Lagrangian mean curvature flow is potentially a powerful tool in solving problems in symplectic topology. One of the key challenges is the understanding of formation of singularities, which is conjectured to have links to J-holomorphic curves, stability conditions and the Fukaya category. Unlike the usual mean curvature flow for hypersurfaces, here one is expected to have to tackle singularities modelled on translating solutions to the flow. I will describe joint work with Felix Schulze and Gabor Szekelyhidi which allows one to recognize a singularity model in Lagrangian mean curvature flow as a translator - this is the first such result in any form of mean curvature flow beyond curves.

It is natural to ask whether an action of a finite cyclic group extends to a circle action. Here, the action is on a symplectic manifold of dimension four. Admitting a circle action implies that a simply connected closed symplectic four-manifold is either the projective plane or obtained from an $S^2$ bundle over $S^2$ by $k$ blowups. I will show that for $k$ small enough, any cyclic action that is trivial on homology extends to a circle action, and present a case in which the action does not extend. I will also discuss how we approach this question for a general $k$. The proofs combine holomorphic and combinatorial methods. The talk is based on a joint work with River Chiang.

The Thomas-Yau conjecture is an open-ended program to relate special Lagrangians to stability conditions in Floer theory, but the precise notion of stability is subject to many interpretations. I will focus on the exact case (Stein Calabi-Yau manifolds), and deal only with almost calibrated Lagrangians. I will attempt a formulation of Thomas-Yau semistability condition (meant to be less ambitious than Joyce’s program), and focus primarily on the symplectic aspects, and the technique of integration over the moduli space of holomorphic discs.

Enumerative geometry sinks its roots many centuries back in time. In the last decades, ideas coming from physics brought innovation to this research area, with both new techniques and the emergence of new rich geometrical structures. As an example, Gromov-Witten theory, focusing on symplectic invariants defined as counting numbers of curves on a target space, led to the notion of quantum cohomology and quantum differential equations (qDEs).

The qDEs define a class of ordinary differential equations in the complex domain, whose study represents a challenging active area in both contemporary geometry and mathematical physics. The qDEs define rich invariants attached to smooth projective varieties. These equations, indeed, encapsulate information not only about the enumerative geometry of varieties but even (conjecturally) of their topology and complex geometry. The way to disclose such a huge amount of data is through the study of the asymptotics and monodromy of their solutions. This talk will be a gentle introduction to the study of qDEs, their relationship with derived categories of coherent sheaves (in both non-equivariant and equivariant settings), and a theory of integral representations for its solutions. Overall, the talk will be a survey of the results of the speaker in this research area.

We introduce a notion of staticity for non-compact spaces which encompasses several known examples including any domain in hyperbolic space whose boundary is a non-compact totally umbilical hypersurface. For a (time-symmetric) initial data set modeled at infinity on any of these latter examples, we formulate and prove a positive mass theorem in the spin category under natural dominant energy conditions (both on the interior and along the boundary) whose rigidity statement in particular retrieves a recent result by Souam to the effect that no such umbilical hypersurface admits a compactly supported deformation keeping the original lower bound for the mean curvature. A key ingredient in our approach is the consideration of a new boundary condition on spinors which somehow interpolates between chirality and MIT bag boundary conditions. Joint work with S. Almaraz (arXiv:2206.09768, to appear in ASNS Pisa).

The proof for the resolution of singularities in characteristic $0$ is built on a complicated induction frame, doing one main job: it allows one to define at each stage of the resolution process and at each singular point of the current variety a local invariant $\operatorname{inv}_a(X) $ — a string of natural integers $(n_1,n_2,...,n_k)$, considered lexicographically.

This invariant, in turn, performs two jobs:

1. It defines the center of the blowups to which the singularities are submitted as the locus of points where the invariant assumes its maximal value.
2. It drops after each blowup at the points which have been modified by the blowup.

As the lexicographic order is a well ordering, one arrives in finitely many steps at the minimal value of the invariant, corresponding to a smooth variety.

In the talk, which is for a general audience, we will explain the main ideas of how to construct the invariant. There are some basic principles to observe, and putting these together, everything then evolves quite systematically.

I will briefly recall the notion of stable surfaces and of the corresponding moduli space. Then I will outline a partial description of the boundary points in the case of surfaces with $K^2=1$, $p_g=2$ (joint work with Stephen Coughlan, Marco Franciosi, Julie Rana and Soenke Rollenske, in various combinations) and, time permitting, in the case of Campedelli and Burniat surfaces (joint work with Valery Alexeev).

In many areas of mathematics there are theorems of the following kind: Any map in a suitable class has a big fiber. The classes of maps and the notions of size vary from field to field. In my talk I'll present several examples of this phenomenon. I'll show how Gromov's notion of ideal valued-measures derived from cohomology can be used to prove some of them. I'll also introduce objects which are a suitable generalization of ideal-valued measures in the context of symplectic geometry, called ideal-valued quasi-measures, indicate how they can be constructed using relative symplectic cohomology, a tool recently introduced by U. Varolgunes, and demonstrate how they can be used to obtain new symplectic rigidity results. Based on joint work with A. Dickstein, Y. Ganor, and L. Polterovich.

The symplectic non-squeezing theorem, discovered by Gromov in 1985, has been the first result showing a fundamental difference between symplectic transformations and volume preserving ones. A similar but more subtle phenomenon in contact topology was found by Eliashberg, Kim and Polterovich in 2006, and refined by Fraser in 2016 and Chiu in 2017: in this case non-squeezing depends on the size of the domains, and only appears above a certain quantum scale.

In my talk I will outline the geometric ideas behind a proof of this general contact non-squeezing theorem that uses generating functions, a classical method based on finite dimensional Morse theory. This is a joint work with Maia Fraser and Bingyu Zhang.

Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? This question was originally asked by Poincaré and was initially studied in the 60s. However, various facets of it remain largely open. Recently, several advances were made in the context of Hamiltonian and contact flows. I will discuss a connection between this problem and Gromov-Witten invariants, which are "counts" of holomorphic curves. This is based on a joint work with Julian Chaidez.

Given a degenerating family of Kähler-Einstein metrics it is natural to study from a differential geometric perspective the collection of all metric limits at all possible scales, a typical example being the emergence of Kronheimer’s ALE spaces near the formation of orbifold singularities for Einstein 4-manifolds. In this talk, I will describe, focusing on the discussion of some concrete and elementary examples, how it should be possible to use algebro geometric tools to investigate such problem for algebraic families, leading in the non-collapsing case to an inductive argument identifying the so-called metric bubble tree at a singularity (made of a collection of asymptotically conical Calabi-Yau varieties) with a subset of the non-Archimedean Berkovich analytification of the family. Based on joint work with M. de Borbon.

I will explain recent work on relationships among geometric quantization, deformation quantization, Berezin-Toeplitz quantization and brane quantization.

Certain simple symplectic manifolds (symplectic vector space, Milnor fibres of certain complex surface singularities,...) contain sets of symplectically distinct Lagrangian tori which have the following remarkable property: they remain symplectically distinct under embeddings into any reasonable (i.e. geometrically bounded) symplectic manifold. This leads to a vast extension of the class of spaces in which the existence of exotic tori is known, especially in dimensions six and above. In this talk we mainly focus on recent joint work with Johannes Hauber and Joel Schmitz which treats the more intricate case of dimension four.

We introduce the classical results of de Rham and Berger on the holonomy of a Riemannian manifold. We compare these to the situation of parallel skew-torsion, where we obtain Riemannian submersions from reducible holonomy. *If time permits I will give an introduction to 3-$(\alpha, \delta)$-Sasaki manifolds and their submersion onto quaternionic Kähler manifolds.

In this talk I will explain a general mechanism, based on derived symplectic geometry, to glue the local invariants of singularities that appear naturally in Donaldson-Thomas theory. Our mechanism recovers the categorified vanishing cycles sheaves constructed by Brav-Bussi-Dupont-Joyce, and provides a new more evolved gluing of Orlov’s categories of matrix factorisations, answering a conjecture of Kontsevich-Soibelman and Y. Toda. This is a joint work with B. Hennion (Orsay) and J. Holstein (Hamburg). The talk will be accessible to a general audience.

We will discuss some classification results for Hermitian non-Kähler gravitational instantons. There are three main results: (1) Non-existence of certain Hermitian non-Kähler ALE gravitational instantons. (2) Complete classification for Hermitian non-Kähler ALF/AF gravitational instantons. (3) Non-existence of Hermitian non-Kähler gravitational instantons under suitable curvature decay condition, when there is more collapsing at infinity (ALG, ALH, etc.). These are achieved by a thorough analysis of the collapsing geometry at infinity and compactifications.

Lagrangian submanifold rigidity has been a fundamental topic in symplectic topology, contributing to key theories like the Arnold-Givental conjecture and Lagrangian Floer theory. These theories often show that intersections between Lagrangian submanifolds are unavoidable via symplectic maps, exemplified by Biran's concept of Lagrangian Barriers (2001).
Conversely, submanifolds not containing Lagrangian submanifolds usually exhibit flexibility, and can often be symplectically displaced. In this joint work with Richard Hind and Yaron Ostrover, we introduce what appears to be the first illustration of Symplectic Barriers, demonstrating necessary intersections of symplectic embeddings with symplectic (non-Lagrangian) submanifolds. The key point is that Lagrangian submanifolds are not the sole barriers, and there exist situations where a symplectic submanifold is not flexible.

In our work, we also answer a question by Sackel–Song–Varolgunes–Zhu and calculate the optimal symplectic ball embedding in the ball after removing a codimension 2 hyperplane with a prescribed Kähler angle.

The logarithm map from complex algebraic torus to the Euclidean space, sends an n-tuple of nonzero complex numbers to the logarithms of their absolute values. The image of a subvariety in the torus under the logarithm map is called "amoeba" and it contains geometric information about the variety. In this talk we explore the extension of the notion of logarithm map and amoeba to the non-commutative setting, that is for a spherical homogeneous space G/H where G is a connected complex reductive algebraic group. This is related to Victor Batyrev's question of describing K-orbits in G/H.

The talk is based on a joint work with Victor Batyrev, Megumi Harada and Johannes Hofscheier.

In this talk, I will present an instanton-type PDE associated with a Poisson manifold M. After reviewing its role in an underlying field theory, we present the main theorem showing existence and classification of its solutions. Finally, we discuss its geometric significance leading to a generating function for a symplectic groupoid, Lie-theoretic, integration of M.

Infinite-time convergence of geometric flows, as even for finite-dimensional gradient flows, is a notoriously subtle problem. The best (or only) bet is to get a Łojasiewicz(-Simon) inequality stating that a power of the gradient dominates the distance to the critical energy value. I'll introduce a Łojasiewicz inequality between the tension field and Dirichlet energy of a map from the 2-sphere to itself, removing the technical restrictions from an estimate of Topping (Annals ‘04). The inequality guarantees convergence of weak solutions of harmonic map flow from $S^2$ to $S^2$ assuming that the body map is nonconstant.

Sharp restriction theory and the finite field extension problem have both received much attention in the last two decades, but so far they have not intersected. In this talk, we discuss our first results on sharp restriction theory on finite fields. Even though our methods for dealing with paraboloids and cones borrow some inspiration from their euclidean counterparts, new phenomena arise which are related to the underlying arithmetic and discrete structures. The talk is based on recent joint work with Cristian González-Riquelme (https://arxiv.org/abs/2405.16647)

In the present talk I will introduce a new class of almost contact metric manifolds, called anti-quasi-Sasakian (aqS for short). They are non-normal almost contact metric manifolds $(M,φ,ξ,η,g)$, locally fibering along the 1-dimensional foliation generated by $ξ$ onto Kähler manifolds endowed with a closed 2-form of type (2,0). Various examples of anti-quasi-Sasakian manifolds will be provided, including compact nilmanifolds, $\mathbb{S}^1$-bundles and manifolds admitting a $\operatorname{Sp}(n)\times \{1\}$-reduction of the structural group of the frame bundle. Then, I will discuss some geometric obstructions to the existence of aqS structures, mainly related to curvature and topological properties. In particular, I will focus on compact manifolds endowed with aqS structures of maximal rank, showing that they cannot be homogeneous and they must satisfy some restrictions on the Betti numbers.

This is based on joint works with Giulia Dileo (Bari) and Ivan Yudin (Coimbra).

References

1. D. Di Pinto, On anti-quasi-Sasakian manifolds of maximal rank J. Geom. Phys. 200 (2024), Paper no. 105174, 10 pp.
2. D. Di Pinto, G. Dileo, Anti-quasi-Sasakian manifolds, Ann. Global Anal. Geom. 64 (1), Article no. 5 (2023), 35 pp.

This talk is based on a joint work with Yang Li. Motivated by collapsing Calabi-Yau 3-folds and G2-manifolds with Lefschetz K3 fibrations in the adiabatic setting, Donaldson and Scaduto conjectured the existence of a special Lagrangian pair-of-pants in the Calabi-Yau 3-fold $X \times \mathbb R^2$, where $X$ is either a hyperkähler K3 surface (global version) or an A2-type ALE hyperkähler 4-manifold (local version). After a brief introduction to the subject, we discuss the significance of this conjecture in the study of Calabi-Yau 3-folds and G2-manifolds, and then prove the local version of the conjecture, which in turn implies the global version for an open subset of the moduli of K3 surfaces.

Special Lagrangians form an important class of minimal submanifolds in Calabi-Yau manifolds. In this talk, we will consider the Calabi-Yau 3-folds with a K3-fibration and the size of the K3-fibres are small. Motivated by tropical geometry, Donaldson-Scaduto conjectured that special Lagrangian collapse to “gradient cycles” when the K3-fibres collapse. This phenomenon is similar to holomorphic curves in Calabi-Yau manifolds with collapsing special Lagrangian fibrations converging to tropical curves. Similar to the realization problem in tropical geometry, one might expect to reconstruct special Lagrangians from gradient cycles. In this talk, I will report the first theorem of this kind based on a joint work with Shih-Kai Chiu.

A crucial problem in geometric quantization is to understand the relationship among quantum spaces associated to different polarizations. Two types of polarizations on toric varieties, Kähler and real, have been studied extensively. This talk will focus on the quantum spaces associated with mixed polarizations and explore their relationships with those associated with Kähler polarizations on toric varieties.

We present a systematic study of kinematic formulas in convex geometry. We first give a classical presentation of kinematic formulas for integration with respect to the rotation group $SO(n)$, where Steiner's Formula, the intrinsic volumes and Hadwiger's Characterization Theorem play a crucial role. Then we will show a new extension to integration along the general linear group $GL(n)$. Using the bijection of matrix polar decomposition and a Gaussian measure to integrate along positive definite matrices, a new formula is obtained, for which the classical $SO(n)$ formula is a particular case. We also reference the unitary group $U(n)$ case and its corresponding extension to the symplectic group $Sp(2n,\mathbb{R})$.

3d mirror symmetry is a mysterious duality for certain pairs of hyperkähler manifolds, or more generally complex symplectic manifolds/stacks. In this talk, we will describe its relationships with 2d mirror symmetry. This could be regarded as a 3d analog of the paper Mirror Symmetry is T-Duality by Strominger, Yau and Zaslow which described 2d mirror symmetry via 1d dualities.