Spring 2019 Room 705 Hill Center, Tuesdays 11.45 AM - 12.45 PM


Guillemin-Sternberg, 1982, Inventiones Mathematicae
CONVEXITY PROPERTIES OF THE MOMENT MAPPING
January, Tuesday 22 - Marco Castronovo


We quickly recap the basics of Lie theory, and how to use it to generalize the notion of Hamiltonian circle action to arbitrary compact Lie groups G via the moment map. Then we outline Guillemin-Sternberg's proof of two convexity statements about the image of this map. For abelian G the image is globally convex, because the moment map is remarkably Morse-Bott in every direction. For nonabelian G the image is locally convex, and the proof reduces to the abelian case by constructing a symplectic submanifold crossing most orbits and invariant under the action of a maximal torus.

Parker, 2015, Advances in Mathematics
HOLOMORPHIC CURVES IN EXPLODED MANIFOLDS
February, Tuesday 19 - Chris Woodward [TALK IN THE GRADUATE LOUNGE]


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Thurston, 1976, Proceedings of the AMS
SOME SIMPLE EXAMPLES OF SYMPLECTIC MANIFOLDS
February, Tuesday 26 - Soham Chanda 
[TALK IN THE GRADUATE LOUNGE]

In this paper Thurston gives an example of a symplectic manifold which is not Kähler. We discuss the construction of this example and check that indeed it doesn't have a Kähler structure. We also discuss a method of constructing a symplectic structure on some special fiber bundles such that each fiber is a symplectic submanifold of the bundle.

Auroux, 2014, Bolyai Society Mathematical Studies
A BEGINNER'S INTRODUCTION TO FUKAYA CATEGORIES
March, Tuesday 12 - Xindi Zhang [TALK IN THE GRADUATE LOUNGE]


I will talk about the first half of the paper. We will learn the definition Floer homology, and mainly focus on the spaces with a pair of Lagrangians. We will discuss some examples when the Floer differential is or is not well defined.

Harada-Kaveh, 2015, Inventiones Mathematicae
INTEGRABLE SYSTEMS AND TORIC DEGENERATIONS
April, Tuesday 02 - Joey Palmer 
[TALK IN THE GRADUATE LOUNGE]

We will talk about the first part of "Integrable systems, toric degenerations, and Okounkov bodies" by Harada-Kaveh which equips a smooth projective variety X with an integrable system when X is part of a toric degeneration with certain properties. The idea is to use the Hamiltonian torus action on the special fiber (i.e. the toric fiber) of the degeneration to define such an action on an open dense subset of X. Some background material on toric degenerations may also be taken from "An invitation to toric degenerations" by Gross-Siebert.

Kronheimer, 1990, Journal of the LMS
A HYPER-KÄHLERIAN STRUCTURE ON COADJOINT ORBITS
April, Tuesday 30 - Marco Castronovo


After a quick review of connections on principal G-bundles, we focus on the special class of G-instantons on R^4 with a pole at the origin. We explain how instantons invariant under a natural SU(2) action can be thought as solutions of a system of ordinary differential equations with values in the Lie algebra g of G, the Nahm's equations, and how to interpret them as gradient flow equations for a simple functional on g^3. We outline Kronheimer's arguments showing that the set of instantons with fixed asymptotic behavior at the pole has a hyper-Kähler manifold  structure, and how to use the asymptotic data to define a diffeomorphism with certain coadjoint orbits in the complexification of g.