A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere $S^{2n+1}$ has at least $n+1$ simple periodic Reeb orbits. In this talk, I will consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least $n+1$ simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex. This is joint work with Miguel Abreu and Hui Liu.

The moduli space of Higgs bundles over a curve is a well known (singular) variety with an extremely rich geometry, in particular it is hyperKähler and becomes an integrable system after being equipped with the so called Hitchin morphism which, to any Higgs bundle, associates a finite cover of the base curve named spectral curve. Associated to the hyperKähler structure, Kapustin and Witten introduced in 2007, BBB and BAA-branes, predicting that they occur in pairs dual under mirror symmetry. An example of BBB-brane is a hyperKähler bundle supported on hyperKähler subvariety, and an example of BAA-brane is a flat bundle supported on a complex Lagrangian subvariety. Hitchin described in 2019 a family of subintegral systems lying on the critical loci of the Hitchin integrable system parametrized by spectral curves with a fixed number of singularities. The critical subsystem obtained by considering spectral curves with maximal number of singularities is a hyperKähler subvariety and the author, along with Oliveira, Peón-Nieto and Gothen, studied the BBB-branes constructed over it, and their image under Fourier-Mukai transform, which are supported on complex Lagrangian subvarieties. Surprinsingly, Hitchin showed that the critical subsystem obtained by considering spectral curves with 1 singularity is not a hyperKähler subvariety and he conjectured that only the critical subsystem with a maximal number of singularities is hyperKähler.

In this work, joint with Hanson, Horn and Oliveira, we study the critical subsystems with any number of singularities, showing that their image under Fourier-Mukai is supported on a certain family of complex Lagrangian subvarieties which we describe.

We discuss the typical behavior of two important quantities on compact manifolds with a Riemannian metric g: the number, c(T, g), of primitive closed geodesics of length smaller than T, and the error, E(L, g), in the Weyl law for counting the number of Laplace eigenvalues that are smaller than L. For Baire generic metrics, the qualitative behavior of both of these quantities has been understood since the 1970’s and 1980’s. In terms of quantitative behavior, the only available result is due to Contreras and it says that an exponential lower bound on c(T, g) holds for g in a Baire-generic set. Until now, no upper bounds on c(T, g) or quantitative improvements on E(L, g) were known to hold for most metrics, not even for a dense set of metrics. In this talk, we will introduce the concept of predominance in the space of Riemannian metrics. This is a notion that is analogous to having full Lebesgue measure in finite dimensions, and which, in particular, implies density. We will then give stretched exponential upper bounds for c(T, g) and logarithmic improvements for E(L, g) that hold for a predominant set of metrics. This is based on joint work with J. Galkowski.

This is a survey talk of current progress of mirror symmetry of Fano varieties. For a given smooth Fano variety X, it has been conjectured that there exists a Laurent polynomial called a (weak) Landau-Ginzburg mirror (or weak LG mirror shortly) which encodes a quantum cohomology ring structure of X. Tonkonog proved that one can find a weak LG mirror using a monotone Lagrangian torus in X. In this talk I will explain how to find a monotone Lagrangian torus using a Fano toric degeneration of X. If time permits, I will also describe a monotone Lagrangian torus in a given flag variety.

After some basic recalls on the notion of Gromov width of a symplectic manifold, I will focus on the case of toric manifolds. I shall explain how this symplectic capacity can be estimated and even computed. This is a joint work with C. Bonala.

– Europe/Lisbon
Room P3.10, Mathematics Building — Online

Gonçalo Oliveira, Instituto Superior Técnico, Universidade de Lisboa

Motivated by some conjectures originating in the Physics literature, I have recently been looking for closed geodesics in the K3 surfaces constructed by Lorenzo Foscolo. It turns out to be possible to locate several such with high precision and compute their index (their length is also approximately known). Interestingly, in my view, the construction of these geodesics is related to an open problem in electrostatics posed by Maxwell in 1873.

– Europe/Lisbon
Room P3.10, Mathematics Building — Online

Quantum Hall effect is well known physics experiment, featuring precise quantization of the Hall conductance in materials with imprecisely known characteristics. It is said to be one of the most striking examples of macroscopic manifestation of quantum phenomena.

One of the approaches to explain QHE is by constructing explicit N-particle wave functions, called Quantum Hall states. It was laid out and explored in the famous works of Laughlin, Haldane, Haldane-Rezayi, Wen-Niu, Avron-Seiler-Zograf and others. It became customary to study QH states analytically on geometric backgrounds, e.g. compact Riemann surfaces. This approach turned out to be unexpectedly fruitful, leading in particular, to the discovery of topological phases of matter, anyons, non-abelian statistics, topological quantum computing and much more.

I will review this approach and talk about the program to set this approach on a rigorous mathematical footing, and to prove various conjectures in the field. I will also talk about what in my opinion are the most exciting things to do going forward. The keywords for the lectures include holomorphic line bundles, Riemann surfaces, moduli spaces, Bergman kernel, determinant point processes, Coulomb gas, etc.

Some references

R. B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50, 1395 (1983).

F. D. M. Haldane and E. H. Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect, Phys. Rev. B 31, 2529 (1985).

X. G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B41, 9377 (1990).

J. E. Avron, R. Seiler, and P. G. Zograf, Adiabatic Quantum Transport: Quantization and Fluctuations, Phys. Rev. Lett. 73, 3255 (1994).

S. Klevtsov, X. Ma, G. Marinescu, and P. Wiegmann, Quantum Hall effect and Quillen metric, Commun. Math. Phys. 349, 819 (2017).

S. Klevtsov, Laughlin states on higher genus Riemann surfaces, Commun. Math. Phys. 367, 837 (2019).

S. Klevtsov and D. Zvonkine, Geometric Test for Topological States of Matter, Phys. Rev. Lett. 128, 036602 (2021).

Supported by CAMGSD UIDB/04459/2020 and UIDP/04459/2020.

– Europe/Lisbon
Room P3.10, Mathematics Building — Online

Quantum Hall effect is well known physics experiment, featuring precise quantization of the Hall conductance in materials with imprecisely known characteristics. It is said to be one of the most striking examples of macroscopic manifestation of quantum phenomena.

One of the approaches to explain QHE is by constructing explicit N-particle wave functions, called Quantum Hall states. It was laid out and explored in the famous works of Laughlin, Haldane, Haldane-Rezayi, Wen-Niu, Avron-Seiler-Zograf and others. It became customary to study QH states analytically on geometric backgrounds, e.g. compact Riemann surfaces. This approach turned out to be unexpectedly fruitful, leading in particular, to the discovery of topological phases of matter, anyons, non-abelian statistics, topological quantum computing and much more.

I will review this approach and talk about the program to set this approach on a rigorous mathematical footing, and to prove various conjectures in the field. I will also talk about what in my opinion are the most exciting things to do going forward. The keywords for the lectures include holomorphic line bundles, Riemann surfaces, moduli spaces, Bergman kernel, determinant point processes, Coulomb gas, etc.

Some references

R. B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50, 1395 (1983).

F. D. M. Haldane and E. H. Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect, Phys. Rev. B 31, 2529 (1985).

X. G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B41, 9377 (1990).

J. E. Avron, R. Seiler, and P. G. Zograf, Adiabatic Quantum Transport: Quantization and Fluctuations, Phys. Rev. Lett. 73, 3255 (1994).

S. Klevtsov, X. Ma, G. Marinescu, and P. Wiegmann, Quantum Hall effect and Quillen metric, Commun. Math. Phys. 349, 819 (2017).

S. Klevtsov, Laughlin states on higher genus Riemann surfaces, Commun. Math. Phys. 367, 837 (2019).

S. Klevtsov and D. Zvonkine, Geometric Test for Topological States of Matter, Phys. Rev. Lett. 128, 036602 (2021).

Supported by CAMGSD UIDB/04459/2020 and UIDP/04459/2020.

The Minimal Model Program (MMP) is a far reaching conjecture in birational geometry which aims at constructing a good representative (minimal model) of any given complex projective variety W. When such a model exists it might not be unique and so it becomes natural to study the relations between them. In the case when W is covered by rational curves, its minimal model is a Mori fibre space, that is, a fibration whose generic fibre is positively curved, and its uniqueness is encoded in the notion of birational rigidity. In this talk we will give an introduction to the ideas of the MMP with the background of Fano threefold complete intersections.

We will give a complete picture of the metric geometry of Calabi-Yau manifolds along degenerations of complex structures, which holds for all dimensions. In particular, we will classify the Gromov-Hausdorff limits on all scales, describe the singularity formation, and formulate a more general conjecture. This is based on my joint work with Song Sun (arXiv: 1906.03368).

Consider a compact subset K of a closed symplectic manifold M. We say that K is SH-visible if its relative symplectic cohomology does not vanish over the Novikov field. With Cheuk Yu Mak and Yuhan Sun, we recently proved that SH-visibility is equivalent to K being heavy as defined by Entov-Polterovich. l will recall these notions and explain the proof. If time permits I will also discuss some consequences.

There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there might also exist other Einstein metrics on them that are utterly different from the ones we currently know.

In this lecture, I will present two quite different characterizations of the known Einstein metrics on del Pezzo surfaces. These results imply, in particular, that the known Einstein metrics exactly sweep out a single connected component of the Einstein moduli space. I will then briefly indicate the role these results play in current avenues of research.

Donaldson (folklore) asked whether Lagrangian Dehn twists always generate the symplectic mapping class groups in real dimension four. So far, all known examples indicate this is true, even though the symplectic Torelli group is generally much larger than the algebraic one. Yet there are only very few cases people could prove this as a theorem.

We will define a notion of "positive rational surfaces", which is equivalent to the ambient symplectic manifolds of (symplectic) log Calabi-Yau pairs. We compute the symplectic Torelli group for the positive rational surfaces and confirm Donaldson's conjecture as a result. We also answer several other questions about the symplectic Torelli groups in dimension $4$.

Complete Calabi-Yau metrics provide singularity models for limits of Kahler-Einstein metrics. We study complete Calabi-Yau metrics with Euclidean volume growth and quadratic curvature decay. It is known that under these assumptions the metric is always asymptotic to a unique cone at infinity. Previous work of Donaldson-S. gives a 2-step degeneration to the cone in the algebro-geometric sense, via a possible intermediate object (a K-semistable cone). We will show that such intermediate K-semistable cone does not occur. This is in sharp contrast to the case of local singularities. This result together with the work of Conlon-Hein also give a complete algebro-geometric classification of these metrics, which in particular confirms Yau’s compactification conjecture in this setting. I will explain the proof in this talk, and if time permits I will describe a conjectural picture in general when the curvature decay condition is removed. Based on joint work with Junsheng Zhang (UC Berkeley).

The notion of a generalized Kahler (GK) structure was introduced in the early 2000's by Hitchin and Gualtieri in order to provide a mathematically rigorous framework of certain nonlinear sigma model theories in physics. Since then, the subject has developed rapidly. It is now realized, thanks to more recent works of Hitchin, Goto, Gualtieri, Bischoff and Zabzine, that GK structures are naturally attached to Kahler manifolds endowed with a holomorphic Poisson structure. Inspired by Calabi's program in Kahler geometry, which aims at finding a "canonical" Kahler metric in a fixed deRham class, I will present in this talk an approach towards a “generalized Kahler" version of Calabi's problem motivated by an infinite dimensional moment map formalism, and using the Bismut-Ricci flow introduced by Streets and Tian as analytical tool. As an application, we give an essentially complete resolution of the problem in the case of a toric complex Poisson variety. Based on joint works with J. Streets and Y. Ustinovskiy.

– Europe/Lisbon
Online

Joé Brendel, School of Mathematical Sciences, Tel Aviv University

Toric symplectic manifolds contain an interesting and well-studied family of Lagrangian tori, called toric fibres. In this talk, we address the natural question of which toric fibres are equivalent under Hamiltonian diffeomorphisms of the ambient space. On one hand, we use a symmetric version of McDuff's probes to construct such equivalences and on the other hand, we give certain obstructions coming from Chekanov's classification of product tori in symplectic vector spaces combined with a lifting trick from toric geometry. We will discuss many four-dimensional examples in which a full classification can be achieved.

What does it mean for a category to be endowed with a compatible differentiable structure? In this talk, we will discuss the interplay of a categorical structure with that of a smooth manifold, and show how to describe such categories infinitesimally, similarly as to how we construct the Lie algebra of a Lie group. We will generalise the notion of rank from linear algebra to morphisms of Lie categories, and introduce the notion of an extension of a Lie category to a groupoid. Examples of Lie categories arising in differential geometry and in physics will be highlighted.

I will talk about geometric compactifications of moduli spaces of K3 surfaces, similar in spirit to the Deligne-Mumford moduli spaces of stable curves. Constructions borrow ideas from the tropical and integral-affine geometry and mirror symmetry. The main result is that in many common situations there exists a geometric compactification which is toroidal, and many of these compactifications can be described explicitly using tropical spheres with 24 singular points. Much of this talk is based on the joint work with Philip Engel.

Famous results of N. Hitchin establish existence of an integrable structure of the Hitchin moduli spaces. The goal of this talk will be to discuss a more explicit approach known in the integrable models literature as separation of variables, how it can be applied to the quantisation of the Hitchin system, and how the result is related to the analytic Langlands correspondence studied by Etingof, Frenkel and Kazhdan.

Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

The talk is based on joint work in progress with Anuk Dayaprema (UW-Madison).

Bigness of the cotangent bundle is a negativity property of the curvature which has important complex analytic consequences, such as on the Kobayashi hyperbolicity properties and the GGL-conjecture for surfaces. We present a birational criterion for a surface to have big cotangent bundle that takes in account the singularities present in the minimal model and describe how it improves upon other criterions. The criterion allows certain geographic regions of surfaces of general type to have big cotangent bundle, that other criterions can not reach. In this spirit, we produce the examples with the lowest slope $c_1^2/c_2$ having big cotangent bundle that are currently known.

– Europe/Lisbon
Room P3.10, Mathematics Building — Online

Following A. Beauville, a complex algebraic variety $X$ is said to be symplectic if it admits a holomorphic symplectic form $\omega$ on its smooth locus such that, for every resolution $\pi: Y \to X$, $\pi^*\omega$ extends to a holomorphic $2$-form on $Y$. When this extension is actually non-degenerate (a de facto symplectic form) on $Y$, we call $\pi$ a symplectic (or crepant) resolution.

Let $G$ be a complex reductive group and $A$ an abelian variety of dimension $d$. The aim of this talk is to show that all moduli spaces of $G$-Higgs bundles over $A$ are symplectic varieties, and that, for $G=\mathrm{GL}(n,\mathbb C)$, the canonical Hilbert-Chow morphism is a symplectic resolution if and only if $d=1$.

Moreover, using a little representation theory, we can obtain explicit expressions for the Poincaré polynomials of all Hilbert-Chow resolutions (either $d=1$, all $n$; or $n=1,2,3$ and all $d$). This is joint work with I. Biswas and A. Nozad.

The question whether a symplectic manifold embeds into another is central in symplectic topology. Since Gromov nonsqueezing theorem, it is known that this is a different problem from volume preserving embedding. There are several nice results about symplectic embeddings between open subsets of $\mathbb R^{2n}$ showing that even for those examples the question can be completely nontrivial. The problem is substantially more well understood when the manifolds are toric domains and have dimension $4$, mostly because of obstructions coming from embedded contact homology (ECH). In this talk we are going to discuss symplectic embedding problems in which the target manifold is the disk cotangent bundle of a two-dimensional sphere, i.e., the set consisting of the covectors with norm less than $1$ over a Riemannian sphere. We shall talk about some tools such as ECH capacities and action angle coordinates. Much of this talk is based on joint works with Vinicius Ramos and Alejandro Vicente.

One of the distinctive feature of the $d$-dimensional torus $T^d$ is that it admits a faithful smooth action by $\mathrm{SL}_d(\mathbb Z)$, so one might wonder whether such an action (or any nontrivial action) also exists for exotic tori i.e. smooth $d$-manifolds that are homeomorphic but not diffeomorphic to $T^d$. I will discuss this and related questions in the talk, based on joint work with M. Bustamante, A. Kupers, and B. Tshishiku.

Existence of symplectic embeddings of $k$ disjoint balls of given capacites $c_1,\ldots, c_k$ into a given symplectic manifold is a central problem in symplectic topology. However, beside a few examples, very little is known about the space of all such embeddings. In this talk, I will discuss the case of rational $4$-manifolds of small Euler numbers, with a special attention to the minimal manifolds $\mathbb{C}P^2$ and $S^2\times S^2$. For rational manifolds, a very rich and intricate picture emerges that blends symplectic topology, complex geometry, and algebraic topology.

Studying metrics with special curvature properties on compact Kähler manifolds is a fundamental problem in Kähler geometry. In this talk, I will focus on the existence and uniqueness of singular Kähler-Einstein metrics whose singular behavior is prescribed. These results are based on a series of joint works with T. Darvas and C. Lu.

With Spencer Whitehead, we developed a systematic framework to study instantons on $\mathbb R^4$ that are invariant under groups of isometries. In this presentation, I will describe this framework and some results obtained using it.

Let $X$ be a smooth, complex Fano $4$-fold, and $\rho(X)$ its Picard number. We will discuss the following theorem: if $\rho(X)>12$, then $X$ is a product of del Pezzo surfaces. This implies, in particular, that the maximal Picard number of a Fano $4$-fold is $18$. After an introduction and a discussion of examples, we explain some of the ideas and techniques involved in the proof.

In this talk I will discuss a natural generalization of symplectic toric manifolds, for which the symmetry is given by a (symplectic) torus bundle, rather than a torus. The aim will be to explain how the Abreu-Guillemin theory of toric Kähler metrics extends to this setting. This is based on an ongoing project with Miguel Abreu and Rui Loja Fernandes.

– Europe/Lisbon
Online

Filip Živanović, Simons Center for Geometry and Physics at Stony Brook

We study open symplectic manifolds with pseudoholomorphic $\mathbb C^*$-actions whose $S^1$-part is Hamiltonian, and construct their associated symplectic cohomology. From this construction, we obtain a filtration on quantum/ordinary cohomology that depends on the choice of the $\mathbb C^*$-action. One should think about this filtration as a Floer-theoretic analogue of the Atiyah-Bott filtration. We construct filtration functional on the Floer chain complex, allowing us to compute the aforementioned filtration via Morse-Bott spectral sequence that converges to symplectic cohomology, which is readily computable in examples. We compare our filtration with known ones from algebraic geometry/representation theory literature. Time-allowing, I may present the $S^1$-equivariant picture as well. This is joint work with Alexander Ritter.

In 1987, Gromov and Eliashberg showed that if a sequence of diffeomorphisms preserving a symplectic form C⁰ converges to a diffeomorphism, the limit also preserves the symplectic form — even though this is a C¹ condition. This result gave rise to the notion of symplectic homeomorphisms, i.e. elements of the C⁰-closure of the group of symplectomorphisms in that of homeomorphisms, and started the study of "continuous symplectic geometry".

In this talk, I will present recent progress in understanding the fundamental group of the C⁰-closure of the group of Hamiltonian diffeomorphisms in that of homeomorphisms. More precisely, I will explain a sufficient condition which ensures that certain essential loops of Hamiltonian diffeomorphisms remain essential when seen as "Hamiltonian homeomorphisms". I will illustrate this method (and its limits) on toric manifolds, namely complex projective spaces, rational products of 2-spheres, and rational 1-point blow-ups of CP².

Our condition is based on (explicit) computation of the spectral norm of loops of Hamiltonian diffeomorphisms which is of independent interest. For example, in the case of 1-point blow-ups of CP², I will show that the spectral norm exhibits a surprising behavior which heavily depends on the choice of the symplectic form. This is joint work with Vincent Humilière and Alexandre Jannaud.