SPG Seminar 2021

Sept 12 - Florian Zeiser (UIUC): Stability for Riemannian foliations

Given a foliation F on a smooth manifold M one may ask if any foliation G close to F in some suitable topology is actually conjugate to F with respect to some diffeomorphism on M. This property is called stability. Infinitesimally, this problem is controlled by H1(F,NF), the first foliated cohomology with coefficients in the normal bundle. It is natural to ask whether infinitesimal stability (H1(F,NF)=0) implies stability for F.

This question was first answered to the positive by Hamilton for Hausdorff foliations on closed manifolds. Around the same time Epstein & Rosenberg gave a postive answer for compact Hausdorff Cr -foliations on manifolds without boundary.

In this talk we extend Hamiltons result and give a positive answer for Riemannian foliations on closed manifolds. This is based on joint work with Stephane Geudens.

Sept 19 - NO TALK

Sept 26 - Raj Mehta (UIUC): Frobenius objects in the symplectic category

Roughly, a topological quantum field theory is a functor from a cobordism category to the category of vector spaces. The classical analogue should take values in symplectic manifolds instead of vector spaces. In this talk, I'll describe some ongoing work aimed at trying to characterize such functors in terms of algebraic and geometric data. I'll explain what we mean by `symplectic category', how the categories of spans and relations provide simple models for it, and how Frobenius objects in the categories of spans and relations correspond to simplicial sets satisfying certain conditions. This is based on various joint works with Ivan Contreras, Molly Keller, Adele Long, Sophia Marx, Walker Stern, and Ruoqi Zhang.

Oct 3 - Wenyuan Li (Northwestern): Conjugate fillings and Legendrian weaves. 

In this talk, we compare two different approaches to constructing Lagrangian fillings of Legendrian knots. The first one is conjugate Lagrangian fillings of alternating Legendrians, introduced by Shende-Treumann-Williams-Zaslow, which are characterized using bipartite graphs, and the second one is Lagrangian projections of Legendrian weaves, introduced by Casals-Zaslow, which are depicted by planar graphs encoding their wavefronts. We will develop a diagrammatic calculus to show that conjugate Lagrangian fillings are Hamiltonian isotopic to certain Lagrangian projections of Legendrian weaves. The result includes Legendrian positive braid closures and ideal triangulations on punctured surfaces. We will then explain some implications on Lagrangian mutations and cluster theory. This is joint work in preparation with Roger Casals.

Oct 10 - Gloria Marí Beffa (UW-Madison): W_m lattices, pseudo-difference operators and projective polygons

In this talk I will describe the construction of a Poisson pair using the algebra of pseudo-difference operators similarly to how pseudo-differential operators was used in the definition of W_m algebras. Difference operators are naturally connected to projective polygons and I will use that connection to describe a discrete version of the well-known Drinfel’d-Sokolov reduction for generalized KdV equations. This is on-going work with Anton Izosimov.

Oct 17 - no talk this week

Oct 24 - Jaume Alonso (TU Berlin): Constructing integrable birational systems with a 3D generalisation of QRT special time (TBD)!

When completely integrable Hamiltonian systems are discretised, the resulting discrete-time maps are often no longer integrable themselves. This is the so-called problem of integrable discretisation. Two known exceptions to this situation in 3D are the so-called Kahan discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one β, both birational maps of degree 3. By analysing the geometry of these pencils, we develop a framework that generalises QRT maps and QRT roots to 3D, which allows us to create new integrable maps as a composition of two involutions. We show that under certain geometric conditions, the new maps become of degree 3. We use these results to create new families of discrete integrable maps and we solve the problem of integrable discretisation of the Zhukovski-Volterra gyrostat with two β’s.

This is a joint work with Yuri Suris and Kangning Wei.

Oct 31 - Ioan Marcut (Radboud): Poisson structures with compact support

When compared to symplectic structures, Poisson structures can have very wild properties. Some can be very flexible with huge deformation spaces, others can have very pathological automorphism groups. Many properties are still only poorly understood for general Poisson manifolds.

In this talk, I will discuss two open questions in Poisson geometry: Smoothness of the Poisson diffeomorphism group and the Poisson extension problem. To illustrate some of the phenomena related to these questions, I will give explicit examples of Poisson structures, all with compact supports and constructed in an elementary way. The first examples have a Poisson diffeomorphism group which is not locally path-connected for the Whitney topology. The other examples are cases in which we can solve the Poisson extension problem explicitly.

This talk is based on:

[1] "Poisson structures whose Poisson diffeomorphism group is notlocally path-connected", Ioan Marcut,  Ann. H. Lebesgue 4 (2021), 1521–1529.

[2] "Poisson structures with compact support", Gil R. Cavalcanti, Ioan Marcut, arXiv:2209.14016.

Nov 7 - Michael Gekhtman (Notre Dame): Log-canonical coordinates on Poisson-Lie groups

Log-canonical coordinates provides the simplest from to which a quadratic Poisson bracket can be reduced via a rational transformation. Such coordinates play an important role in a construction of cluster structures on Poisson varieties. We present a construction of log-canonical charts on Poisson- Lie groups and Poisson homogeneous spaces associated with R-matrices arising in the Belavin-Drinfeld classification. The key ingredient is a Poisson map from a simple Lie group equipped with the standard Poisson-Lie group to the same group equipped with a nontrivial Belavin-Drinfeld Poisson bracket. This is a joint project with M. Shapiro and A. Vainshtein.

Nov 14 - Aliakbar Daemi (WashU): Lagrangians, SO(3)-instantons and the Atiyah-Floer Conjecture

A useful tool to study a 3 -manifold is the space of the representations of its fundamental group, a.k.a. the 3 -manifold group, into a Lie group. Any 3 -manifold can be decomposed as the union of two handlebodies. Thus, representations of the 3-manifold group into a Lie group can be obtained by intersecting representation varieties of the two handlebodies. Casson utilized this observation to define his celebrated invariant. Later Taubes introduced an alternative approach to define Casson invariant using more geometric objects. By building on Taubes' work, Floer refined Casson invariant into a graded vector space whose Euler characteristic is twice the Casson invariant. The Atiyah-Floer conjecture states that Casson's original approach can be also used to define a graded vector space and the resulting invariant of 3 -manifolds is isomorphic to Floer's theory. In this talk, after giving some background, I will discuss some recent progress on the Atiyah-Floer conjecture for SO(3) bundles, which is based on a joint work with Kenji Fukaya and Maksim Lipyanskyi. I will only assume a basic background in algebraic topology and geometry.

Nov 21 - Pedram Hekmati (Univ. Auckland): Equivariant index on toric Sasaki manifolds

Sasaki manifolds are the odd dimensional analogues of Kähler manifolds. In this talk, I will discuss the equivariant index of the horizontal Dolbeault complex on toric Sasaki manifolds. We show that the index localizes to certain closed Reeb orbits and can be expressed as a sum over lattice points of the moment cone. This index problem arises for instance in the calculation of partition functions of cohomologically twisted supersymmetric gauge theories. This is joint work with Marcos Orseli.

Nov 28 - Gabriele La Nave (UIUC): J-holomorphic curves in conical symplectic manifolds

I will talk about recent work with G. Jayasinghe and H. Quan on Gromov compactness of J-holomorphic curves in manifolds endowed with singular symplectic forms which have conical singularities along a divisor. The types of singularities we discuss were introduced in Kahler geometry by Tian and used by Donaldson to solve the Donaldson-Tian-Yau conjecture on the existence of Kahler-Einstein metrics on a Fano manifolds. We introduce them in the context of symplectic geometry as a natural generalization  of symplectic forms. This generalization occurs naturally in the context of trying to understand a symplectic version of the Kodaira dimension.

Dec 5 - Soham Chanda (Rutgers): Invariance of Floer cohomology under higher mutation via neck-stretching

Pascaleff-Tonkonog defined higher mutations for monotone toric fibers and proved an invariance of disc potential under a change of local system.  In this talk I will define a  local version of higher mutations for locally mutable Lagrangians and use neck-stretching to show invariance of Lagrangian intersection cohomology under a change of local system which agrees with the mutation formula in Pascaleff-Tonkonog.

Monday, Feb 14th - Sue Tolman (UIUC)

Beyond Semitoric

A compact four dimensional completely integrable system $f : M \to R^2$ is semitoric if it has only non-degenerate singularities, without hyperbolic blocks, and one of the components of $f$ generates a circle action.  Semitoric systems have been extensively studied and have many nice properties: for example, the preimages $f^{-1}(x)$  are all connected.  Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitation.  For example, there are blowups of $S^2 \times S^2$ with Hamiltonian circle actions which cannot be extended to semitoric systems.  We expand the class of semitoric systems by allowing certain degenerate singularities, which we call ephemeral singularities.  We prove that the preimage $f^{-1}(x)$ is still connected for this larger class.  We hope that this class will be large enough to include not only all compact four manifolds with Hamiltonian circle actions, but more generally all complexity one spaces. Based on joint work with D. Sepe.

Monday, Feb 21st - Rui Loja Fernandes (UIUC)

Non-formal deformation quantization of Poisson manifolds

The Kontsevich formality theorem, established more than 20 years ago, implies that every Poisson manifold has a formal deformation quantization. The existence of non-formal deformations, on the other hand, remains largely an open problem. We study star products defined by semi-classical integral Fourier operators. Our main result states that a Poisson manifold which admits such a star product must be integrable by a symplectic groupoid. This is on-going joint work with Alejandro Cabrera (UFRJ).

Monday, Feb 28 - Marius Crainic (Utrecht University) - virtual talk over zoom!

A new approach to codimension one symplectic foliations

The existence of codimension one foliations on compact manifolds has a nice history, starting with Reeb's foliation on $S^3$ that marked the birth of Foliation Theory, then on general 3-manifolds, then Lawson's foliations on $S^5$ and then on all odd-dimensional sphere, up to the general characterization of Thurston via the Euler characteristic. The analogous question for symplectic foliations is far from being understood- it is only rather recently that Lawson's foliation on $S^5$ was turned onto a symplectic one, by Mitsumatsu, but the construction is rather involved.  

On the other hand, the confoliations of Eliashberg-Thurston revealed very close relationship (via actual deformations) between foliations and contact structure. That theory is well developed only in dimension 3 and that is due, I believe (at least in part), to not noticing that when moving to higher dimensions one should be looking not only at foliations, but at symplectic ones. As a side remark: also Mitsumatsu's construction exploits the geometry of contact forms and adapted open book decompositons.

The aim of the talk will be to present a new, we believe much simpler and more conceptual, approach to Lawson's foliation on $S^5$, based on log-symplectic geometry/stable generalized geometry instead of contact geometry. That is joint work with my colleague Gil Cavalcanti.

Monday, March 7 - Olguta Buse (IUPUI)

Homotopic Stability Chambers in Blow-up Ruled Symplectic Surfaces

This talk focuses on homotopic properties of symplectomorphism groups of blow ups of ruled surfaces. The scaffolding that allows to establish homotopic  stability chambers inside their symplectic cones expands from existing literature results on their minimal counterparts.

We extent inflation and J-holomorphic techniques (and other homotopic tools) to spaces of non-rational, non-minimal symplectic ruled surfaces. We will explain new results regarding $\pi_*$ of the symplectomorphism groups  and discuss future results. The results are in collaboration with Jun Li.

Monday, March 14 - No talk (spring break)

Monday, March 21 - Miquel Cueca (Gottingen) - virtual talk over zoom

Integration of Courant algebroids

In this talk I will introduce 2-shifted symplectic Lie 2-groupoids and construct explicit examples that integrate some particular kind of Courant algebroids. This is joint work with Chenchang Zhu.

Monday, March 28 - Nikolay Martynchuk (Groningen) - virtual talk over zoom at special time (noon IL time)

On the symplectic equivalences for parabolic orbits of integrable Hamiltonian systems

Parabolic orbits are the simplest examples of degenerate singularities of integrable two degree of freedom Hamiltonian systems. Yet until recently, their symplectic (and even C^\infty smooth) classification was not known. We fill in this gap and show that the action variables corresponding to such an orbit form a complete set of symplectic invariants (up to the fiberwise symplectic equivalence). This generalises an earlier result by A.V. Bolsinov, L. Guglielmi and E.A. Kudryavtseva proving this in the analytic category. The smooth case that we present is more complicated and has useful consequences for more global symplectic classification problems. 

We shall also discuss a new classification result for parabolic orbits in the analytic category; specifically, we  shall give a simple normal form for such orbits up to the right symplectic equivalence. This type of equivalence is different, but closely related to the usual fiberwise (or right-left) symplectic equivalence and interestingly enough, the normal form that we obtain is not given in terms (of the asymptotics) of the action variables.

This talk is based on a recent work with Prof. E.A. Kudryavtseva.

Monday, April 4 - Joel Villatoro (WashU)

A diffeological approach to solving the integration problem

In the theory of Lie groupoids and Lie algebroids, there is a procedure for differentiting a Lie groupoid to result in a Lie algebroid. This process is very much analogous to the construction of a Lie algebra from a Lie group. From this analogy, it is reasonable to ask whether or not it is possible to construct a Lie groupoid given the data of a Lie algebroid in much the same manner tha you do for Lie algebras. In fact, it turns out tha this is not possible due to the fact tha integration procedure results in something tha is not quite a manifold. In this talk I will discuss an approach to patching this problem using diffeological spaces which are a generalization of smooth manifolds.

Monday, April 11 - David Miyamoto (Toronto)

Quasifold groupoids and diffeological quasifolds

A quasifold is a space that is locally modeled by quotients of R^n by countable group actions. These include orbifolds and manifolds. We approach quasifolds in two ways: by viewing them as diffeological spaces, we form the category of diffeological quasifolds, and by viewing them as Lie groupoids (with bibundles as morphisms), we form the category of quasifold groupoids. We show that, restricting to effictive groupoids, and locally invertible morphisms, these two categories are equivalent. In particular, an effective quasifold groupoid is determined by its diffeological orbit space. This is join work with Yael Karshon.

Monday, April 18 - Maxence Mayrand (Toronto) (canceled)

Monday, April 25 - Jason Liu (UIUC)

Lifting Complexity-1 Spaces to Toric Manifolds

A toric manifold is a 2n dimensional compact connected symplectic manifold equipped with an n-dimensional torus acting effectively in a Hamiltonian manner. In 1980s, Delzant completely classified toric manifolds up to equivariant symplectomorphism by their moment images (Delzant polytopes). Given a toric manifold, we can take an (n-1)-dimensional subtorus and restrict our attention to the action of the subtorus. These spaces are important examples of complexity-1 space. A natural question to ask is: given a complexity-1 space, is there a way to lift it to a toric manifold? In this talk, I will first talk about complexity-1 spaces and present the explicit construction of lifting under certain assumptions. This is the joint work with Joey Palmer and Sue Tolman.

Monday, May 2 - Eva Miranda (UPC) (canceled)

Monday, May 9 - Maarten Mol (Utrecht)

Toric Hamiltonian actions in a Poisson context

This talk concerns toric Hamiltonian actions with momentum maps taking values in regular Poisson manifolds of compact type, or more precisely: toric Hamiltonian actions of regular and proper symplectic groupoids. Examples of these include symplectic toric manifolds, proper Lagrangian fibrations and proper isotropic realizations of Poisson manifolds of compact type. The aim will be to explain the classification of such toric actions in terms of "Delzant polytopes" in the leaf space of the symplectic groupoid. 

Monday, Sept 13th - Ely Kerman (UIUC)

Around Viterbo's Conjecture

A conjecture of Claude Viterbo, from 1997, asserts that the symplectic capacity of a convex body should be bounded from above by its volume, when both are suitably normalized. 

This conjecture is of significant current interest, in part because its validity is now known to imply the Mahler conjecture in convex geometry. In this talk I will describe the proof of a weaker form of the inequality in which the role of the volume is played by a symplectic version of the mean-width. Time permitting, I will also describe some open problems suggested by this work.

Monday, Sept 20th - Eugene Lerman (UIUC)

Differential geometry over C-infinity rings (Part 1)

Abstract: (joint work with Yael Karshon) We develop some basic tools of differential geometry for singular spaces.  There is a variety of approaches to geometry on singular spaces going back to 1960's.   The spaces we work with lie in the intersection of differential spaces of Sikorski and C-infinity schemes of Dubuc. We integrate vector fields to flows and we construct a C-infinity ring analogue of Grothendieck's algebraic de Rham complex.

Monday, Sept 27th - Eugene Lerman (UIUC)

Differential geometry over C-infinity rings (Part 2)

Continuation of talk from the previous week.

Monday, Oct 4th - Wilmer Smilde (UIUC)

Linearization of Poisson groupoids

Motivated by the search for Lie group structures on groups of Poisson diffeomorphisms, we investigate linearizability of the Poisson structure of a Poisson groupoid around the unit section, and present some results in that direction.

Our approach revolves around `lifting' symplectic- and Poisson geometry to Lie algebroids: we will encounter an algebroid version of Weinstein's Lagrangian neighborhood theorem, and the integration of an algebroid-Poisson structure to an algebroid-symplectic groupoid.

Monday, Oct 11th - Eckhard Meinrenken (Toronto)

On the integration of transitive Lie algebroids 

We'll revisit the problem of integrating Lie algebroids A to Lie groupoids, for the special case that the Lie algebroid A is transitive. We obtain a geometric explanation of the Crainic-Fernandes obstructions for this situation, and an explicit construction of the integration whenever these obstructions vanish. Based on arXiv:2007.07120

Monday, Oct 18th - Gonçalo Oliveira (UFF)

Minimal Lagrangian tori and action-angle coordinates

In this talk I will report on joint work with Rosa Sena-Dias studying minimal Lagrangian submanifolds appearing as the fibers of the moment map on a toric Kahler manifold. I will use action-angle coordinates to answer the following questions: How many such minimal Lagrangian tori exist? Can their stability, as critical points of the area functional, be inferred from the ambient geometry? Which sets of such Lagrangian submanifolds can be made minimal with respect to some toric Kahler metric.

If time permits I will also make some comments on Lagrangian mean curvature flow and Hamiltonian stationary Lagrangians.

Monday, Oct 25th - NO MEETING

Monday, Nov 1st - Henrique Bursztyn (IMPA)

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. 

Monday, Nov 8th - Nathaniel Bottman (Max Planck Institute)

The symplectic (A-infinity,2)-category and a simplicial version of the 2D Fulton-MacPherson operad 

The symplectic (A-infinity,2)-category Symp, which is currently under construction by myself and my collaborators, is a 2-category-like structure whose objects are symplectic manifolds and where hom(M,N) := Fuk(M^- x N). Symp is a coherent algebraic structure which encodes the functoriality properties of the Fukaya category. This talk will begin with the following question: what can we say about the part of Symp that knows only about a single symplectic manifold M, and the diagonal Lagrangian correspondence from M to itself? We expect that the answer to this question should be a chain-level algebraic structure on symplectic cohomology, and in this talk I will present progress toward confirming this. Specifically, I will present a "simplicial version" of the 2-dimensional Fulton-MacPherson operad, which may be of independent topological interest. If there is time, I will also explain how this development can be used to give a definition of (A-infinity,2)-categories that involves only finitely many operations of each arity.

Monday, Nov 15th - No meeting

Monday, Nov 22nd - No meeting

Monday, Nov 29th - Umut Varolgunes (SPECIAL TIME: 12 noon)

Locality for relative symplectic cohomology

The talk will be based on my recent preprint with Yoel Groman. I will start by introducing the question of locality for relative (and truncated relative) symplectic cohomologies. Then I will state our main result which involves the notion of a symplectic manifold being geometrically of finite type, e.g. cone completion of a symplectic manifold with convex boundary. I will end with examples coming from singular Lagrangian torus fibrations over complete bases and briefly mention the relevance to mirror symmetry.

Monday, Dec 6th - Rui Fernandes (UIUC)

Non-formal deformation quantization

We introduce a (very general) notion of non-formal deformation quantization through semi-classical Fourier integral operators. The semi-classical limit of a non-formal deformation quantization, just like in the formal case, is a Poisson structure. To each non-formal deformation quantization we show that one can associate a local symplectic groupoid. Using this local groupoid, we obtain obstructions to the existence of a non-formal deformation quantization in the direction of a given Poisson structure. Joint work with Alejandro Cabrera (UF Rio de Janeiro)

Monday, Dec 13th - NO MEETING