10:00am **SCGP:** Poisson Geometry Program Seminar: Martin Kassabov

**Where:** SCGP 102**When:** Wed, Jun 20 10:00am — 12:00pm

**Title:** From the algebra of symplectic derivations to cohomology of automorphism groups of free groups

**Abstract:**

Let V be a vector space with a symplectic form, and let D^\omega(V) be the Lie algebra of derivations of the free Lie algebra on V which preserved the symplectic form.

This Lie algebra has very interesting properties and is related to the cohomology of the automorphism groups of free groups. I will discuss how to compute the abelianization of the Lie algebra (and several other Lie algebras related to it). The connection with cohomology Aut(F_n) leads to a natural assembly construction in the homology for the automorphism groups of free groups. This construction has origins in the theory of (modular) operads but has a very natural and easy geometric interpretation. I will show how to use a few nontrivial homology classes to construct many interesting ones. However, it remains an open problem whether any of the resulting classes are non-trivial

(based on a joint works with J Conant A. Hatcher and K. Vogtmann)

Let V be a vector space with a symplectic form, and let D^\omega(V) be the Lie algebra of derivations of the free Lie algebra on V which preserved the symplectic form.

This Lie algebra has very interesting properties and is related to the cohomology of the automorphism groups of free groups. I will discuss how to compute the abelianization of the Lie algebra (and several other Lie algebras related to it). The connection with cohomology Aut(F_n) leads to a natural assembly construction in the homology for the automorphism groups of free groups. This construction has origins in the theory of (modular) operads but has a very natural and easy geometric interpretation. I will show how to use a few nontrivial homology classes to construct many interesting ones. However, it remains an open problem whether any of the resulting classes are non-trivial

(based on a joint works with J Conant A. Hatcher and K. Vogtmann)

11:15am Mathematics Department Gathering: Theodore Drivas - Informal Survey of 3D Fluid Motion Part I

**Where:** Math Tower Common Room**When:** Wed, Jun 20 11:15am — 12:45pm

**Title:** Informal Survey of 3D Fluid Motion Part I

**Speaker:** Theodore Drivas [Princeton University]

**Abstract:** An invitation to 3D fluids: experiments, particle models and simulations, the continuum model and discussion of scales. With discussions by grad students and all interested parties.

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1:30pm **SCGP:** Poisson Geometry Program Seminar: Florian Naef

**Where:** SCGP 102**When:** Wed, Jun 20 1:30pm — 3:30pm

**Title:** On a moduli space interpretation of the Turaev cobracket

**Abstract:** Given an oriented surface, Goldman defines a Lie bracket on the vector space spanned by free homotopy classes of loops in terms of intersections. This Lie bracket is the universal version of the Atiyah-Bott Poisson structure on the moduli space of flat connections. Using self-intersections Turaev defines a Lie cobracket on loops. We give a possible interpretation of this structure on moduli spaces of flat connections in the form of a natural BV operator.

This is joint work with A. Alekseev, J. Pulmann and P. Ċ evera.

This is joint work with A. Alekseev, J. Pulmann and P. Ċ evera.

2:00pm Mathematics Department Gathering: Theodore Drivas - Informal Survey of 3D Fluid Motion Part II

**Where:** Math Tower Common Room**When:** Wed, Jun 20 2:00pm — 3:30pm

**Title:** Informal Survey of 3D Fluid Motion Part II

**Speaker:** Theodore Drivas [Princeton University]

**Abstract:** An invitation to 3D fluids: experiments, particle models and simulations, the continuum model and discussion of scales. With discussions by grad students and all interested parties.

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10:00am **SCGP:** Poisson Geometry Program Seminar: Mini course by Dennis Sullivan

**Where:** SCGP 102**When:** Thu, Jun 21 10:00am — 12:00pm

**Title:** Algebra and Geometry of String Topology Part II

**Abstract:** The general picture of string topology in terms of the algebraic topology of the stratified space of closed curves in a manifold.

10:00am SCGP Seminars: Dennis Sullivan - Algebra and Geometry of String Topology Part II

**Where:** SCGP 102**When:** Thu, Jun 21 10:00am — 12:00pm

**Title:** Algebra and Geometry of String Topology Part II

**Speaker:** Dennis Sullivan [Stony Brook University]

**Abstract:** The general picture of string topology in terms of the algebraic topology of the stratified space of closed curves in a manifold.

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1:30pm Dynamical Systems Seminar: Roland Roeder - Limiting Measure of Lee-Yang Zeros for the Cayley Tree

**Where:** Math Tower P-131**When:** Thu, Jun 21 1:30pm — 2:30pm

**Title:** Limiting Measure of Lee-Yang Zeros for the Cayley Tree

**Speaker:** Roland Roeder [IUPUI]

**Abstract:** I will explain how to use detailed properties of expanding maps of the circle (Shub-Sullivan rigidity, Ledrappier-Young formula, large deviations principle, ...) to study the limiting distribution of Lee-Yang zeros for the Ising Model on the Cayley Tree. No background in mathematical physics is expected of the audience. This is joint work with Ivan Chio, Caleb He, and Anthony Ji.

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10:00am **SCGP:** Poisson Geometry Program Seminar: Arkady Berenstein

**Where:** SCGP 102**When:** Tue, Jun 26 10:00am — 12:00pm

**Title:** Integrable clusters

**Abstract:** The goal of my talk (based on joint work with Jacob Greenstein and David Kazhdan) is to discuss Poisson cluster algebras such that in one of the clusters all mutable variables Poisson commute which each

other. It turns out that this property is frequently preserved by mutations, i.e., all mutable variables in all clusters commute with each other. We prove the "total integrability" phenomenon for any Poisson cluster algebra whose initial seed is principal. In particular, this gives a large family of Lagrangian foliations of the corresponding cluster variety.

Integrable clusters admit a natural quantization: those quantum clusters in which all mutable variables commute with each other. Once again, if the corresponding quantum seed is principal, then the "quantum

integrability" is preserved by mutations in all directions. Remarkably, this "total quantum integrability" is equivalent to the celebrated sign coherence conjecture proved by Gross, Hacking, Keel and Kontsevich in

2014.

other. It turns out that this property is frequently preserved by mutations, i.e., all mutable variables in all clusters commute with each other. We prove the "total integrability" phenomenon for any Poisson cluster algebra whose initial seed is principal. In particular, this gives a large family of Lagrangian foliations of the corresponding cluster variety.

Integrable clusters admit a natural quantization: those quantum clusters in which all mutable variables commute with each other. Once again, if the corresponding quantum seed is principal, then the "quantum

integrability" is preserved by mutations in all directions. Remarkably, this "total quantum integrability" is equivalent to the celebrated sign coherence conjecture proved by Gross, Hacking, Keel and Kontsevich in

2014.

1:30pm **SCGP:** Poisson Geometry Program Seminar: Ivo Sachs

**Where:** SCGP 102**When:** Tue, Jun 26 1:30pm — 3:30pm

**Title:** Homotopy algebras in string field theory

**Abstract:** Homotopy algebra and its involutive generalisation plays an important role in the consontruction of string field theory.

It ensures consistemcy and also enters crucially in deformation theory and background independence of string theory.

Conversely, world sheet string theory naturally realizes a minimal model map.

I will review recent progress in these applications of homotopy algebra, their operadic description and its relation to moduli spaces.

It ensures consistemcy and also enters crucially in deformation theory and background independence of string theory.

Conversely, world sheet string theory naturally realizes a minimal model map.

I will review recent progress in these applications of homotopy algebra, their operadic description and its relation to moduli spaces.

10:00am **SCGP:** Poisson Geometry Program Seminar: Matteo Felder

**Where:** SCGP 102**When:** Wed, Jun 27 10:00am — 11:00am

**Title:** Higher genus Grothendieck-Teichmüller Lie algebras

**Abstract:** The Grothendieck-Teichmüller Lie algebra grt was introduced by Drinfeld and is a mysterious object which has many applications in algebra, geometry and topology. An important result by Willwacher identifies grt with the degree zero cohomology of Kontsevich's graph complex GC, itself an interesting combinatorial object whose cohomology in positive degrees is unknown.

In this talk we will discuss a possible ``higher genus" analogue of this result. More precisely, we will recall Enriquez' elliptic Grothendieck-Teichmüller Lie algebra grt_ell and how to assign to a closed surface S of genus g a graph complex GC_S which generalises Kontsevich's GC. Strikingly, in genus one the degree zero cohomology of GC_S coincides with the elliptic Grothendieck-Teichmüller Lie algebra. For higher genus, it should be possible to express the zeroth cohomology of GC_S in similar terms as grt and grt_ell, and we will give possible candidates for what could then be called ``higher genus Grothendieck-Teichmüller Lie algebras". This is joint work in progress with Thomas Willwacher.

In this talk we will discuss a possible ``higher genus" analogue of this result. More precisely, we will recall Enriquez' elliptic Grothendieck-Teichmüller Lie algebra grt_ell and how to assign to a closed surface S of genus g a graph complex GC_S which generalises Kontsevich's GC. Strikingly, in genus one the degree zero cohomology of GC_S coincides with the elliptic Grothendieck-Teichmüller Lie algebra. For higher genus, it should be possible to express the zeroth cohomology of GC_S in similar terms as grt and grt_ell, and we will give possible candidates for what could then be called ``higher genus Grothendieck-Teichmüller Lie algebras". This is joint work in progress with Thomas Willwacher.

11:30am **SCGP:** Poisson Geometry Program Seminar: Elise Raphael

**Where:** SCGP 102**When:** Wed, Jun 27 11:30am — 12:30pm

Title : On elliptic versions of the Kashiwara-Vergne Lie algebra and mould theory

Using the theory of moulds developed by Ecalle, we will define a linearized and an elliptic versions of the Kashiwara Vergne Lie algebra. This elliptic version of krv_ell is very close to the elliptic version of the double shuffle Lie algebra coming from number theory.

Another elliptic Kashiwara Vergne Lie algebra was defined by Alekseev, Kawazumi, Kuno and Naef using topological tools. We will translate their definition in the moulds language and show that the two definitions are equivalent.

This is joint work with Leila Schneps.

Title : On elliptic versions of the Kashiwara-Vergne Lie algebra and mould theory

Using the theory of moulds developed by Ecalle, we will define a linearized and an elliptic versions of the Kashiwara Vergne Lie algebra. This elliptic version of krv_ell is very close to the elliptic version of the double shuffle Lie algebra coming from number theory.

Another elliptic Kashiwara Vergne Lie algebra was defined by Alekseev, Kawazumi, Kuno and Naef using topological tools. We will translate their definition in the moulds language and show that the two definitions are equivalent.

This is joint work with Leila Schneps.

2:00pm **SCGP:** Poisson Geometry Program Seminar: Travis Schedler

**Where:** SCGP 102**When:** Wed, Jun 27 2:00pm — 4:00pm

**Title:** Holonomic Poisson manifolds and deformations of elliptic algebras

**Abstract:** I will introduce the notion of holonomic Poisson manifolds, which can be thought of as a refinement of the log symplectic condition, describing "minimally degenerate" compactifications of symplectic manifolds, which we expect to arise in representation theoretic contexts. These manifolds are characterized by having finite-dimensional spaces of local deformations and quantizations. The notion is closely related to the flow of the modular vector field (a local symmetry discovered by Brylinski--Zuckerman and Weinstein, which also measures the failure of Hamiltonian flow to preserve volume). As an application, I will prove that the the first families of

Feigin-Odesski elliptic algebras quantizing P^{2n} (and the corresponding Poisson structures) are universal deformations. This is joint work with Brent Pym.

Feigin-Odesski elliptic algebras quantizing P^{2n} (and the corresponding Poisson structures) are universal deformations. This is joint work with Brent Pym.