-dimensional rotationally symmetric manifold with radial sectional curvature $-G(r)$, where $G$ is a smooth even function on $\mathbb{R}$ is defined as the space \[ \mathbb{M}^m_{h}=[0,R_0)\times\mathbb{S}^{m-1}/\sim \] with ($(t, \theta)\sim (s,\beta)\Leftrightarrow t=s=0$ and $\forall \theta, \beta\in \mathbb{S}^{m-1}$ or $ t=s$ and $\theta=\beta$, endowed with the metric $ds^2_{h}(t,\theta)=dt^2+h^2(t)d\theta^2$ where $\sigma$ denotes the unique solution of the Cauchy problem on $[0,R_h)$ \begin{eqnarray}\label{ubs2} \left\{\begin{array}{l} h''-Gh=0,\\ h(0)=0, h'(0)=1 \end{array}\right. \end{eqnarray}and $R_h$ is the largest positive real number such that $h>0$. Our main result in this talk is the following theorem.

Let $B_r(o)\subset \mathbb{M}_{h}^{n}$ be the geodesic ball of radius $r$ centred at the origin $o$. Let $\sigma^{\rm rad}(B_r(o))=\{\lambda_{1}^{\rm rad}(B_r(o))\leq \lambda_{2}^{\rm rad}(B_r(o)) \leq \cdots\}$ be the radial spectrum of $B_r(o)$. The following identity is true. \begin{equation}\label{eqMain}\sum_{i=1}^{\infty}\frac{1}{\lambda_{i}^{\rm rad}(B_r(o))}=\int_{0}^{r}\frac{V(s)}{S(s)}ds\cdot \end{equation}If $\mathbb{M}_{h}^{n}$ is stochastically incomplete then \begin{equation}\label{eqMain2}\sum_{i=1}^{\infty}\frac{1}{\lambda_{i}^{\rm rad}(\mathbb{M}_{h}^{n})}=\int_{0}^{\infty}\frac{V(s)}{S(s)}ds<\infty\cdot \end{equation}Here $V(r)={\rm vol}(B_r(o))$ and $S(r)={\rm vol}(\partial B_r(o))$.

There are examples of non-rotationally symmetric geodesic balls with non-empty radial spectrum.