### 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. He has held several visiting positions, most recently at the California Institute of Technology in 2013. From 2006-2011 he was the director of the Center for the Topology and Quantisation of Moduli Spaces (CTQM).

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

- 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 - 10:30am)

__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

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

__Abstract__: TBA

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

__Abstract__: TBA

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

__Abstract__: TBA

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

__Abstract__: TBA