In this talk, we will review the notions of non-commutative divergence and log-Jacobian 1-cocycles for free associative algebras. It turns out that these constructions become more rigid on (completed) group rings of free groups. This result is in some way analogous to existence of Haar measures on Lie groups. As an application, we consider fundamental groups of compact oriented surfaces with boundary. We show that for each framing of the surface there is a canonical (divergence type) 1-cocycle on derivations of the corresponding (completed) group ring. The talk is based on a joint work in progress with N. Kawazumi, Y. Kuno and F. Naef.

This talk will introduce a version of factorization homology tailored for 1-dimensional tangles. An instance of this construction is the Skein module for a 3-manifold. The talk will go on to outline a proof of the Tangle Hypothesis for 1-dimensional tangles in 3-manifolds. A key technical feature is the construction of an action of the group of based loops in the orthogonal group O(3) on a braided monoidal category with duals. This talk will synthesize and extend the lecture series that will take place the weekend prior. This is a report on joint work with John Francis.

I will discuss a construction of the shifted symplectic structure on the derived moduli space of flat bundles on a topological threefold using the Dupont contraction. This is joint work in progress with Elliot Cheung.

The heat equation method in index theory gives an explicit local formula for the index of a Dirac operator. Its Lagrangian counterpart involves supersym- metric path integrals. Getzler’s rescaling method has made this correspondence very explicit. Similar methods can be developed to give a geometric formula for semi- simple orbital integrals associated with the Casimir operator of a reductive group, this computation being related to Selberg’s trace formula. The analogue of the heat equation method is now a suitable deformation of the Laplacian by a family of Fokker-Planck operators Lb|b>0 that interpolates between the Casimir operator and the geodesic flow. Connections to Langevin’s approach to Brownian motion will be explained, b2 being identified with the mass m in the Langevin equation.

We discuss analytic aspects of dg-categories arising in differential geometry.

The Barr-Beck theorem gives conditions under which an adjunction F -| G is monadic. Monadicity, in turn, means that G can be computed in terms of the data of F and its endomorphism GF. I will present joint work-in-progress with Abouzaid, in which we consider this theorem in the case of the functors between Fuk(M1) and Fuk(M2) associated to a Lagrangian correspondence L12 and its transpose. These functors are often adjoint, and under the hypothesis that a certain map to symplectic cohomology hits the unit, the hypotheses of Barr–Beck are satisfied. This can be interpreted as an extension of Abouzaid's generation criterion, and we hope that it will be a useful tool in the computation of Fukaya categories.

Given (a certain kind of) 2CY structure on a category C, its derived moduli stack of objects acquires a 0-shifted symplectic structure. Via a general formality result for such categories, if the underlying classical stack has a good moduli space, the stack may be etale-locally modelled as the stack of representations of a preprojective algebra. Furthermore, it is possible to show that the derived direct image of the dualizing complex along the morphism to the good moduli space satisfies the famous BBDG decomposition theorem, and is furthermore pure, when considered as a mixed Hodge module. I will explain all this, as well as applications to Kac polynomials and (time permitting) nonabelian Hodge theory for stacks.

I will explain how generalized Kahler classes are actually holomorphic symplectic bibundles, whose Lagrangian bisections define generalized Kahler metrics. This “higher” or “1-shifted” point of view leads to a GIT/Hamiltonian reduction approach to generalized Kahler geometry analogous to that of Fujiki and Donaldson for the usual Kahler case. this is work in progress with Lennart Döppenschmitt.

PROBs are braided analogs of PROPs, so they are data describing algebraic structures on objects of braided monoidal categories involving multiplications and comultiplications. For a commutative monoid $L$ satisfying some conditions we compare two objects: (1) The colored PROB $B(L$) describing $L$-graded bialgebras. It splits into blocks $B_n(L)$ labelled by $nin L$. (2) The space $Z(C,L)$ of divisors (0-cycles) on the complex line $C$ with coefficients in $L$. It splits into components $Z(C,L)_n$ labelled by $nin L$. This includes the symmetric products of $C$ (obtained for $L=Z_+$) and the Ran space of $C$ (obtained for $L$ being the ``Boolean algebra of truth values"). We consider the category of perverse sheaves on $Z(C,L)_n$ (with respect to the natural stratification) and identify it with the category of representations of $B_n(L)$. The key role in this construction is played by a cell decomposition of $Z(C,L)$ labelled by ``contingency matrices" with coefficients in $L$. Joint work in progress with V. Schechtman.

The underlying geometry of a gauged linear sigma model (GLSM) consists of a GIT quotient of a complex vector space by the linear action of a reductive algebraic group G (the gauge group), and a regular function (the superpotential) on the GIT quotient. The Higgs-Coulomb correspondence relates (1) GLSM invariants which are essentially virtual counts of curves in the critical locus of the superpotential (Higgs branch), and (2) Mellin-Barnes type integrals on the Lie algebra of G (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the stability condition. This is based on joint work with Konstantin Aleshkin.

Inspired by work of Getzler-Jones, I will present a view on the homology of linear operads which easily extends to oo-operads, relates to the homology of categories, and allows for a simple and explicit proof of bar-cobar ("Koszul") duality.

I will explain an important extension of Hamiltonian and Quasi-Hamiltonian reduction which uses derived geometry in an essential way. This extension has built-in flexibility and provides a universal construction of many known and new symplectic structures in algebraic geometry. The generalized reduction construction relies on the notion of a relative shifted symplectic structure along the stalks of a constructible sheaf of derived stacks on a stratified space. As applications I will discuss a construction of symplectic structures on derived moduli of Stokes data on smooth varieties. This is a joint work with Dima Arinkin and Bertrand Toën.

Descendent classes on moduli spaces of sheaves are defined via the Chern characters of the universal sheaf. I will present several conjectures and results concerning stable pairs descendent invariants for 3-folds: rationality of generating functions, functional equations, cobordism classes, and Virasoro constraints.

I will explain a calculation of the stable cohomology of the hyperelliptic mapping class group with coefficients in an arbitrary symplectic representation. The result is closely related to, and provides a geometric interpretation of, a series of conjectures on asymptotics of moments of families of quadratic L-functions. (Joint with J. Bergström, A. Diaconu and C. Westerland)

The idea behind Grothendieck-Teichmüller theory is to study the absolute Galois group via its actions on (the collection of all) moduli spaces of genus $g$ curves. In practice, this is often done by studying an intermediate object: The Grothendieck-Teichmüller group, GT. In this talk, I’ll describe an algebraic gadget built from simple decomposition data of Riemann surfaces. This gadget, called an infinity modular operad, provides a model for the collection of all moduli spaces of genus $g$ curves with $n$ boundaries, which we justify by showing that the automorphisms of this algebraic object is isomorphic to a subgroup of Grothendieck-Teichmüller group. This is joint work with L. Bonatto

We will describe an approach to the Batalin-Vilkovisky formalism using derived symplectic geometry, specifically using a derived version of coisotropic reduction in the context of (-1)-shifted symplectic stacks. In the process, we obtain a modular interpretation of the master equation and give a geometric description of BRST cohomology.

I will report on recent work with Olga Trapeznikova on a new proof for the parabolic Verlinde formula based on a comparison of wall-crossings in Geometric Invariant Theory and certain iterated residue functionals.

In this talk, I will introduce "quasi-smooth foliations", which are (holomorphic) foliations with possibly singular leaves. The typical example of a quasi-smooth foliation is given by the fibers of a flat morphism between complex manifolds. Under mild conditions, I will explain the construction of a monodromy/holonomy groupoid integrating a given quasi-smooth foliation. When the quasi-smooth foliation is algebraic, I will state a foliated Riemann-Hilbert correspondance, which can be used in order to understand (part of) monodromy or holonomy in purely algebraic data. Joint with Vezzosi.

I will report on a joint work with Yiannis Loizides and Paul-Emile Paradan, and work in progress with Yiannis Loizides. Let G be a compact Lie group acting on a Hamiltonian way on a compact symplectic manifold with Kostant line bundle L. We give a formula for the semi-classical asymptotics of the equivariant index of the Dirac operator on M tensored with powers of L. When M is the projective space, the formula coincides with the Euler-MacLaurin formula. We indicate some application of this formula to geometric quantization of possible non compact Hamiltonian manifolds

L_infty algebras are a higher homotopical analogue of Lie algebras which arise in geometry and physics. Building on Getzler's work on integrating nilpotent L_infty algebras, as well as work of Henriques and Rogers-Zhu, we prove that every finite type L_infty algebra integrates to a finite dimensional Lie infty-group. This is joint work with Chris Rogers.