### Distinguished IGA Lecture Series by

### Jorgen Ellegaard Andersen (University of Aarhus)

and

### Frances Kirwan (University of Oxford)

#### Short Biographies

Professor Andersen is a leading expert on the gauge theoretic approach to quantum invariants of 3-manifolds and their underlying conformal and topological quantum field theories.

He is the director of the Center for Quantum Geometry of Moduli Spaces (QGM), Aarhus University. Following his PhD at the University of Oxford in 1992, he was appointed a C.B. Morrey Jr. assistant professor at the University of California, Berkeley from 1992-1994. In 2007 he was appointed a Professor of Mathematics at the University of Aarhus. In 2009 he was appointed director of the Center of Excellence, Centre for Quantum Geometry of Moduli Spaces (QGM). He has held several visiting positions, in particular he was Visiting Clay Professor of Mathematics at Berkeley in 2000-2001 and since 2013 he has been a Distinguished Visiting Professor at California Institute of Technology.

Professor Kirwan is a leading expert on moduli spaces in algebraic geometry, geometric invariant theory (GIT), and the link between GIT and moment maps in symplectic geometry. Her work endeavours to understand the structure of geometric objects by investigation of their algebraic and topological properties.

From 1983-85 she held a Junior Fellowship at Harvard. From 1983-86 she held a Fellowship at Magdalen College, Oxford, before later becoming a Fellow of Balliol College, Oxford. She is an honorary fellow at Clare College, Cambridge. In 1996 she was appointed a University Professor of Mathematics. From 2004-06 she was President of the London Mathematical Society, the second-youngest president in the society's history. In 2005, she received a five-year EPSRC Senior Research Fellowship, to support her research on the moduli spaces of complex algebraic curves. In 2017, she was elected to the Savilian Professorship of Geometry at the University of Oxford.

#### Professor Kirwan's lectures on "Non-reductive GIT"

- Lecture 1 (Monday 4 December, 9:45am - 10:30am)

__Title__: Moduli spaces and classical Geometric Invariant Theory__Abstract__: Geometric invariant theory (GIT) was developed by Mumford in the 1960s in order to construct and study quotients of algebraic varieties by actions of reductive linear algebraic groups. His main motivation was that many interesting moduli spaces in algebraic geometry can be constructed in this way.

- Lecture 2 (Monday 4 December, 10:45am - 11:30am)

__Title__: Non-reductive Geometric Invariant Theory__Abstract__: In general GIT for non-reductive linear algebraic group actions is much less well behaved than for reductive actions. However when the unipotent radical U of a linear algebraic group H is graded, in the sense that a Levi subgroup has a central one-parameter subgroup which acts by conjugation on U with all weights strictly positive, then GIT for a linear action of H on a projective scheme is almost as well behaved as in the reductive setting, provided that we are willing to multiply the linearisation by an appropriate rational character.

- Lecture 3 (Tuesday 5 December, 9:45am - 10:30am)

__Title__: Generalising symplectic implosion__Abstract__: The symplectic reduction of a Hamiltonian action of a Lie group on a symplectic manifold plays the role of a quotient construction in symplectic geometry. It has been understood for several decades that symplectic reduction can be used to describe the quotients for complex reductive group actions in algebraic geometry provided by Mumford's GIT. There is an analogue of this description for GIT quotients by suitable non-reductive actions, which generalises the symplectic implosion construction of Guillemin, Jeffrey and Sjamaar.

- Lecture 4 (Tuesday 5 December, 10:45am - 11:30am)

__Title__: Moduli spaces of unstable objects__Abstract__: Non-reductive GIT can be applied to the construction of moduli spaces in cases when classical GIT is not applicable. These include moduli spaces of 'unstable' objects of prescribed type, such as sheaves of fixed Harder-Narasimhan type, unstable projective curves or projective schemes of dimension greater than 1.

#### Professor Andersen's lectures on "Geometric Recursion" (joint work with G. Borot and N. Orantin)

- Lecture 1 (Monday 4 December, 1:30pm - 2:15pm)

__Abstract__: We propose a general theory whose main component are functorial assignments for a large class of functors from a certain category of bordered surfaces to a suitable a target category of topological vector spaces. The construction is done by summing appropriate compositions of the initial data over all homotopy classes of successive excisions of embedded pair of pants.

- Lecture 2 (Monday 4 December, 2:30pm - 3:15pm)

__Abstract__: We provide sufficient conditions to guarantee these infinite sums converge and as a result, we can generate mapping class group invariant vectors which we call amplitudes. The initial data encode the amplitude for pair of pants and tori with one boundary, as well as the "recursion kernels" used for glueing. We give this construction the name of "geometric recursion", abbreviated GR.

- Lecture 3 (Tuesday 5 December, 1:30pm - 2:15pm)

__Abstract__: As an illustration, we show how to apply our formalism to various spaces of continuous functions over Teichmüller spaces, as well as continuous functions on Teichmüller space with values in Poisson structures on the moduli space of flat connections. The theory has a wider scope than that and one expects that many functorial objects in low-dimensional geometry and topology should have a GR construction. The geometric recursion has various projections to topological recursion (TR) and we in particular show it retrieves all previous variants and applications of TR.

- Lecture 4 (Tuesday 5 December, 2:30pm - 3:15pm)

__Abstract__: We show that, for any initial data for topological recursion, one can construct initial data for GR with values in Frobenius algebra-valued continuous functions on Teichmüller space, such that the ω_{g,n} of TR are obtained by integration of the GR amplitudes over the moduli space of bordered Riemann surfaces and Laplace transform with respect the boundary lengths. In this regard, the structure of the Mirzakhani-McShane identities served as a prototype of GR.