Extremal Kähler metrics were introduced by Calabi in the 80’s as a type of canonical Kähler metric on a Kähler manifold, and are a generalisation of constant scalar curvature Kähler metrics in the case when the manifold admits automorphisms. A natural question is when the blowup of a manifold in a point admits an extremal Kähler metric. We completely settle the question in terms of a finite dimensional moment map/GIT condition, generalising work of Arezzo-Pacard, Arezzo-Pacard-Singer and Székelyhidi. Our methods also allow us to deal with a certain semistable case that has not been considered before, where the original manifold does not admit an extremal metric, but is infinitesimally close to doing so. As a consequence of this, we solve the first non-trivial special case of a conjecture of Donaldson. This is joint work with Ruadhaí Dervan.

In this talk I will consider a moduli space of projective varieties enhanced with a certain frame of its cohomology bundle. In many examples such as elliptic curves, abelian varieties and Calabi-Yau varieties, and conjecturally in general, this moduli space is a quasi-affine variety. There are certain vector fields on this moduli which are algebraic incarnation of differential equations of automorphic forms. Using these vector fields one can construct foliations with algebraic leaves related to Hodge loci. The talk is based on my book Modular and Automorphic Forms & Beyond, Monographs in Number Theory, World Scientific (2021) in which the Tupi name ibiporanga (pretty land) for such a moduli space is suggested.

Together with Javier Reyes, in https://arxiv.org/abs/2110.10629 we have been able to construct compact 4-manifolds $3\mathbb{CP}^2\#(19-K^2)\overline{\mathbb{CP}}^2$ with complex structures for $K^2=1,2,3,4,5,6,7,8,9$. The cases $K^2=7,9$ are completely new in the literature, and this finishes with the whole range allowed by the technique of Q-Gorenstein smoothing (rational blow-down). But one can go further: Is it possible to find minimal exotic $3\mathbb{CP}^2\#(19-K^2)\overline{\mathbb{CP}}^2$ for $K^2\geq10$? Here it would be much harder to prove the existence of complex structures, but, as a motivation, there is not even one example for $K^2 > 15$, and very few for $10 \leq K^2 \leq 15$ (see e.g. works by Akhmedov, Park, Baykur). In this talk I will explain the constructions in connection with the geography of spheres arrangements in $K3$ surfaces, where the question of the title arises. We do not have an answer. So far we have been implementing what we know in computer searches, finding these very rare exotic surfaces for $K^2=10,11,12$. This is a new and huge world which promises more findings, we have explored very little.

(joint work with M. Möller) The complexity of singular fibers of the Hitchin system stems from the variety of singularities of the spectral curve. In this talk I will explain how to modify the rank 2 Hitchin base, such that the family of spectral curves can be resolved to a family of semi-stable nodal curves. This allows to extend the Hitchin system to the singular locus of the modified Hitchin base by well-understood compactified Jacobians of semi-stable curves

(Joint with Y. Lekili) If someone gives you a variety with a singular point, you can try and get some understanding of what the singularity looks like by taking its “link”, that is you take the boundary of a neighbourhood of the singular point. For example, the link of the complex plane curve with a cusp $y^2 = x^3$ is a trefoil knot in the 3-sphere. I want to talk about the links of a class of 3-fold singularities which come up in Mori theory: the compound Du Val (cDV) singularities. These links are 5-dimensional manifolds. It turns out that many cDV singularities have the same 5-manifold as their link, and to tell them apart you need to keep track of some extra structure (a contact structure). We use symplectic cohomology to distinguish the contact structures on many of these links.

Questions can be motivated from dynamical systems about the size of complements of a disjoint collection of Lagrangian tori in a symplectic manifold. We will discuss the simplest case, namely the complement of the integral product Lagrangians, $L(k,l)$ with $k,l \in \mathbb{N}$, inside $\mathbb{C}^2$. Here $L(k,l) = \{ |z_1| = k, |z_2|=l \}$. We will make some computations of the Gromov width and then describe joint work with Ely Kerman on the existence of Lagrangian tori in the complement.

A celebrated conjecture of Yau states that the existence of a Kähler metric of constant scalar curvature on a projective manifold should be equivalent to a purely algebraic stability condition. Much progress have been done on this conjecture, which culminated in what is now called the Yau-Tian-Donaldson program. In this talk, I will explain the key role played by quantization methods in this program, and how they can be improved by a semiclassical study of the quantum noise of Berezin-Toeplitz quantization.

This is partly based on joint works in collaboration with Victoria Kaminker, Leonid Polterovich and Dor Shmoish.

Wobbly bundles are the complement to very stable bundles, a dense open set of the moduli space of vector bundles. This notion was generalised to arbitrary fixed points of the $C^\ast$ action on the moduli space of Higgs bundles by Hausel and Hitchin. In this talk, after introducing the meaningful notions and motivating them, I will analyse the geometry of higher wobbly components in rank three. In particular, I will focus on an extension of Drinfeld's conjecture about pure codimensionality of the wobbly locus, as well as the relation with real forms. This is joint work with Pauly.

Let K be a knot in the 3-sphere. I will explain how one can count minimal discs in hyperbolic 4-space which have ideal boundary equal to K, and in this way obtain a knot invariant. In other words the number of minimal discs depends only on the isotopy class of the knot. I think it should actually be possible to define a family of link invariants, counting minimal surfaces filling links, but at this stage this is still just a conjecture. “Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how these counts of minimal discs can be seen as Gromov-Witten invariants for the twistor space of hyperbolic 4-space. Whilst Gromov-Witten theory suggests the overall strategy for defining the minimal surface link-invariant, there are significant differences in how to actually implement it. This is because the geometry of both hyperbolic space and its twistor space become singular at infinity. As a consequence, the PDEs involved (both the minimal surface equation and J-holomorphic curve equation) are degenerate rather than elliptic at the boundary. I will try and explain how to overcome these complications.

The theory of Gromov-Witten invariants is a curve counting theory defined by integration on the moduli of stable maps. Varying the stability condition gives alternative compactifications of the moduli space and defines similar invariants. One example is epsilon-stable quasimaps, defined for a large class of GIT quotients. When epsilon tends to infinity, one recovers Gromov-Witten invariants. When epsilon tends to zero, the invariants are closely related to the B-model in physics. The space of epsilon's has a wall-and-chamber structure. In this talk, I will explain how wall-crossing helps to compute the Gromov-Witten invariants and sketch a proof of the wall-crossing formula.

The symplectomorphism groups $\operatorname{Symp}(M, \omega)$ of ruled surfaces have been started by Gromov, McDuff, and Abreu, etc, using J-holomorphic techniques. For rational ruled surfaces, the topological structure of $\operatorname{Symp}(M, \omega)$ is better understood, while for irrational cases our only knowledge is for minimal ruled surfaces. In this talk, we apply the J-inflation techniques of Anjos-Li-Li-Pinsonnault to irrational non-minimal ruled surfaces and prove a stability result for $\operatorname{Symp}(M, \omega)$. As an application, we find symplectic mapping classes that are smoothly but not symplectically isotopic to identity. The talk is based on joint works with Olguta Buse.

A very stable vector bundle over a curve is a vector bundle having no non-zero nilpotent Higgs fields. They were introduced by Drinfeld and studied by Laumon in connection with the nilpotent cone of the Hitchin system. According to Drinfeld non-very stable bundles, also called wobbly bundles, form a divisor in the moduli space of vector bundles. In this talk I will try to explain the motivations for studying the properties of wobbly divisors, with a special focus on the rank-2 (joint work with S. Pal) and rank-3 case (joint work with A. Peon-Nieto).

A very popular model in machine learning is the feedforward neural network (FFN). After a brief introduction to machine learning, we describe FFNs which represent sections of holomorphic line bundles on complex manifolds, and software which uses them to get numerical approximations to Ricci flat Kähler metrics.

A symplectic capacity is a functor that to each symplectic manifold (possibly in a restricted subclass) assigns a nonnegative number. The Lagrangian capacity is an example of such an object. In this talk, I will state a conjecture concerning the Lagrangian capacity of a toric domain. Then, I will present two results concerning this conjecture. First, I will explain a proof of the conjecture in the case where the toric domain is convex and 4-dimensional. This proof makes use of the Gutt-Hutchings capacities as well as the McDuff-Siegel capacities. Second, I will explain a proof of the conjecture in full generality, but assuming the existence of a suitable virtual perturbation scheme which defines the curve counts of linearized contact homology. This second proof makes use of Siegel's higher symplectic capacities.

A Riemannian metric on a closed manifold is called Zoll when all of its geodesics are closed and have the same period. An infinite dimensional family of Zoll metrics on the two-dimensional sphere were constructed by Otto Zoll in the beginning of 1900's, but many questions about them remain unanswered. In this talk, I will explain my motivation to look for higher dimensional analogues of Zoll metrics, where closed geodesics are replaced by embedded minimal spheres of codimension one. Then, I will discuss some recent results about the construction and geometric understanding of these new geometries. This is a joint project with F. Marques (Princeton) and A. Neves (UChicago).

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams. The main goal of this talk is to describe how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to the Ehrhart (quasi-)polynomial of its toric diagram. This is part of joint work with Leonardo Macarini and Miguel Moreira (arXiv:2202.00442).

Joint with the Workshop Fuchsian Differential Equations.

In the 1860's, Fuchs looked at linear ordinary differential equations with variable coefficients, say, holomorphic functions. The issue was that the highest derivative could have a coefficient which vanished at some point. Just think of the equation $x^2 y' + y = 0$ with singularity at $x = 0$, or the hypergeometric equations.

So he asked, quite naturally, when the solutions would have a moderate behaviour at these singular points: he was willing to allow meromorphic functions, roots of them, and logarithms (examples suggest these are a good class), but not functions with worse behaviour (the precise definition will be given in the talk).

And — surprise — he was able to give an algebraic and effective criterion to characterize these "regular" singularities (the terminology is pityful). His discovery and the proof of it was like an explosion, with many subsequent works (Frobenius, Thomé, Fabry, ...) and many important results, applications and generalizations.

After describing the approach of Fuchs, we will briefly address Grothendieck's conjecture in this context: He proposes to understand the differential equation (when defined over $\mathbb{Q}$) by means of its reduction modulo prime numbers $p$. Fortunately, his conjecture is still wide open, so researchers have something to think about.

Albert Fathi proved in the late 70's that the group of volume preserving homeomorphisms of the n-sphere is simple for n at least 3, but the case of the 2-sphere remained open until recently. In this talk, I will present results obtained in several works with Dan Cristofaro-Gardiner, Cheuk-Yu Mak, Sobhan Seyfaddini and Ivan Smith on the structure of the group of area preserving homeomorphisms of surfaces, which include in particular a solution of this problem. Even if the considered objects are not smooth (they are just homeomorphisms), the tools we use come from symplectic topology.

We will discuss a simple proof that the symplectic mapping class groups of many K3s are infinitely generated, extending a recent result of Sheridan and Smith. The argument will be based on some basic family Seiberg-Witten theory and algebraic geometry.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's K-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's K-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

The Gale correspondence provides a duality between sets of n general points in projective spaces $P^s$ and $P^r$ when $n$ equals $r + s + 2$. By a result of Mukai, the blow-up of $P^4$ at $8$ points say $X$, can be realized as a moduli space of torsion-free rank $2$ semi-stable sheaves (with certain fixed Chern class datum) on the blow-up of $P^2$ in $8$ Gale dual points. In a recent work, Casagrande, Codogni and Fanelli use this to describe the Mori chamber decomposition of the effective cone of divisors of $X$. It was shown by Castravet and Tevelev that the blow-up of $P^r$ at $n$ points for the case when $r\geq 5$ and $n\geq r + 4$ is no longer a Mori dream space. In joint work with Carolina Araujo, Ana-Maria Castravet and Diletta Martinelli we show that even in this case it is possible to give a Mori chamber type decomposition for a part of the effective cone.

The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation to a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectomorphic to balls. This is joint work with Y. Ostrover and D. Sepe.

The Yamabe invariant is a real-valued diffeomorphism invariant coming from Riemannian geometry. Using Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a Kähler surface is determined by its Kodaira dimension. We consider the extent to which this remains true when the Kähler hypothesis is removed. This is joint work with Claude LeBrun.

I will discuss joint work with Jonas Stelzig in which we consider the beginnings of a bigraded analogue of rational homotopy theory adapted to complex manifolds, in a somewhat different fashion than that of Neisendorfer-Taylor which appeared in the 1970’s soon after Sullivan’s Infinitesimal Computations in Topology. Taking cues from an additive decomposition theorem for double complexes, we define two natural notions of formality for our basic objects — commutative bigraded bidifferential algebras — which place both bigraded components of the de Rham differential on equal footing. These notions are related by the ddbar-lemma (the additive property used to show formality, in the usual sense, of compact complex manifolds admitting a Kähler metric). We consider obstructions to these notions of formality, taking in Bott-Chern cohomology classes and outputting classes in a chain complex of Demailly-Schweitzer, whose construction mimics those of classical Massey products and extends the triple products landing in Aeppli cohomology considered by Angella-Tomassini; we also touch upon their behavior under blow-ups and more generally positive-degree holomorphic maps.

It is well known that for surfaces the positivity property of the cotangent bundle $\Omega^1_X$ called bigness implies hyperbolic properties. We give a criterion for bigness of $\Omega^1_X$ involving the singularities of the canonical model of $X$ and compare it with other criterions. The criterion involves invariants of the canonical singularities whose values were unknown. We describe a method to find the invariants and obtain formulas for the $A_n$ singularities. An application of this work is to determine for which degrees do hypersurfaces in $\mathbb {P}^3$ have deformations with big cotangent bundles and have symmetric differentials of low degrees.

Gromov-Witten invariants of a given Kahler target space are defined as suitable intersection numbers in moduli spaces of stable maps of complex curves into the target space. Their K-theoretic analogues are defined as holomorphic Euler characteristics of suitable vector bundles over these moduli spaces.

We will describe how the Kawasaki-Riemann-Roch theorem expressing holomorphic Euler characteristics in cohomological terms leads to the adelic formulas for generating functions encoding K-theoretic Gromov-Witten invariants.

Kontsevich suggested that enumerative predictions of Mirror Symmetry should follow directly from Homological Mirror Symmetry. This requires a natural construction of analogues of Gromov-Witten invariants associated to any A-infinity Calabi-Yau category, with some extra choices. I will explain what these choices are and survey two approaches to this construction, one in genus zero and another (conjectural) in all genera.

I will report on joint work in progress with Aleksandar Milivojevic (MPIM Bonn) on the elementary topology of the space of almost complex structures on a manifold. First I will describe a certain natural parametrization and associated stratification of the space of linear complex structures on a vector space and give a lower bound for the number of complex k-planes jointly preserved by two linear complex structures. Then I will focus on dimension 6 and prove a formula for the homological intersection of two orthogonal almost complex structures on a Riemannian 6-manifold when these are regarded as sections of the twistor space.

I will discuss some of the unusual properties, in geometry and physics, of a family of Calabi-Yau threefolds fibered by elliptic curves. I will compare it to a construction by Elkies and a classical results of Burkhardt. This leads to some open questions.

Virasoro constraints for Gromov-Witten invariants have a rich history tied to the very beginning of the subject, but recently there have been many developments on the sheaf side. In this talk I will survey those developments and talk about joint work with A. Bojko and W. Lim where we propose a general conjecture of Virasoro constraints for moduli spaces of sheaves and formulate it using the vertex algebra that D. Joyce recently introduced to study wall-crossing. Using Joyce's framework we can show compatibility between wall-crossing and the constraints, which we then use to prove that they hold for moduli of stable sheaves on curves and surfaces with $h^{0,1}=h^{0,2}=0$. In the talk I will give a rough overview of the vertex algebra story and focus on the ideas behind the proof in the case of curves.

The Yau-Tian-Donaldson (YTD) conjecture predicts that the existence of an extremal metric (in the sense of Calabi) in a given Kähler class of Kähler manifold is equivalent to a certain algebro-geometric notion of stability of this class. In this talk, we will discuss the resolution of this conjecture for a certain class of toric fibrations, called semisimple principal toric fibrations. After an introduction to the Calabi Problem for general Kähler manifolds, we will focus on the toric setting. Then we will see how to reduce the Calabi problem on the total space of a semisimple principal toric fibration to a weighted constant scalar curvature Kähler problem on the toric fibers. If the time allows, I will give elements of proof.

Every smooth fiber bundle admits a complete Ehresmann connection. I will talk about the story of this theorem and its relation with Riemannian submersions. Then, after discussing some foundations of Riemannian geometry of Lie groupoids and stacks, I will present a generalization of the theorem into this framework, which somehow answers an open problem on linearization. Talk based on collaborations with my former student M. de Melo.

Let $K$ be a compact Lie group. I will review the construction of Mabuchi geodesic families of $K\times K$–invariant Kähler structures on $T^\ast K$, via Hamiltonian flows in imaginary time generated by a strictly convex invariant function on $\operatorname{Lie}K$, and the corresponding geometric quantization. At infinite geodesic time, one obtains a rich mixed polarization of $T^\ast K$, the Kirwin-Wu polarization, which is then continuously connected to the vertical polarization of $T^\ast K$. The geometric quantization of $T^\ast K$ along this family of polarizations is described by a generalized coherent state transform that, as geodesic time goes to infinity, describes the convergence of holomorphic sections to distributional sections supported on Bohr-Sommerfeld cycles. These are in correspondence with coadjoint orbits $O_{\lambda+\rho}$. One then obtains a concrete (quantum) geometric interpretation of the Peter-Weyl theorem, where terms in the non-abelian Fourier series are directly related to geometric cycles in $T^\ast K$. The role of a singular torus action in this construction will also be emphasized. This is joint work with T. Baier, J. Hilgert, O. Kaya and J. Mourão.

In an influential article from the 1970s, Albert Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n\geq 3$, asked if the same statement holds in dimension $2$. In a joint work with Cristofaro-Gardiner and Humilière, we proved that the group of compactly supported area-preserving homeomorphisms of the $2$-disc is not simple. This answers Fathi's question and settles what is known as "the simplicity conjecture" in the affirmative.

In fact, Fathi posed a more general question about all compact surfaces: is the group of "Hamiltonian homeomorphisms" (which I will define) simple? In my talk, I will review recent joint work with Cristofaro-Gardiner, Humilière, Mak and Smith answering this more general question of Fathi. The talk will be for the most part elementary and will only briefly touch on Floer homology which is a crucial ingredient of the solution.

We give an effective characterization of quasi-abelian surfaces extending to the quasi-projective setting results of Enriques and Chen-Hacon. This is a joint work with M. Mendes Lopes and R. Pardini.

The four dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. In 2012, McDuff and Schlenk computed this function for a ball. The function has a delicate structure known as an infinite staircase. This implies infinitely many obstructions are needed to know when an embedding can exist. Based on work with McDuff, Pires, and Weiler, we will discuss the classification of which Hirzebruch surfaces have infinite staircases. We will focus on the part of the argument where symplectic embeddings are constructed via almost toric fibrations.

Given a symplectic manifold, one can ask what Lagrangian submanifolds it contains. I will discuss this question for one of the simplest examples of a non-trivial symplectic manifold, namely the cotangent bundle of the 2-sphere. Specifically, I will present a result about monotone Lagrangian tori as objects in the Fukaya category. If time permits, I will also discuss the problem of classifying Lagrangian tori up to Hamiltonian isotopy.