Join the conference via zoom

Seven of our 28 speakers will address the Savannah audience remotely via zoom, and we intend to broadcast most talks from Savannah on zoom as well. The zoom link will be shared on the mailing list <gonefishing-group@georgiasouthern.edu>, to which you can subscribe by filling the registration form.

Wi-Fi

If your institution has eduroam, you may use that channel to connect. Otherwise, select the gsguest channel and follow the onscreen instructions to create an account. That account will then work for the conference’s duration.

Titles & Abstracts

Anton Alekseev (Geneva, via zoom):

Virasoro Hamiltonian spaces. We develop a theory of Hamiltonian actions of the canonical central extension of the group of diffeomorphisms of the circle. It turns out that Virasoro Hamiltonian spaces (this is another name for such Hamiltonian actions) are in bijective correspondence with group valued Hamiltonian spaces with moment map taking values in (a certain part of) the universal cover of the group SL(2, R). Among other things, this correspondence allows to recover the classical result of Lazutkin-Pankratova, Kirillov, Segal, Witten (and others) on classification of coadjoint orbits of the Virasoro algebra. Interesting examples of Virasoro Hamiltonian spaces arise as moduli spaces of conformally compact hyperbolic metrics on oriented surfaces with boundary.

The talk is based on a joint work in progress with Eckhard Meinrenken. SLIDES

Ana Balibanu (Harvard):

Steinberg slices and group-valued moment maps. We define a natural class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth partial compactification of Z by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic. SLIDES

Jeffrey Carlson (Imperial College, via zoom):

The topology of the Gelfand–Zeitlin fiber. It is known due to Cho–Kim–Oh that the fibers of unitary and orthogonal Gelfand–Zeitlin systems are determined as iterated pullbacks by the combinatorics of joint eigenvalues of systems of truncated matrices, but the resulting expressions could be argued to be somewhat inexplicit. We provide a new interpretation of GZ fibers as balanced products of Lie groups (or biquotients), and pursue both viewpoints to a determination of the cohomology rings and low-dimensional homotopy groups of a GZ fiber which can be read transparently off of the combinatorics.

This all represents joint work with Jeremy Lane. SLIDES

David Clausen (UC Irvine):

Cohomology and Morse Theory on Symplectic Manifolds. On symplectic manifolds, there are intrinsincally symplectic cohomologies of differential forms that are analogous to the Dolbeault cohomology on complex manifolds. These cohomologies are isomorphic to the de Rham cohomologies on odd-dimensional sphere bundles over the symplectic manifold. In this talk, I will describe how we can use this sphere bundle perspective to define a novel Morse-type theory on symplectic manifolds associated with the symplectic cohomologies. This is joint work with Li-Sheng Tseng and Xiang Tang.

Mark Colarusso (South Alabama):

Complex Gelfand-Zeitlin Integrable Systems. We give an overview of the complex Gelfand-Zeitlin (GZ) integrable systems on the general linear and orthogonal Lie algebras. We describe the geometry of the Zariski open set where the joint flows of the system are Lagrangian as well as regular levels of the GZ moment map. We also describe how to construct an étale, Poisson covering of this open set and algebraically integrate the GZ vector fields on the covering. If time permits, we will discuss recent work in understanding the orbits of a certain spherical subgroup of GL(n,C) (resp SO(n,C)) on the flag variety and how these orbits can be used to construct a category of modules closely related to GZ modules.

Ivan Contreras (Amherst College):

Frobenius objects in the category of spans and the symplectic category. It is well known that Frobenius algebras are in correspondence with 2-dimensional TQFT. In this talk, we introduce Frobenius objects in any monoidal category and in particular, in the category where objects are sets and morphisms are spans of sets. We prove the existence of a simplicial set that encodes the data of the Frobenius structure in this category. This serves as a (simplicial) toy model of the Wehrheim-Woodward construction for the symplectic category.

This is part of a program that intends to describe, in terms of category theory, the relationship between symplectic groupoids and topological field theory, via the Poisson sigma model. Based on joint work with Rajan Mehta and Molly Keller (arXiv:2106.14743), and ongoing work with Rajan Mehta and Walker Stern. SLIDES

Mark Hamilton (Mount Allison):

Toric degenerations and quantization of Gelfand-Zeitlin systems. Guillemin-Sternberg cite the G-Z system as an example of "independence of polarization" in geometric quantization, based on the equality of two numbers that can be interpreted as the dimensions of the quantizations with respect to real and Kahler polarizations. I will describe a way to obtain a direct correspondence between elements of the two quantizations, via a deformation of the complex structure on the underlying flag manifold. SLIDES

Jonas Hartwig (Iowa State):

Galois orders and deformation quantization. We study a family of noncommutative algebras called Galois orders, from the point of view of deformation quantization. The most important examples of Galois orders (enveloping algebras, finite W-algebras, spherical Hecke algebras) are quantizations of some Poisson algebras. These are related to the complex Gelfand-Tsetlin integrable system. We discuss to what extent such Poisson algebras can be obtained for more general classes of Galois orders. In particular we describe a Poisson algebra analog of the so called standard Galois orders. SLIDES

Yoosik Kim (Pusan, via zoom):

Exotic monotone Lagrangian tori in flag manifolds. As a part of classification problems of Lagrangian submanifolds in a symplectic manifold, constructing monotone Lagrangian tori that are not Hamiltonian isotopic to each other is important. In this talk, I discuss how to construct infinitely many monotone Lagrangian tori and distinguish them in complete flag manifolds Fl(n) (n > 5) by using completely integrable systems. SLIDES

Ralph Klaasse (Brussels, via zoom):

Characteristic classes of Dirac structures and the modular foliation. In this talk we discuss how Lie algebroid representations of Dirac structures allow one to construct Dirac structures on their total space. Similar to the Poisson setting, this allows for the definition of a modular foliation. We discuss this construction, including some of its properties and consequences. This is based on joint work with Charlotte Kirchhoff-Lukat.

Yiannis Loizides (Cornell):

Transversely symplectic Riemannian foliations and [Q,R]=0. I will discuss a quantization commutes with reduction ([Q,R]=0) theorem for Riemannian foliations equipped with a transverse symplectic structure. This is joint work in progress with Yi Lin, Reyer Sjamaar, and Yanli Song. SLIDES

Alessandro Malusà (Toronto):

Quantisation on Sp(1)-symmetric hyper-Kähler manifolds. If one is interested in geometric quantisation on a hyper-Kähler space, there is a problem they immediately need to face: there are too many symplectic structures to choose from! Even more, they are often permuted by group actions. This suggests that one should proceed by quantising all symplectic forms at once, or at least all those within a given orbit.

In a joint work with J.E. Andersen and G. Rembado (arXiv:2111.03584), we consider the case of a transitive group action. Under suitable assumptions, we use the group action to decompose the resulting family of quantum Hilbert spaces as a direct sum of Hermitian line bundles and compute the curvature of natural connections defined on them. SLIDES

Mykola Matviichuk (McGill):

Poisson deformations of three transverse hyperplanes. In a joint work with Brent Pym and Travis Schedler, we described the Poisson deformations of a log symplectic form whose polar divisor has only normal crossings singularities. In this talk, I will discuss an illustrative example: the case when the polar divisor in question is a union of 3 transverse hyperplanes in an affine space of dimension at least 4. The possible singularities for the deformed polar divisor are the Whitney umbrella, elliptic singularities, and (non-isolated) stem singularities. I will explain how to detect which of the singularities occur based on the de Rham class of the log symplectic form we are deforming.

Eckhard Meinrenken (Toronto):

On the integration of transitive Lie algebroids. We revisit the problem of integrating Lie algebroids to Lie groupoids, for the special case that the Lie algebroid 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. SLIDES

David Miyamoto (Toronto):

Basic Forms on Foliated manifolds. Given a regular foliation F on a manifold M, we consider: the complex of basic differential forms on M, and of diffeological differential forms on the quotient M/F. Using the holonomy groupoid, pseudogroups, and diffeological tools, we prove the pullback by the quotient M → M/F induces an isomorphism of these complexes. We get a similar result for certain singular foliations. SLIDES

Nicholas Ovenhouse (Minnesota):

Noncommutative Poisson Structures from Networks. To a network drawn on a disc or cylinder, one can associate a noncommutative ring, and an associated noncommutative Poisson bracket. We show that this noncommutative Poisson bracket satisfies a version of the Sklyanin R-matrix formula. This has applications in "cluster integrable systems", most notably the pentagram map. This is joint work with Michael Shapiro and Semeon Arthamonov. SLIDES

Joey Palmer (Urbana-Champaign):

Extending compact Hamiltonian S^1 spaces to integrable systems in dimension 4. We study the relationship between Hamiltonian S^1-actions and integrable systems on compact symplectic 4-manifolds. In particular, we show that any such S^1-action can be extended to an integrable system such that all singularities are non-degenerate, except possibly for finitely many degenerate orbits of parabolic type, and we study various properties of the resulting system. This is joint with Sonja Hohloch. SLIDES

Gus Schrader (Northwestern, via zoom):

Bifundamental Baxter operators and DT transformations for Coulomb branches. Given a finite dimensional representation N of a complex reductive group G, Braverman, Finkelberg and Nakajima construct an affine Poisson variety called the Coulomb branch of the corrresponding 3d N=4 gauge theory. In my talk I will describe some interesting discrete symmetries of these Poisson varieties when G = GL_n x GL_m and N is the bifundamental representation. These symmetries are constructed with the help of the cluster structure on the (K-theoretic) Coulomb branch, and allow us to compute the Donaldson-Thomas transformation encoding the BPS spectrum of a quiver gauge theory when the quiver has no self-loops. SLIDES

Seokbong Seol (Penn State):

Formal exponential map of dg manifolds. Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to Poincare–Birkhoff–Witt isomorphism and the complete symbols of differential operators. We will investigate the question on how to extend these maps to dg manifolds. As an application, we will show there is an L-infinity structure on the space of vector fields in connection with Atiyah class of a dg manifold. In particular, for the dg manifold arising from a foliation, we induce an L-infinity structure on the deRham complex associated to the foliation. As a special case, for the dg manifold arising from a complex manifold, we show that the L-infinity structure is quasi-isomorphic to the Kapranov’s L-infinity structure on the Dolbeault complex. This is a joint work with Mathieu Stienon and Ping Xu. SLIDES

Alexander Shapiro (Edinburgh, via zoom):

Gelfand-Zeitlin systems and cluster varieties. In a joint work with Gus Schrader, we have described a realization of the quantum group as a quantum cluster variety. It turns out that the Gelfand-Zeitlin subalgebra has a very natural cluster algebraic description, and is closely related to Coxeter-Toda integrable systems, Fenchel-Nielsen coordinates in higher Teichmüller theory, and K-theoretic Coulomb branches of 3d N=4 theories recently described by Braverman, Finkelberg, and Nakajima. I will discuss some of these topics. SLIDES

Wilmer Smilde (Urbana-Champaign):

Linearization of Poisson groupoids. This talk concerns the linearization of Poisson structures on Poisson groupoids around the unit section. We show that the dual integration of triangular Lie bialgebroid always comes with a linearizable Poisson structure. Based on ArXiV:2108.11491. SLIDES

Margaret Symington (Mercer University):

Coordinate charts for Lagrangian-fibered K3s. Lagrangian fibrations of K3 surfaces arising from mirror symmetry and toric degenerations can be represented by a marked reflexive polytope. I will describe the relationship between those representations and integral affine coordinate charts that are suitable for exploring the geometry of Lagrangian fibrations of the K3 surface that need not be compatible with any complex algebraic structure.

Xiudi Tang (Beijing Institute of Technology, via zoom):

Symplectic and smooth excision. A symplectic/smooth excision is a symplecto/diffeomorphism between a noncompact manifold and the complement of a closed subset. Smooth excisions are not only a more general concept of symplectic ones, but also our key step to constructing them. We will discuss the existence of those excisions by explicit constructions and examples. SLIDES

Kurt Trampel (Notre Dame):

Quantum cluster algebras and discriminants. The setup of quantum cluster algebras will be reviewed, and the case of roots of unity will be discussed. Fundamental structure theorems will be given in this case, such as the embedding of classical cluster algebras into quantum cluster algebras as a central subalgebra. In the case of quantum Schubert cells at roots of unity, this recovers the canonical central subalgebra of De Concini-Kac-Procesi. Discriminants of these algebras will be discussed with regards to cluster structure.

Joel Villatoro (St Louis):

Diffeological Solutions to the Integration Problem. In this talk I will talk about recent work towards developing a diffeological version of Lie theory which is permissive enough to study non-integrable algebroids. SLIDES

Jordan Watts (Central Michigan):

Bicategories of Diffeological Groupoids. Diffeological groupoids have become important in recent years in the study of group actions, foliations, and Lie algebroids. A new paper of van der Schaaf shows that, similar to the Lie groupoid case, there is a bicategory of diffeological groupoids with principal bibundles as one-arrows and biequivariant diffeomorphisms as two-arrows. However, there remained an open question: does a diffeological Morita equivalence between Lie groupoids imply a Lie Morita equivalence?

In this talk, we answer this in the affirmative, and one can obtain this answer by jumping between three bicategories of diffeological groupoids: Pronk's bicategory of fractions, Roberts' anafunctor bicategory, and that above; moreover, this procedure seems to be a form of "optimization". SLIDES

Alan Weinstein (Berkeley/Stanford):

A Lie-Rinehart algebra in general relativity. Blohmann, Schiavina, and I have found a Lie-Rinehart algebra on a graded extension of the space of initial values for the Einstein equations whose bracket relations match those of the constraints on the initial values. PREPRINT

Florian Zeiser (Urbana-Champaign):

The Poisson cohomology of 3-dimensional Lie algebras. Poisson cohomology is a natural invariant of every Poisson manifold, obtained from the complex of multivector fields and the Schouten-Nijenhuis bracket of the Poisson bivector field. The cohomology groups play an important role in questions such as linearization or deformations, but are quite difficult to compute in general. For the linear Poisson structure on the dual of Lie algebra, Poisson cohomology can be understood as Lie algebra cohomology with coefficients in the smooth functions on the dual. In this talk we present a description of the Poisson cohomology groups associated to all 3-dimensional Lie algebras. This is based on joint work with D. Hoekstra and I. Marcut.

Workshop Dinner

All participants are invited to the Workshop Dinner which will take place after the Friday afternoon talks, in the Marriott Courtyard hotel in downtown Savannah. (See hotels.)