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UIUC symplectic and poisson geometry seminar
This seminar meets Mondays at 3-4pm (central US time), typically in person. Some talks will occur via zoom if it is more convenient for the speakers and possibly at an alternate time (usually Mondays 9-10am) if the speaker is outside of our time zone.
The art is The Ancient of Days setting a Compass to the Earth by William Blake (1794).
Spring 2023
Jan 20 - Isabelle Charton: Monotone Symplectic Manifolds and Fano Varieties with Symmetries
A compact symplectic manifold (M, ω) is called positive monotone if its first Chern class is a positive multiple of [ω] in H2 dR(M). A Fano variety is a smooth complex variety that admits a holomorphic embedding into CP N (for some N). Such a variety can be endowed with a symplectic form such that it is a positive monotone symplectic manifold. For this reason, Fano varieties are considered the algebraic counterparts of monotone symplectic manifolds. A general outstanding issue in symplectic geometry is the question of whenever a positive monotone Fano manifold is diffeomorphic to a Fano variety. In low dimensions, namely two and four, it is proven by McDuff-Gromov-Taubes that any positive monotone symplectic manifold is diffeomorphic to a Fano variety. In higher dimensions analogues results are not known.
In this talk I will explain what is known about the difference between Fano varieties and positive monotone symplectic manifolds endowed with Hamiltonian torus action. In particular, I will represent new results for the case that the complexity of the action is one, i.e., the dimension of the torus is equal to dim(M)/2−1.
This is joint work with Liat Kessler, Silvia Sabatini and Daniele Sepe.
Jan 30 - Donghoon Jang: Almost complex torus manifolds - graphs, Hirzebruch genera, and a problem of Petrie type
Let a k-dimensional torus act on a 2n-dimensional compact connected almost complex manifold M with isolated fixed points. There is a multigraph that contains information on weights at the fixed points and isotropy submanifolds. If k=n, that is, M is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. For an almost complex torus manifold, the coefficients of its Hirzebruch genus are non-zero. Using these ideas, we show that if k=n and there are n+1 fixed points, many invariants of M agree with those of a linear action on the complex projective space; if in addition the action is equivariantly formal, their equivariant cohomologies also agree.
Feb 6 - no talk
Feb 13 - Alejandro Cabrera
In this talk, we discuss generalized geometric reduction procedures which lead to Poisson (and related) structures on the reduced space but in which the starting "reduction data" is of more general type. The idea behind it is based on the characterization of the relevant geometric structures as Lie algebroids equipped with compatible 2-forms. The main results involve quotients of such Lie algebroids, their corresponding Lie theoretic integration to Lie groupoids endowed with multiplicative 2-forms, and the infinitesimal characterization of non-degeneracy conditions. Finally, we discuss several concrete applications, recovering particular cases and showing how these results can be applied in various problems. This talk is based on joint work with C. Ortiz.
Feb 20 - Ely Kerman - Mean width, symplectic capacities and Viterbo's conjecture
In this talk I will discuss the relationship between some symplectic measurements of convex sets and a classical geometric measurement, the mean width. More precisely, I will describe an inequality between the symplectic capacity of a convex subset and a symplectic refinement of its mean width. This inequality is formally weaker than a much-studied inequality between the capacity and volume of convex sets that was conjectured by Viterbo. It is not yet clear whether it is actually weaker. I will lay out the evidence, state some open questions, and present some new results concerning a linear version of the problem. This is joint work in progress with Jonghyeon Ahn.
Feb 27 - Andrew Hanlon: Toric Frobenius and generation through mirror symmetry
We will use symplectic geometry to prove some new results on the algebraic geometry of toric varieties. Namely, we will revisit ideas of Bondal for viewing the toric Frobenius morphism through the lens of homological mirror symmetry. This perspective and recent structural results for Fukaya categories allow us to produce a natural resolution of the structure sheaf of any toric subvariety in a smooth toric variety. In fact, this resolution is built from a finite set of line bundles and has length equal to the codimension of the subvariety, which implies, in particular, that the Rouquier dimension of the derived category of any toric variety coincides with the dimension of the variety. This is joint work in progress with Jeff Hicks and Oleg Lazarev.
March 6 - Alexey Bolsinov (virtual talk at 9am!): Nijenhuis Geometry
This talk is an introduction to Nijenhuis Geometry, a new challenging area in Differential Geometry that studies local and global properties of geometric structures given by a field ofendomorphisms with vanishing Nijenhuis torsion. This topic is located on the crossroad of Geometry, Mathematical Physics and Algebra as Nijenhuis structures naturally appear in many seemingly unrelated research areas such as bi-Hamiltonian integrable systems (both finite and infinite-dimensional), projective geometry, theory of left-symmetric algebras and others.
March 13 - spring break (no talk)
March 20 - Eric Zaslow: The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures
I will describe how the moduli space of sheaves microsupported on a Legendrian surface forms a Lagrangian patch in a chart of a Fock-Goncharov cluster variety, and how the structure is compatible with quantization and mutation. The classical geometry conjecturally determines open disk invariants of Lagrangian fillings, while the quantum geometry conjecturally encodes open higher-genus invariants. I will also describe dualities relating quiver invariants and refined Kac polynomials with these numbers. This is joint work with Gus Schrader and Linhui Shen.
March 27 - no talk this week
April 3 - Sonja Hohloch: Fiber classification for hypersemitoric systems
This talk starts with motivating the definition of hypersemitoric systems which are two degree of freedom integrable Hamiltonian systems on 4-dimensional symplectic manifolds with possibly mild degeneracies where one of the integrals gives rise to an effective Hamiltonian S^1-action. Then we give a brief overview of their main features and some examples before we sketch a topological classification of their fibers using 'labeled graphs'.
April 10 - Yael Karshon: Symplectic excision
Removing a properly embedded ray from a (noncompact) manifold does not affect the topology nor the diffeotype. What about the symplectic analogue of this fact? And can we go beyond rays? I will show how to use incomplete Hamiltonian flows to excise interesting subsets: the product of a ray with a Cantor set, a "box with a tail", and - more generally - epigraphs of lower semicontinuous functions. This is based on joint work with Xiudi Tang, in which we answer a question of Alan Weinstein.
April 17 - Yohann Le Floch: Families of (hyper)semitoric systems
A semitoric system is a four-dimensional integrable system whose momentum map has a component generating an effective S^1-action and possesses mild singularities. These systems have been classified by Pelayo and Vũ Ngọc thanks to several invariants, but the construction of a system with given invariants involves symplectic gluing of some local normal forms. My goal will be to review these notions and to explain some attempts to answer the following question: can one construct a fully explicit (i.e. whose momentum map is given by global explicit formulas) semitoric system with a given partial list of invariants in a relatively simple way? This will lead us to some more general systems with S^1-symmetry called hypersemitoric systems. This is based on past and ongoing work with J. Palmer (University of Illinois Urbana-Champaign).
April 24 - Arturo Fernandez Perez (Federal University of Minas Gerais): Holomorphic Poisson structures: an approach through holomorphic foliations
I will to explain some results where holomorphic foliations have been applied in the classification of holomorphic Poisson structures on projective manifolds. Moreover, motivated by these topics we will establish some open problems.
May 1 - Xiang Tang
May 8 - first day of exam week (probably no talk)
Fall 2022
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.
Previous talks Spring 2022 (click to expand)
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.
Previous talks Fall 2021 (click to expand)
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
Previous talkS 2020/2021 (click to expand)
Thursday, May 6th, 11am-12pm: Marco Zambon (KU Leuven)
Deformations of symplectic foliations
Symplectic foliations and regular Poisson structures are the same thing. Taking the latter point of view, we exhibit an algebraic structure that governs the deformations of symplectic foliations, i.e. which allows to describe the space of symplectic foliations nearby a given one. Using this, we will address the question of when it is possible to prolong a first order deformation to a smooth path of symplectic foliations. We will be especially interested in the relation to the underlying foliation. This is joint work in progress with Stephane Geudens and Alfonso Tortorella.
Thursday, April 29th, 11am-12pm: Silvia Sabatini (Univ. of Cologne)
Elliptic genera and circle actions
Consider a compact symplectic manifold with nonzero first Chern class. Its index is defined to be the largest integer dividing the first Chern class in the second cohomology group (modulo torsion). In the algebraic geometry setting there is a question of relating the index to the Betti numbers of the manifold in the case in which the manifold is a Fano variety, which we refer to as a ``positivity condition on the first Chern class''. This question is not fully answered and there are still open conjectures about it, for instance the Mukai conjecture.
In the first part of this talk I will introduce the audience to results which are already known in the symplectic setting, relating the index to the Betti numbers for manifolds admitting some special Hamiltonian circle action. In the second part of the talk I will introduce elliptic genera and show how their behaviour can be used to deduce relations between the Betti numbers and the index without assuming the aforementioned positivity condition on the first Chern class.
Thursday, April 22nd, 11am-12pm: Hendrik Seuss (Univ. Manchester)
Normalised volumes of singularities
The notion of the normalised volume of a singularity has been introduced relatively recently, but plays a crucial role in the context of Einstein metrics and K-stability. After introducing this invariant my plan is to specialise quickly to the case of toric singularities and show that even in this relatively simple setting interesting phenomena occur.
Monday, April 12th, 3-4pm: Joseph Palmer (UIUC)
Integrable systems and Hamiltonian S^1 spaces
A Hamiltonian S^1 space is a four dimensional symplectic manifold equipped with an action of S^1 generated by a real-valued function known as the Hamiltonian. If there exists an additional real-valued function which is independent from the Hamiltonian function and Poisson commutes with it, then we say that the given Hamiltonian S^1 space can be "lifted" to a completely integrable system. We study when Hamiltonian S^1 spaces can be lifted to four dimensional completely integrable systems, and, depending on the properties of the original S^1 space, we study the properties of the resulting integrable systems. In particular, we ask the question: what is the "nicest" class of integrable systems to which all Hamiltonian S^1 spaces can be lifted? This talk is based on joint work with S. Hohloch.
Thursday April 1st, 11am-12pm: Alvaro del Pino
On subRiemannian geodesics and billiard trajectories
SubRiemannian Geometry studies triples consisting of a smooth manifold, a subbundle of the tangent bundle (a "tangent distribution"), and a metric along the distribution. Such a triple can be regarded as a degenerate limit of Riemannian manifolds, where the directions not in the distribution have been infinitely penalised.
In such a setup, which is the framework in which Geometric Control Theory is phrased, a central question is to study the properties of the minimising curves. In many ways, this resembles the usual theory of geodesics in Riemannian Geometry, but various exotic behaviours appear. The underlying reason behind these behaviours is that, upon dualising, the subRiemannian metric becomes a degenerate Hamiltonian with fibrewise non-compact level sets.
I will review the basic theory behind this setup, particularly the cotangent formulation, which goes back to work of Pontryagin. I aim to give a (very biased) overview of the area, emphasising various results about global topological properties of subRiemannian geodesics, as well as some intriguing open questions. If time allows, I will comment on recent work, joint with L. Dahinden, in which we study the subRiemannian billiard flow, which is the natural generalisation to manifolds with boundary.
Monday, March 22nd, 3-4pm: Joseph Palmer
Immersed Floer cohomology, mean curvature flow, and Lagrangian surgery
We study the behavior of (immersed) Floer cohomology under coupled mean curvature and Kähler-Ricci flow. Given an unobstructed immersed Lagrangian we prove (under some conditions) a lower bound on the time for which the immersed Floer cohomology is invariant under the flow, as long as the flow exists. Furthermore, in some cases when the Lagrangian becomes obstructed we show how performing a Lagrangian surgery allows the flow to be continued in such a way that the Floer cohomology instead remains unobstructed and is invariant. This surgery prevents certain geometric singularities in the mean curvature flow before they can form. This is partially motivated by a conjecture of Joyce. This work is joint with Chris Woodward, see arxiv.org/abs/1804.06799 and arxiv.org/abs/1903.01943.
Monday, March 15th, 3-4pm: Reyer Sjamaar
Stacky Hamiltonian actions and symplectic reduction
Extending work of Lerman and Malkin, we introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of some basic theorems of symplectic geometry in this context, such as the Kirwan convexity theorem and the Mayer-Marsden-Weinstein symplectic reduction theorem. This is joint work with Benjamin Hoffman.
Monday, March 8th, 3-4pm: Surena Hozoori (Georgia Tech)
Symplectic geometry of Anosov flows in dimension 3 and bi-contact topology
We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss a characterization of Anosovity based on Reeb flows and its consequences.
Monday, March 1st, 3-4pm: Luka Zwaan
Poisson manifolds of strong compact type over 2-tori
An integrable Poisson manifold is said to be of strong compact type if the source 1-connected groupoid integrating it is compact. A trivial class of such manifolds is that of compact symplectic manifolds with finite fundamental group, but beyond that finding examples is difficult. The first non-trivial example was given by D. Martínez Torres in 2014. The construction there is inspired by D. Kotschick’s construction of a free symplectic circle action with contractible orbits. In this talk I will go over the original construction, recalling the relevant results on Poisson manifolds of compact types as well as the geometry of the moduli spaces of K3 surfaces, and then modify the construction to obtain more examples. In the end, we will have for every strongly integral affine 2-torus (i.e. integral affine 2-torus with integral translational part) a Poisson manifold of strong compact type having said torus as its leaf space.
Monday Feb 22, 3-4pm: John Etnyre
Symplectic fillings and cobordisms of lens spaces
Tight contact structures on lens space break into two types: universally tight and virtually overtwisted. In 2008, Lisca classified all symplectically fillable contact structures of universally tight contact structures on lens spaces. There have been several partial results for the larger class of virtually overstated contact structures on lens spaces, most notably Plamenevskaya and Van Horn-Morris classification of symplectic fillings of all tight contact structures on L(p,1). In this talk I will discuss joint work with Agniva Ray that classifies all minimal symplectic fillings of all tight contact structures on lens spaces. If time permits we will also discuss some interesting cobordisms between them and related open problems.
Monday, Nov 30: Jeremy Lane (McMaster University)
Canonical bases, toric degenerations, and collective integrable systems
There are three important settings for studying actions of reductive Lie groups: modules, algebraic group actions, and Hamiltonian group actions. In the study of modules one encounters various constructions of nice bases which are in some sense canonical (e.g. Gelfand-Zeitlin, Lusztig). In the study of algebraic group actions canonical bases give rise to toric degenerations; deformations of the G-variety to a toric variety (cf. Caldero, Alexeev-Brion).
In this talk I will discuss the symplectic analogue of these constructions: integrable systems. We show how toric degenerations give rise to integrable systems on arbitrary symplectic manifolds equipped with Hamiltonian group actions. This generalizes a family of well-known examples called Gelfand-Zeitlin integrable systems due to Guillemin and Sternberg. As a by-product, we generalize results of Harada and Kaveh on construction integrable systems from toric degenerations.
This talk is based on joint work with Benjamin Hoffman. arXiv:2008.13656
Monday, Nov 9: San Vũ Ngọc (Université de Rennes 1)
One can hear semitoric systems
I will report on an ongoing work with Yohann Le Floch about inverse spectral theory for quantum integrable systems. The question, adapted from the famous "Can one hear the shape of a drum?", is to fully reconstruct a semi-toric system from the joint spectrum of its quantum counterpart. It was proposed as a conjecture about 10 years ago. We show that the conjecture is true, by analyzing the quantum versions of all "symplectic invariants" of semi-toric systems.
Monday, Oct 26, 11am-noon (special time!): Marine Fontaine (University of Antwerp)
Real forms and Hamiltonian dynamics
We present a theory of real forms for holomorphic Hamiltonian systems which behaves well under integrability: given a real analytic integrable system, one can (under some assumptions) complexify the system and obtain other real integrable systems on different real forms. These systems are dynamically different but they do share the same complexification. As an example, we explain how we can apply this theory and use hyperkähler geometry to find a compact integrable real form of the spherical pendulum on S2 x S2. This is based on joint work with P. Arathoon.
Monday, Oct 19, 3-4pm: Rajan Mehta (Smith College)
Frobenius objects in categories of relations and spans
Frobenius algebras can be given a category-theoretic definition in terms of the category of vector spaces. This leads to a more general definition of Frobenius object in any monoidal category. In this talk, I will describe Frobenius objects in categories where the objects are sets and the morphisms are relations or spans. These categories can be viewed as toy models for the symplectic category. The main result is that, in both cases, it is possible to construct a simplicial set that encodes the data of the Frobenius structure.
This work is a very small first step in a bigger program aimed at better understanding the relationship between Poisson geometry and two-dimensional topological field theory. Part of the talk will be devoted to giving an overview of this question as well as its analogue in dimension 3. This is based on work with Ruoqi Zhang and work in progress with Ivan Contreras and Molly Keller.
Monday, Oct 12, 3-4pm: Sonja Hohloch (University of Antwerp)
The semitoric octagon
Roughly speaking, a semitoric system is a completely integrable Hamiltonian system on a 4-dimensional symplectic manifold that admits only nondegenerate singularities without hyperbolic components and whose flow gives rise to an (S^1×R)-action. Semitoric systems have been symplectically classified a decade ago by Pelayo & Vu Ngoc by means of five invariants. Recently, there has been made considerable progress by various authors concerning the computation of these invariants.
In this talk, we will give a brief introduction to semitoric systems before focusing on a recent, intuitive family of semitoric systems that allows for explicit observation of bifurcation behaviour such as ‘nodal trades’ between focus-focus and elliptic-elliptic singularities and other interesting geometric-topological features related to singularities and bifurcations. The latter part is based on a joint work with A. De Meulenaere.
Monday, Sept 28, 3-4pm: Jesse Huang (UIUC)
Toric Mirror symmetry via GIT windows
Every toric variety is a GIT quotient of an affine space by an algebraic torus. In this talk, I will discuss a way to understand and compute the symplectic mirrors of toric varieties from this universal perspective using the concept of window subcategories. The talk is based on results from a work of myself and a joint work in progress with Peng Zhou.
Monday, Oct 5, 3-4pm: Dogancan Karabas (Northwestern)
Microlocal sheaves on pinwheels and Fukaya categories of rational homology balls
It is shown by Kashiwara and Schapira (1980s) that for every constructible sheaf on a smooth manifold, one can construct a closed conic Lagrangian subset of its cotangent bundle, called the microsupport of the sheaf. This eventually led to the equivalence of the category of constructible sheaves on a manifold and the Fukaya category of its cotangent bundle by the work of Nadler and Zaslow (2006), and Ganatra, Pardon, and Shende (2018) for partially wrapped Fukaya categories. One can try to generalise this and conjecture that Fukaya category of a Weinstein manifold can be given by constructible (microlocal) sheaves associated with its skeleton. In this talk, I will briefly explain these concepts and confirm the conjecture for a family of Weinstein manifolds which are certain quotients of A_n-Milnor fibres. I will outline the computation of their wrapped Fukaya categories and microlocal sheaves on their skeleta, called pinwheels.
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