1999 seminars

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Thibaut Delcroix

Thibaut Delcroix, Université de Montpellier
On the Yau-Tian-Donaldson conjecture for spherical varieties

I will present how uniform K-stability translates into a convex geometric problem for polarized spherical varieties. From this, we will derive a combinatorial sufficient condition of existence of constant scalar curvature Kahler metrics on smooth spherical varieties, and a complete solution to the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

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Dusa McDuff

Dusa McDuff, Columbia University
Counting curves and stabilized symplectic embedding conjecture

This is a report on joint work with Kyler Siegel that develops new ways to count $J$-holomorphic curves in $4$-dimensions, both in the projective plane with multi-branched tangency constraints, and in noncompact cobordisms between ellipsoids. These curves stabilize, i.e. if they exist in a given four dimensional target manifold $X$ they still exist in the product $X \times {\mathbb R}^{2k}$. This allows us to establish new cases of the stabilized embedding conjecture for symplectic embeddings of an ellipsoid into a ball (or ellipsoid).

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Justin Sawon

Justin Sawon, University of North Carolina at Chapel Hill
Lagrangian fibrations by Prym varieties

Lagrangian fibrations on holomorphic symplectic manifolds and orbifolds are higher-dimensional generalizations of elliptic K3 surfaces. They are fibrations whose general fibres are abelian varieties that are Lagrangian with respect to the symplectic form. Markushevich and Tikhomirov described the first example whose fibres are Prym varieties, and their construction was further developed by Arbarello, Ferretti, and Sacca and by Matteini to yield more examples. In this talk we describe the general framework, and consider a new example. We describe its singularities and show that it is a 'primitive' symplectic variety. We also construct the dual fibration, using ideas of Menet. This is joint work with Chen Shen.

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Cristiano Spotti

Cristiano Spotti, Aarhus University
On relations between K-moduli and symplectic geometry

A natural intriguing question is the following: how much the moduli spaces of certain polarized varieties know about the symplectic geometry of the underneath manifold? After giving a general overview, I will discuss work-in-progress with T. Baier, G. Granja and R. Sena-Dias where we investigate some relations between the topology of the moduli spaces of certain varieties, of the symplectomorphism group and of the space of compatible integrable complex structures. In particular, using results of J. Evans, we show that the space of such complex structures for monotone del Pezzo surfaces of degree four and five is weakly homotopically contractible.

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Spotti slides.pdf

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Paul Biran

Paul Biran, ETH Zurich
Persistence and Triangulation in Lagrangian Topology

Both triangulated categories as well as persistence homology play an important role in symplectic topology. The goal of this talk is to explain how to put the two structures
together, leading to the notion of a triangulated persistence category. The guiding principle comes from the theory of Lagrangian cobordism.

The talk is based on ongoing joint work with Octav Cornea and Jun Zhang.

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Vicente Muñoz

Vicente Muñoz, Universidad de Málaga
A Smale-Barden manifold admitting K-contact but not Sasakian structure

Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions, with K-contact manifolds corresponding to symplectic manifolds. In this talk, we give the first example of a simply connected compact 5-manifold (Smale-Barden manifold) which admits a K-contact structure but does not admit any Sasakian structure, settling a long standing question of Boyer and Galicki.

For this, we translate the question about K-contact 5-manifolds to constructing symplectic 4-orbifolds with cyclic singularities containing disjoint symplectic surfaces of positive genus. The question on Sasakian 5-manifolds translates to the existence of algebraic surfaces with cyclic singularities containig disjoint complex curves of positive genus. A key step consists on bounding universally the number of singular points of the algebraic surface.

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Alexandru Oancea

Alexandru Oancea, Institut de Mathématiques de Jussieu, Sorbonne Université
Duality and coproducts in Rabinowitz-Floer homology

The goal of the talk is to explain a duality theorem between Rabinowitz-Floer homology and cohomology. These are Floer homology groups associated to the contact boundary of a Liouville domain, and the duality isomorphism is compatible with canonically defined product structures. Dual to the cohomological product is a homology coproduct which satisfies a remarkable compatibility relation with the product structure. We will also discuss the relationship to loop spaces and Chas-Sullivan/Goresky-Hingston products.

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Andrew Neitzke

Andrew Neitzke, Yale University
Aspects of abelianization: exact WKB, classical Chern-Simons theory and knot invariants

There is a long history of attacking problems involving nonabelian Lie groups by reducing to a maximal abelian subgroup. It has been understood in the last decade that

1) the exact WKB method for studying linear ODEs,
2) the computation of classical Chern-Simons invariants of flat connections,
3) the study of some link invariants, such as the Jones polynomial,

can all be understood as aspects of this general idea. I will describe this point of view, trying to emphasize the common features of all three problems, and (briefly) their common origin in supersymmetric quantum field theory. Parts of the talk are a report of joint work with Dan Freed, Davide Gaiotto, Greg Moore, Lotte Hollands, and Fei Yan.

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Leonor Godinho

Leonor Godinho, Instituto Superior Técnico and CAMGSD
On the number of fixed points of periodic flows

Finding the minimal number of fixed points of a periodic flow on a compact manifold is, in general, an open problem. We will consider almost complex manifolds and see how one can obtain lower bounds by retrieving information from a special Chern number.

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Giulia Saccà

Giulia Saccà, Columbia University
Compact Hyper-Kählers and Fano Manifolds

Projective hyper-Kähler (HK) manifolds are among the building blocks of projective manifolds with trivial first Chern class. Fano manifolds are projective manifolds with positive first Chern class.

Despite the fact that these two classes of algebraic varieties are very different (HK manifolds have a holomorphic symplectic form which governs all of its geometry, Fano manifolds have no holomorphic forms) their geometries have some strong ties. For example, starting from some special Fano manifolds one can sometimes construct HK manifolds as parameter spaces of objects on the Fano. In this talk I will explain this circle of ideas and focus on some recent work exploring the converse: given a projective HK manifold, how to recover a Fano manifold from it?

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Lorenzo Foscolo

Lorenzo Foscolo, University College London
Twistor constructions of non-compact hyperkähler manifolds

The talk is based on joint work with Roger Bielawski about twistor constructions of higher dimensional non-compact hyperkähler manifolds with maximal and submaximal volume growth. In the first part of the talk, based on arXiv:2012.14895, I will discuss the case of hyperkähler metrics with maximal volume growth: in the same way that ALE spaces are closely related to the deformation theory of Kleinian singularities, we produce large families of hyperkähler metrics asymptotic to cones exploiting the theory of Poisson deformations of affine symplectic singularities. In the second part of the talk, I will report on work in progress about the construction of hyperkähler metrics generalising to higher dimensions the geometry of ALF spaces of dihedral type. We produce candidate holomorphic symplectic manifolds and twistor spaces from Hilbert schemes of hypertoric manifolds with an action of a Weyl group. The spaces we define are closely related to Coulomb branches of 3-dimensional supersymmetric gauge theories.

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Marco Mazzuchelli

Marco Mazzuchelli, École normale supérieure de Lyon
What does a Besse contact sphere look like?

A closed connected contact manifold is called Besse when all of its Reeb orbits are closed (the terminology comes from Arthur Besse's monograph "Manifolds all of whose geodesics are closed", which deals indeed with Besse unit tangent bundles). In recent years, a few intriguing properties of Besse contact manifolds have been established: in particular, their spectral and systolic characterizations. In this talk, I will focus on Besse contact spheres. In dimension 3, it turns out that such spheres are strictly contactomorphic to rational ellipsoids. In higher dimensions, an analogous result is unknown and seems out of reach. Nevertheless, I will show that at least those contact spheres that are convex still "resemble" a contact ellipsoid: any stratum of the stratification defined by their Reeb flow is an integral homology sphere, and the sequence of their Ekeland-Hofer capacities coincides with the full sequence of action values, each one repeated according to a suitable multiplicity. This is joint work with Marco Radeschi.

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Brian Collier

Brian Collier, University of California Riverside
Global Slodowy slices for moduli spaces of λ-connections

The moduli spaces of Higgs bundles and holomorphic connections both have important affine holomorphic Lagrangian subvarieties, these are the Hitchin section and the space of opers, respectively. Both of these spaces arise from the same Lie theoretic mechanism, namely a regular nilpotent element of a Lie algebra. In this talk we will generalize these parameterizations to other nilpotents. The resulting objects are not related by the nonabelian Hodge correspondence, but by an operation called the conformal limit. Time permitting, we will also discuss their relation to Higher Teichmuller spaces.

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Tommaso Pacini

Tommaso Pacini, Università degli Studi di Torino
Minimal Lagrangian submanifolds, totally real geometry and the anti-canonical line bundle.

The category of totally real (TR) submanifolds was traditionally of interest mainly to complex analysts. We will present a survey of recent work towards a "TR geometry" and explain its relevance to the study of minimal Lagrangians and of convexity properties of the volume functional.

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Laura Schaposnik

Laura Schaposnik, University of Illinois at Chicago
On generalized hyperpolygons

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system.

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Yu-Shen Lin

Yu-Shen Lin, Boston University
Correspondence theorem between holomorphic discs and tropical discs on (Log)-Calabi-Yau Surfaces

Tropical geometry is a useful tool to study the Gromov-Witten type invariants, which count the number of holomorphic curves with incidence conditions. On the other hand, holomorphic discs with boundaries on the Lagrangian fibration of a Calabi-Yau manifold play an important role in the quantum correction of the mirror complex structure. In this talk, I will introduce a version of open Gromov-Witten invariants counting such discs and the corresponding tropical geometry on (log) Calabi-Yau surfaces. Using Lagrangian Floer theory, we will establish the equivalence between the open Gromov-Witten invariants with weighted count of tropical discs. In particular, the correspondence theorem implies the folklore conjecture that certain open Gromov-Witten invariants coincide with the log Gromov-Witten invariants with maximal tangency for the projective plane.

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Marcos Jardim

Marcos Jardim, Universidad Estadual de Campinas
Walls and asymptotics for Bridgeland stability conditions on 3-folds

We consider a class of geometric Bridgeland stability conditions for 3-folds of Picard rank one, parametrized by the upper half plane. We study the geometry of numerical walls, applying our results to prove that Gieseker semistability is equivalent to a strong form of asymptotic semistability along a class of paths in the upper half plane.
Finally, we compute all of the walls and describe the Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex projective 3-space in a suitable region of the upper half plane. Joint work with Antony Maciocia.

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Camilla Felisetti

Camilla Felisetti, Università di Trento
P=W conjectures for character varieties with a symplectic resolution

Character varieties parametrise representations of the fundamental group of a curve. In general these moduli spaces are singular, therefore it is customary to slightly change the moduli problem and consider smooth analogues, called twisted character varieties. In this setting, the P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In a joint work with M. Mauri we establish (and in some cases formulate) analogous P=W phenomena in the singular case.

In particular we show that the P=W conjecture holds for character varieties which admit a symplectic resolution, namely in genus 1 and arbitrary rank and in genus 2 and rank 2. In the talk I will first mention basic notions of non abelian Hodge theory and introduce the P=W conjecture for smooth moduli spaces; then I will explain how to extend these phenomenas to the singular case, showing the proof our results in a specific example.

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Alberto Abbondandolo

Alberto Abbondandolo, Ruhr Universität Bochum
Systolic questions in metric and symplectic geometry

The prototypical question in metric systolic geometry is to bound the length of a shortest closed geodesic on a closed Riemannian manifold by the volume of the manifold. This question has been extensively studied for non simply connected manifolds, but in the recent years there has been some progress also for simply connected manifolds, on which closed geodesics cannot be found simply by minimizing the length. This progress involves extending systolic questions to Reeb flows, a class of dynamical systems generalising geodesic flows. On the one hand, this extension and the use of symplectic techniques provide some answers to classical questions within metric systolic geometry. On the other hand, new questions arise from the more general setting and relate seemingly distant fields such as the study of rigidity properties of symplectomorphisms and the integral geometry of convex bodies. I will give a non-technical panoramic view of some of these recent developments.

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Carolina Araujo

Carolina Araujo, Instituto de Matemática Pura e Aplicada
Higher Fano Manifolds

Fano manifolds, i.e., complex projective manifolds having positive first Chern class, play a key role in higher dimensional algebraic geometry. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and describe special geometric features of these manifolds.

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Mirko Mauri

Mirko Mauri, Max Planck (Bonn)
On the geometric $P=W$ conjecture

The geometric $P=W$ conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In a new version of a joint work with Enrica Mazzon and Matthew Stevenson, we prove the full geometric $P=W$ conjecture for elliptic curves: this is the first non-trivial evidence of the conjecture for compact Riemann surfaces. As a byproduct, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds. We also explain how the geometric $P=W$ conjecture by Katzarkov-Noll-Pandit-Simpson is related to the cohomological version formulated by de Cataldo-Hausel-Migliorini.

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Antoine Song

Antoine Song, UC Berkeley
The essential minimal volume of manifolds

One way to measure the complexity of a smooth manifold M is to consider its minimal volume, denoted by MinVol, introduced by Gromov, which is simply defined as the infimum of the volume among metrics with sectional curvature between -1 and 1. I will introduce a variant of MinVol, called the essential minimal volume, defined as the infimum of the volume over a closure of the space of metrics with sectional curvature between -1 and 1. I will discuss the main properties of this invariant, and present estimates for negatively curved manifolds, Einstein 4-manifolds and most complex surfaces.

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Yael Karshon

Yael Karshon, University of Toronto
Bott canonical basis?

Together with Jihyeon Jessie Yang, we are resurrecting an old idea of Raoul Bott for using large torus actions to construct canonical bases for unitary representations of compact Lie groups. Our methods are complex analytic; we apply them to families of Bott-Samelson manifolds parametrized by $\mathbb C^n$. Our construction requires the vanishing of higher cohomology of sheaves of holomorphic sections of certain line bundles over the total spaces of such families; this vanishing is conjectural, hence the question mark in the title.

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Siu-Cheong Lau

Siu-Cheong Lau, Boston University
Kaehler geometry of quiver moduli in application to machine learning

Neural network in machine learning has interesting similarity with quiver representation theory. In this talk, I will build an algebro-geometric formulation of a computing machine, which is well-defined over the moduli space of representations. The main algebraic ingredient is to extend noncommutative geometry of Connes, Cuntz-Quillen, Ginzburg to near-rings, which capture the non-linear activation functions in neural network. I will also explain a uniformization between spherical, Euclidean and hyperbolic moduli of framed quiver representations.

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Lau slides.pdf

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Olivia Dumitrescu

Olivia Dumitrescu, University of North Carolina at Chapel Hill
On stratifications and moduli

There exist two approaches to the conformal limit mechanism: first was defined by Gaiotto using Analysis techniques and the method of computing was first established for the Hitchin Section and Opers in 2016. The second approach to conformal limits as algebraic shifts via extension classes of vector bundles was established by Dumitrescu and Mulase in 2017 for the lagrangians Hitchin section and opers. In this talk I will report on work in progress with Jennifer Brown and Motohico Mulase of the algebraic approach of conformal limits to a family of Lagrangians covering the Dolbeault and the De Rham moduli space of Higgs bundles and irreducible connections over a curve in rank 2.

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Umberto Hryniewicz

Umberto Hryniewicz, Aachen University
Contact three-manifolds with exactly two simple Reeb orbits

The goal of this talk is to present a complete characterization of Reeb flows on closed 3-manifolds with precisely two periodic orbits. The main step consists in showing that a contact form with exactly two periodic Reeb orbits is non-degenerate. The proof combines the ECH volume formula with a study of the behavior of the ECH index under non-degenerate perturbations of the contact form. As a consequence, the ambient contact 3-manifold is a standard lens space, the contact form is dynamically convex, the Reeb flow admits a rational disk-like global surface of section and the dynamics are described by a pseudorotation of the 2-disk. Moreover, the periods and rotation numbers of the closed orbits satisfy the same relations as (quotients of) irrational ellipsoids, and in the case of $S^3$ the transverse knot-type of the periodic orbits is determined. Joint work with Cristofaro-Gardiner, Hutchings and Liu.

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Daniele Alessandrini

Daniele Alessandrini, Columbia University
The nilpotent cone in rank one and minimal surfaces

I will describe two interesting and closely related moduli spaces: the nilpotent cone in the moduli spaces of Higgs bundles for $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{PSL}_2(\mathbb C)$, and the moduli space of equivariant minimal surfaces in the hyperbolic 3-space.

A deep understanding of these objects is important because of their relations with several fundamental constructions in geometry: singular fibers of the Hitchin fibration, branes, mirror symmetry, branched hyperbolic structures, minimal surfaces in hyperbolic 3-manifolds and so on.

A stratification of the nilpotent cone is well known and was rediscovered by many people. The closures of the strata are the irreducible components of the nilpotent cone. The talk will focus on describing the intersections between the different irreducible components.

This is joint work with Qiongling Li and Andrew Sanders.

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Christine Breiner

Christine Breiner, Brown University and Fordham University
Harmonic branched coverings and uniformization of CAT(k) spheres

Consider a metric space $(S,d)$ with an upper curvature bound in the sense of Alexandrov (i.e.~via triangle comparison). We show that if $(S,d)$ is homeomorphically equivalent to the $2$-sphere, then it is conformally equivalent to the $2$-sphere. The method of proof is through harmonic maps, and we show that the conformal equivalence is achieved by an almost conformal harmonic map. The proof relies on the analysis of the local behavior of harmonic maps between surfaces, and the key step is to show that an almost conformal harmonic map from a compact surface onto a surface with an upper curvature bound is a branched covering. This work is joint with Chikako Mese.

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Barbara Fantechi

Barbara Fantechi, SISSA
Smoothability of non normal stable Gorenstein Godeaux surfaces

This is joint work with Marco Franciosi and Rita Pardini.

Godeaux surfaces, with $K^2=1$ and $p_g=q=0$, are the (complex projective) surfaces of general type with the smallest possible invariants. A complete classification, i.e. an understanding of their moduli space, has been an open problem for many decades.

The KSBA (after Kollár, Sheperd-Barron and Alexeev) compactification of the moduli includes so called stable surfaces. Franciosi, Pardini and Rollenske classified all such surfaces in the boundary which are Gorenstein (i.e., not too singular).

We prove that most of these surfaces corresponds to a point in the moduli which is nonsingular of the expected dimension 8. We expect that the methods used (which include classical and recent infinitesimal deformation theory, as well as algebraic stacks and the cotangent complex) can be applied to all cases, and to more general moduli as well.

The talk is aimed at a non specialist mathematical audience, and will focus on the less technical aspects of the paper.

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slidesBarbara.pdf

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Felix Schlenk

Felix Schlenk, Université de Neuchâtel
On the group of symplectomorphisms of starshaped domains

Take a simply connected compact domain $K$ in $\mathbb R^{2n}$ with smooth boundary. We study the topology of the group $\mathrm{Symp} (K)$ of those symplectomorphisms of $K$ that are defined on a neighbourhood of $K$. A main tool is a Serre fibration $\mathrm{Symp} (K) \to \mathrm{SCont} (\partial K)$ to the group of strict contactomorphisms of the boundary. The fiber is contractible if $K$ is 4-dimensional and starshaped, by Gromov's theorem. The topology (or at least the connectivity) of the group $\mathrm{SCont} (\partial K)$ can be understood in many examples. In case this group is connected, so is $\mathrm{Symp} (K)$. This has applications to the problem of understanding the topology of the space of symplectic embeddings of $K$ into any symplectic manifold. If $\mathrm{Symp} (K)$ is connected, then for embeddings that are not related by an ambient symplectomorphism there is not even an ambient symplectomorphism that maps one image to the other.

The talk is based on work with Joé Brendel and Grisha Mikhalkin.

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SCont.pdf

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Luca Asselle

Luca Asselle, Ruhr University Bochum
A Morse complex for the Hamiltonian action in cotangent bundles

Critical points having infinite Morse index and co-index are invisible to homotopy theory, since attaching an infinite dimensional cell does not produce any change in the topology of sublevel sets. Therefore, no classical Morse theory can possibly exist for strongly indefinite functionals (i.e. functionals whose all critical points have infinite Morse index and co-index). In this talk, we will briefly explain how to instead construct a Morse complex for certain classes of strongly indefinite functionals on a Hilbert manifold by looking at the intersection between stable and unstable manifolds of critical points whose difference of (suitably defined) relative indices is one. As a concrete example, we will consider the case of the Hamiltonian action functional defined by a smooth time-periodic Hamiltonian $H: S^1 \times T^*Q \to \mathbb R$, where $T^*Q$ is the cotangent bundle of a closed manifold $Q$. As one expects, in this case the resulting Morse homology is isomorphic to the Floer homology of $T^*Q$, however the Morse complex approach has several advantages over Floer homology which will be discussed if time permits. This is joint work with Alberto Abbondandolo and Maciej Starostka.

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Lisbon.pdf

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André Neves

André Neves, University of Chicago
Minimal surfaces in hyperbolic manifolds

The study of geodesics in negatively curved manifolds is a rich subject which has been at the core of geometry and dynamical systems. Comparatively, much less is known about minimal surfaces on those spaces. I will survey some of the recent progress in that area.

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Pranav Chakravarthy

Pranav Chakravarthy, Hebrew University of Jerusalem
Homotopy type of equivariant symplectomorphisms of rational ruled surfaces

In this talk, we present results on the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once, under the presence of Hamiltonian group actions of either $S^1$ or finite cyclic groups. For Hamiltonian circle actions, we prove that the centralizers are homotopy equivalent to either a torus or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. We can show that the same holds for the centralizers of most finite cyclic groups in the Hamiltonian group. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczynski smooth classification of $\mathbb Z_n$-actions on Hirzebruch surfaces.

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Carlos Florentino

Carlos Florentino, Faculty of Sciences - University of Lisbon
The geometry of commuting varieties of reductive groups

Let $R_r(G)$ be the (connected component of the identity of the) variety of commuting $r$-tuples of elements of a complex reductive group $G$. We determine the mixed Hodge structure on the cohomology of the representation variety $R_r(G)$ and of the character variety $R_r(G)/G$, for general $r$ and $G$. We also obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomials, Poincaré polynomials and Euler characteristics of these representation and character varieties. In the character variety case, this gives the counting polynomial over finite fields, and some results also apply to character varieties of nilpotent groups.

This is joint work with S. Lawton and J. Silva (arXiv:2110.07060).

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CVRG.pdf

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Henrique Bursztyn

Henrique Bursztyn, Instituto Nacional de Matemática Pura e Aplicada
Revisiting and extending Poisson-Nijenhuis structures

Poisson-Nijenhuis structures arise in various settings, such as the theory of integrable systems, Poisson-Lie theory and quantization. By revisiting this notion from a new viewpoint, I will show how it can be naturally extended to the realm of Dirac structures, with applications to integration results in (holomorphic) Poisson geometry.

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PNtalkLisboa.pdf

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Ely Kerman

Ely Kerman, University of Illinois Urbana-Champaign
On symplectic capacities and their blind spots

In this talk I will discuss a joint work with Yuanpu Liang in which we establish some results concerning the symplectic capacities defined by Gutt and Hutchings using $S^1$-equivariant symplectic homology. Our primary result settles a version of the recognition question in the negative. We prove that the Gutt-Hutchings capacities, together with the volume, do not constitute a complete set of symplectic invariants for star-shaped (in fact convex) domains with smooth boundary. We also prove that, even for star-shaped domains with smooth boundaries, these capacities are mutually independent and are independent from the volume. The constructions that demonstrate these independence properties​ are not exotic. They are convex and concave toric domains. The new tool used here is a significant simplification of the formulae of Gutt and Hutchings for the capacities of convex/concave toric domains, that holds under an additional symmetry assumption. This allows us to identify new mutual blind spots of the capacities which are then used to construct the desired examples.

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Talk 11.23.pdf

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Alfonso Zamora

Alfonso Zamora, Polytechnic University of Madrid
E-polynomials and geometry of character varieties

Quasi-projective complex algebraic varieties (not necessarily smooth or compact) carry, following Deligne, mixed Hodge structures whose numerical information is encoded in the $E$-polinomial, which provides substantial geometrical (topological plus complex) information about these varieties; important examples being character varieties. Let $G$ be a complex algebraic group and let $\mathcal{X}_{\Gamma}G$ be the $G$-character variety of a finitely presented group $\Gamma$. Exploring a new path between the previously known arithmetical and geometrical methods to compute $E$-polynomials we provide a concrete relation, in terms of plethystic functions, stratifications of the character variety as an affine GIT quotient and combinatorics, between the generating series for $E$-polynomials of $\mathcal{X}_{\Gamma}G$ for $G=GL(n,\mathbb{C})$ and the series of the subset of irreducible representation conjugacy classes $\mathcal{X}_{\Gamma}^{irr}G$. This yields explicit (stratified) expressions for the $E$-polynomials of several groups $\Gamma$, for low values of $n$. For the case $\Gamma=F_{r}$, the free group of rank $r$, we prove the equality of the $E$-polynomials for $\mathcal{X}_{r}SL_{n}$ and $\mathcal{X}_{r}PGL_{n}$, for any $n,r\in\mathbb{N}$. This settles a conjecture of Lawton-Muñoz and provides evidence on topological mirror symmetry conjectures for Langlands dual groups.

These results are joint work with Carlos Florentino and Azizeh Nozad.

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Ciprian Manolescu

Ciprian Manolescu, Stanford University
Khovanov homology and the search for exotic 4-spheres

A well-known strategy to disprove the smooth 4D Poincare conjecture is to find a knot that bounds a disk in a homotopy 4-ball but not in the standard 4-ball. Freedman, Gompf, Morrison and Walker suggested that Rasmussen’s invariant from Khovanov homology could be useful for this purpose. I will describe three recent results about this strategy: that it fails for Gluck twists (joint work with Marengon, Sarkar and Willis); that an analogue works for other 4-manifolds (joint work with Marengon and Piccirillo); and that 0-surgery homeomorphisms provide a large class of potential examples (joint work with Piccirillo).

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search.pdf

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Eva Miranda

Eva Miranda, Universitat Politècnica de Catalunya
Looking at the Euler flows through a contact mirror

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [6, 7, 8] launched a programme to address the global existence problem for the Euler and Navier-Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, in the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension [1].

A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Etnyre and Ghrist in [4] more than two decades ago. We end this talk addressing a question raised by Moore in [5] : “Is hydrodynamics capable of performing computations?”. The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere. Can this result be improved? In [2] we construct a Turing complete steady Euler flow in dimension 3. Time permitting, we discuss this and other generalizations for t-dependent Euler flows contained in [3].

In all the constructions above, the metric is seen as an additional "variable" and thus the method of proof does not work if the metric is prescribed.

Is it still possible to construct a Turing complete Euler flow on a 3-dimensional space with the standard metric? Yes, see our recent preprint https://arxiv.org/abs/2111.03559 (joint with Cardona and Peralta).

This talk is based on several joint works with Cardona, Peralta-Salas and Presas.

[1] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Universality of Euler flows and flexibility of Reeb embeddings, arXiv:1911.01963.
[2] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler flows in dimension 3. PNAS May 11, 2021 118 (19) e2026818118; https://doi.org/10.1073/pnas.2026818118.
[3] R. Cardona, E. Miranda and D. Peralta-Salas, Turing universality of the incompressible Euler equations and a conjecture of Moore, International Mathematics Research Notices, rnab233, https://doi.org/10.1093/imrn/rnab233
[4] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture. Nonlinearity 13 (2000) 441–458.
[5] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4 (1991) 199–230.
[6] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.
[7] T. Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete Cont. Dyn. Sys. A 38 (2018) 1553–1565.
[8] T. Tao. Searching for singularities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.

Video

Additional file

geometriaemlisboa.pdf