The geometry, topology and intersection theory of moduli spaces of stable vector bundles on curves have been topics of interest for more than 50 years. In the 90s, Jeffrey and Kirwan managed to prove a formula proposed by Witten for the intersection numbers of tautological classes on such moduli spaces. In this talk, I will explain a different way to calculate those numbers and, more generally, intersection numbers on moduli of parabolic bundles. Enriching the problem with a parabolic structure gives access to powerful tools, such as wall-crossing, Hecke transforms and Weyl symmetry. If time allows, I will explain how this approach gives a new proof of (a generalization to the parabolic setting of) a vanishing result conjectured by Newstead and proven by Earl and Kirwan.

A cobordism W between compact manifolds M and M’ is an h-cobordism if the inclusions of M and M’ into W are both homotopy equivalences. These sort of cobordisms play an important role in the classification of high-dimensional manifolds, as h-cobordant manifolds are often diffeomorphic.

With this in mind, given two h-cobordant manifolds M and M’, how different can their diffeomorphism groups Diff(M) and Diff(M’) be? The homotopy groups of these two spaces are the same “up to extensions” in a range of strictly positive degrees. Contrasting this, I will present examples of h-cobordant manifolds in high-dimensions with different mapping class groups. In doing so, I will review the classical theory of h-cobordisms and introduce several moduli spaces of manifolds that shed light on this question.

The symplectic version of the problem of packing K balls into a ball in the densest way possible (in 4 dimensions) can be extended to that of symplectically embedding an ellipsoid into a ball as small as possible. A classic result due to McDuff and Schlenk asserts that the function that encodes this problem has a remarkable structure: its graph has infinitely many corners, determined by Fibonacci numbers, that fit together to form an infinite staircase.

This ellipsoid embedding function can be equally defined for other targets, and this talk will be about other targets for which the function has and does not have an infinite staircase. Firstly we will see how in the case when these targets have lattice moment polygons, the targets with infinite staircases seem to be exactly those whose polygon is reflexive (i.e., has one interior lattice point). Secondly, we will look at the family of one-point blowups of $CP^2$, where the answer involves self-similar behaviour akin to the Cantor set.

This talk is based on various projects, joint with Dan Cristofaro-Gardiner, Tara Holm, Alessia Mandini, Maria Bertozzi, Tara Holm, Emily Maw, Dusa McDuff, Grace Mwakyoma, Morgan Weiler, and Nicki Magill.

Mirror symmetry is a somewhat mysterious phenomenon that relates the geometry of two distinct Calabi-Yau manifolds. In the realm of trying to understand this relationship several conjectures on the existence of so-called special Lagrangian submanifolds appeared. In this talk, I will report on joint work with Jason Lotay on which we prove versions of the Thomas and Thomas-Yau conjectures regarding the existence of these special Lagrangian submanifolds and the role of Lagrangian mean curvature flow as a way to find them. I will also report on some more recent work towards proving more recent conjectures due to Joyce.

When M is a Fano variety and D is an anticanonical divisor in M, mirror symmetry suggests that the quantum cohomology of M should be a deformation of the symplectic cohomology of M \ D. We prove that this holds under even weaker hypotheses on D (although not in general), and explain the consequences for mirror symmetry. We also explain how our methods give rise to interesting symplectic rigidity results for subsets of M. Along the way we hope to give a brief introduction to Varolgunes’ relative symplectic cohomology, which is the key technical tool used to prove our symplectic rigidity results, but which is of far broader significance in symplectic topology and mirror symmetry as it makes the computation of quantum cohomology “local”. This is joint work with Strom Borman, Mohamed El Alami, and Umut Varolgunes.

An important geometric invariant of a hypersurface singularity is its Fukaya–Seidel category. In this talk, I will motivate and describe the study of two special families of isolated singularities. Time permitting, I will introduce “type A symplectic Auslander correspondence”, a purely geometrical construction which realises a notable result in representation theory.

Mirror Symmetry predicts a correspondence between the complex geometry (the B-side) and the symplectic geometry (the A-side) of suitable pairs of objects. In this talk I will consider certain orbifold del Pezzo surfaces falling outside of the standard mirror symmetry constructions. I will describe the derived category of coherent sheaves of the surfaces (their B-side), and discuss early results on the A-side. This is joint work with Franco Rota.

We fix an arbitrary symplectic toric manifold M. Its real toric lagrangians are the lagrangian submanifolds of M whose intersection with each torus orbit is clean and an orbit of the subgroup of elements that square to the identity of the torus (basically that subgroup is $\{ 1 , -1\}^n$). In particular, real toric lagrangians are transverse to the principal torus orbits and retain as much symmetry as possible.

This talk will explain why any two real toric lagrangians in M are related by an equivariant symplectomorphism and, therefore, any real toric lagrangian must be the real locus for a real structure preserving the moment map. This is joint work with Yael Karshon.

I will talk about recent joint work with Daniel Fadel (University of São Paulo) and Luiz Lara (Unicamp), where we study the SU(2) Yang-Mills-Higgs functional with positive coupling constant on 3-manifolds. Motivated by the work of Alessandro Pigati and Daniel Stern (2021) on the U(1)-version of the functional, we also include a scaling parameter.

When the 3-manifold is closed and the parameter is small enough, by adapting to our context the min-max method used by Pigati and Stern, we construct non-trivial critical points satisfying energy upper and lower bounds that are natural from the point of view of scaling.

Then, over 3-manifolds with bounded geometry, we show that, in the limit as the parameter tends to zero, and under the above-mentioned energy upper bound, a sequence of critical points exhibits concentration phenomenon at a finite collection of points, while the remaining energy goes into an $L^2$ harmonic 1-form. Moreover, the concentrated energy at each point is accounted for by finitely many "bubbles", that is, non-trivial critical points on $R^3$ with the scaling parameter set to 1.

We prove that the space of symplectic embeddings of $n\geq 1$ standard balls, each of capacity at most $\frac{1}{n}$, into the standard complex projective plane $\mathbb{CP}^2$ is homotopy equivalent to the configuration space of $n$ points in $\mathbb{CP}^2$. Our techniques also suggest that for every $n \geq 9$, there may exist infinitely many homotopy types of spaces of symplectic ball embeddings.

A toric Lagrangian fibration is a Lagrangian fibration whose singular fibers are all of elliptic type. I will begin by explaining how such fibrations can be viewed as Hamiltonian spaces associated with symplectic torus bundles. I will then discuss a generalization to this class of fibrations of the Abreu–Guillemin–Donaldson theory of extremal Kähler metrics on toric symplectic manifolds. Integral affine geometry plays a central role in this generalization, as the Delzant polytope is replaced by a more general domain contained in an integral affine manifold. This talk is based on on-going work with Miguel Abreu and Maarten Mol.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

In this series of lectures, I will discuss how methods from modern symplectic geometry (e.g. holomorphic curves or Floer theory) can be made to bear on the classical (circular, restricted) three body problem. I will touch upon theoretical aspects, as well as practical applications to space mission design. This is based on my recent book, available in https://arxiv.org/abs/2101.04438, to be published by Springer Nature.

I will report on the work in progress with S. Anjos and M. Pinsonnault concerning configuration spaces of symplectic balls in the standard complex projective plane. A few weeks ago Martin showed that when the balls are small their configuration space is homotopy equivalent to the configuration space of points. I will discuss what is happening if the balls are bigger. I will also try to put it into a more general context of configuration of rigid balls in domains of a Euclidean space.

The idea is to construct numerical integrator methods for Hamiltonian type of ODE’s which are defined in an ambient Poisson geometry. The goal is to approximate the exact dynamical solutions of this ODE while, at the same time, preserve the Poisson structure to a certain controlled degree. This is a non-trivial and long-range generalization of the notion of symplectic method in which the Poisson geometry is non-degenerate, thus, symplectic. We first outline a first approach to such methods which uses the geometry of so-called approximate symplectic realizations based on recent joint work with D. Martín de Diego and M. Vaquero. Finally, we describe a second approach based on theoretical results coming from Lie-theoretic aspects and which use an underlying groupoid multiplication, based on work in progress with D. Iglesias and J.C. Marrero.

Magnetic flows are generalizations of geodesic flows that describe the motion of a charged particle in a magnetic field. While every closed Riemannian manifold admits at least one closed geodesic, the analogous problem for magnetic orbits (also known as magnetic geodesics) is significantly more challenging and has received considerable attention in recent decades. I will present a result establishing that every low energy level of any magnetic flow admits at least one contractible closed orbit, assuming only that the magnetic strength is not identically zero, has a compact strict local maximum K, and that the cohomology class of the magnetic field is spherically rational. Moreover, this magnetic geodesic can be localized within an arbitrarily small neighborhood of K. This is joint work with Valerio Assenza and Gabriele Benedetti.

In this talk I will survey some results on the existence of periodic points of symplectomorphisms defined on closed orientable surfaces of positive genus g. Namely, I will describe some symplectic flows on such surfaces possessing finitely many periodic points and describe a non-Hamiltonian variant of the Hofer-Zehnder conjecture for symplectomorphisms defined on surfaces; this conjecture provides a quantitative threshold on the number of fixed points (possibly counted homologically) which forces the existence of infinitely many periodic points. This is joint work in progress with Marcelo Atallah and Brayan Ferreira.

A natural question bridging the celebrated Gromov–Eliashberg theorem and the C⁰-flux conjecture is whether the identity component of the group of symplectic diffeomorphisms is C⁰-closed in Symp(M,ω). Beyond surfaces and the cases in which the Torelli subgroup of Symp(M,ω) coincides with the identity component, little is known. In joint work with Cheuk Yu Mak and Wewei Wu, we show that, for all but a few positive rational surfaces, the group of Hamiltonian diffeomorphisms is the C⁰-connected component of the identity in Symp(M,ω), thereby giving a positive answer in this setting. Here, positive rational surface essentially means a k-point blow-up of CP² whose symplectic form evaluates positively on the first Chern class.

It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic maps into the hyperbolic plane $H^2$, while in the Riemannian setting, 4d Ricci-flat metrics with torus symmetry can also be locally reduced to such harmonic maps satisfying a tameness condition. We study such harmonic maps. Applications include a construction of infinitely many new complete, asymptotically flat, Ricci-flat 4-manifolds with arbitrarily large $b_2$. Joint work with Song Sun.

A classical result by McDuff shows that the space of symplectic ball embeddings into many simple symplectic four-manifolds is connected. In this talk, on the other hand, we show that the space of symplectic bi-disk embeddings often has infinitely many connected components, even for simple target spaces like the complex projective plane, or the symplectic ball. This extends earlier results by Gutt-Usher and Dimitroglou-Rizell. The proof uses almost toric fibrations and exotic Lagrangian tori. Furthermore, we will discuss natural quantitative questions arising in this context. This talk is based on joint work in progress with Grigory Mikhalkin and Felix Schlenk.

The Mobius strip; collapsing the equator; exploding a point in the plane; geometric definition of blowups; the secant construction; pull-backs of curves under blowup.

The algebraic definition of blowups; the blowup in affine charts; birational maps; the strict and total transforms of varieties and vector fields.

Axiomatic definition of blowups; Cartier divisors; base change property; persistence properties.

First examples of resolutions; curves and surfaces; effect of blowups on polynomials; singularity invariants; lexicographic orderings.

Choice of centers of blowup; descent in dimension; lexicographic decrease of invariant; transversality; obstructions in positive characteristic; resolution of planar vector fields.

Dimension 4 is the next horizon for applications of Ricci flow to topology, where the main goal is to understand the topological operations that Ricci flow generically performs at singular times. Shrinking Ricci solitons model these topological operations, and only the stable ones should arise generically.

I will present recent joint works with Olivier Biquard and Keaton Naff that determine the stability of all of the currently known shrinking Ricci solitons in dimension 4. The arguments use structural features unique to dimension four, in particular self-duality.