### Schedule of talks for 2015/16:

##### Semester 2

January 29, 2:30pm | Zoe Wyatt | Minimal surfaces in general relativity |

Abstract: In this talk I will present some of the links between minimal surfaces and general relativity. I will start by explaining concepts from differential geometry and relativity that are needed to formulate the Einstein equations. This will be very much aimed at people who haven't met these ideas before. Next I will discuss so-called trapped surfaces, when these can be minimal surfaces, and their relation to the formation of black holes. Lastly I will discuss the positive mass theorem, and outline where minimal surfaces appear in its proof. Along the way, I hope to explain some of the broad questions and conjectures that relativists and cosmologists think about. | ||

February 5, 2:30pm | Chunyi Li | Representations of the symmetric group |

Abstract: In this talk, we will have a brief review of the representation theory of the symmetric group Sn, with an emphasis on its relation between permutation actions of the symmetric group. We will also give examples from the Sn-action on the moduli space of curves with marked points. | ||

March 4, 2:30pm | Chris Campbell | Deforming rings with automorphisms of surfaces |

Abstract: The classification of "noncommutative surfaces” is still an open problem in dimension 3 - in dimension 4 we have basically no idea what can happen. In this talk I will present some of my work in attempting to generate new examples of these surfaces using automorphisms of surfaces birational to projective plane. I will explain all of the background so no knowledge beyond basic commutative algebra and geometry will be assumed. | ||

March 18, 2:30pm | Soheyla Feyzbakhsh | Stability on K3 surfaces |

Abstract: I will start by describing the space geometric stability conditions on an algebraic K3 surface, then explain how this space helps us to prove some classical result. I will focus on proving slope stability of the tangent bundle of projective space restricted to a surface.
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March 25, 2:30pm | Juliet Cooke | Surgery theory and cobordisms |

Abstract: In this talk I will give a basic introduction to surgery and its connection to Morse theory and cobordisms. I will also talk about the s/h-cobordism theorem and the application to the classification of surfaces of surgery theory including exotic spheres. | ||

April 1, 2:30pm | Igor Krylov | Introduction to MMP |

April 22, 2:30pm | Sjoerd Beentjes | The Hilbert scheme and what it’s good for |

Abstract: Constructed by Grothendieck in the 60’s, the Hilbert scheme classifies closed subvarieties of projective space. It is one of the few moduli schemes that is, in fact, a scheme, and most other moduli spaces rely on its existence in some way or another. In this talk, I will define and motivate the Hilbert functor of a fixed complex projective variety. Admitting the Hilbert scheme’s existence (i.e. not doing the hard work), we then quickly move on to concrete examples and some of its many applications. Buzzwords: Grassmannian, rationality, curve-counting. | ||

April 29, 2:30pm | Martin Kalck | TBD |

##### Semester 1

September 25, 2:30pm | Krylov Igor | Generators and finite subgroups of Cremona group |

Abstract: Cremona n-group Cr_n(C) is a group of automorphisms of a field of rational functions k(x_1,...,x_n) or, equivalently, a group of a birational transformations of a projective space of dimension n. I will give basic definitions for birational geometry of algebraic surfaces. Then I will discuss generation of Cremona group in dimension 2. I will also discuss finite subgroups of Cremona group, in particular simple subgroups and the conjugacy classes of simple subgroups. Notes | ||

October 2, 11:00am | Noah White | Geometric invariant theory: quotients in algebraic geometry |

This will be the first talk of a reading group on spherical varieties. I will give an introduction to some basic properties of algebraic groups and explain how one can construct (approximations of) quotient spaces. I will focus on providing lots of examples. | ||

October 9, 2:30pm | Carmen Rovi | The signature of a fibration modulo 8. |

In this talk we shall be concerned with the residues modulo $4$ and modulo $8$ of the signature $\sigma(M) \in \mathbb{Z}$ of an oriented $4k$-dimensional geometric Poincaré complex 4M^{4k}$. The precise relation between the signature modulo $8$, the Arf invariant and the Brown-Kervaire invariant will be given. Furthermore we shall discuss how the relation between these invariants can be applied to the study of the signature of a fibration modulo $8$. | ||

October 16, 2:30pm | Salvatore Dolce | Invariants and Polynomial identities |

Two weeks ago Noah introduced us to GIT and we discussed a lot of examples. In this talk I will take a more algebraic point of view and, always discussing concrete examples, I will show how GIT naturally appears in other areas of Mathematics. In particular, we will look at spaces of matrices and identities between them. Technically speaking we will describe the structure of the space of G-equivariant matrix-valued alternating multilinear maps on spaces of matrices, where G is some classical group. More technically speaking we study the structure of the space of covariants for a certain class of infinitesimal symmetric spaces such that the space of invariants is an exterior algebra. We prove that it is a free model over a special sub algebra of the invariants and we give an explicit basis for that. Further, we deduce some identities of Caley-Hamilton type. | ||

October 23, 2:30pm | Carlos Zapata-Carratala | Geometric Mechanics and a brief introduction to Quantum Physics |

As the first talk of the series by Tim and myself on Geometric Quantization, I decided to present the physical background of this branch of Differential Geometry. A completely mathematically rigorous approach will be taken and the language, except for the occasional physical concept, will be that of modern mathematics. There are two fundamental ideas I want to convey: (i) why symplectic manifolds (or more generally Poisson manifolds) are usually considered to represent phase spaces of physical systems, and (ii) what is the mathematical framework of quantum mechanics and where did it come from. | ||

October 30, 2:30pm | Tim Weelinck | Geometric Quantization & The Quantum Harmonic Oscillator |

We continue the talk of Carlos by introducing geometric quantization: a bridge to walk from the classical world to the quantum world. As geometric quantization in all its generality needs many technical notions (Hermitian line bundles, polarizations, canonical line bundle, etc) we will restrict our attention to the quantum harmonic oscillator (Q.H.O). Concretely, we will perform geometric quantization in Euclidean space which make the necessary machinery much more manageable. We motivate the different steps of performing geometric quantization such as `pre-quantization' and `quantization' by looking at the Q.H.O. If time permits we will discuss how to obtain the correct energy spectrum of the Q.H.O. by using half-density quantization. Geometric quantization is by no means perfect; we will point out its benefits, but also its shortcomings along the way. | ||

November 6, 2:30pm | Cancelled | |

November 13, 2:30pm | Martina Lanini | Combinatorics of flag varieties. |

Abstract: Appeared for the first time in the 19th Century to encode questions in enumerative geometry, flag varieties and their Schubert varieties had been intensively studied since then, constituting an important investigation object in topology, geometry, representation theory and algebraic combinatorics. In this talk, I will recall their definition and mostly focus on the underlying combinatorics. | ||

November 20, 2:30pm | Jenny August | Resolving Kleinian Singularities Using the SL(2) McKay Correspondence |

Abstract: A classical problem in geometry is trying to resolve singularities. In this talk, I will explain how this can be done for Kleinian singularities using tools from algebra and the McKay Correspondence which relates finite subgroups of SL(2) with Dynkin diagrams. | ||

November 27, 2:30pm | Matthew Woolf | The moduli space of curves |

Abstract: In this talk, we will have a gentle introduction to the algebraic geometry of moduli spaces of curves, with an emphasis on the detailed study of low genus examples. We will also discuss how one can use the moduli space and its Deligne-Mumford compactification to study the geometry of curves. | ||

December 11, 2:30pm | Tim Large | Floer theory, to A-infinity and beyond |

Floer theory is a powerful tool now ubiquitous in symplectic and low-dimensional topology. In its incarnation as Lagrangian Floer cohomology, it plays a central role in various mathematical formulations of string theory, and modern formulations of mirror symmetry link it to the study of coherent sheaves on algebraic varieties. We will set up Floer cohomology in its simplest setting, on oriented (Riemann) surfaces, where the hard analysis that characterises the higher-dimensional theory disappears, and the theory becomes wonderfully visual and essentially combinatorial. However all the essential algebraic features of the theory are clear and very geometric in this formulation, including the surprising appearance of A-infinity categories: "categories" where the composition of morphisms is not strictly associative, but this failure is tightly controlled by "higher order composition" maps. Absolutely no knowledge of symplectic geometry is required- we will be starting from scratch and working entirely in an elementary setting! |