### Talks in 2017/18:

##### Semester 2

January 26, 1:45pm | Ruth Reynolds | Classifying Non-Commutative Projective Surfaces |

Abstract: In 1995, Stafford and Artin classified non-commutative projective curves, that is to say finitely generated, connected, graded k-algebras of GK dimension 2. In this talk we describe how to think of non-commutative projective surfaces and the '95 classification. We also cover some of the progress made towards a more general classification of non-commutative projective surfaces. | ||

February 2, 1:45pm | Matt Booth | Flops and the (derived) contraction algebra |

Abstract: I'll talk about how birational geometry links to noncommutative algebra via the Homological MMP. In particular, given a threefold flopping contraction X \to Y, one can associate a certain noncommutative algebra, the contraction algebra, that is conjectured to control the geometry of the flop. I'll talk about a deformation-theoretic interpretation, and generalise the construction to that of the derived contraction algebra. I'll give an alternate interpretation of the derived contraction algebra as a certain `derived quotient', and explain how this gives rise to a semi-orthogonal decomposition of the derived category of X. | ||

February 9, 1:45pm | Carlos Zapata-Carratala | Generalised Phase Spaces: Morphisms and Reductions |

Abstract: This talk will consist of an account of several generalised notions of classical phase spaces (which I take to be symplectic manifolds of finite dimension) presented in the context of Dirac geometry. Lie algebroids and Courant algebroids will be introduced and the general notion of Dirac structure will be shown to be a good candidate for a generalised phase space. Then the natural notions of morphism and reduction for this structures, which directly generalise those of symplectic manifolds, will be discussed. | ||

February 16, 12:00 noon | Tim Weelinck | Representation theory without vectors |

Abstract: What would representation theory look like without vectors? We will take this idea seriously and following work of Etingof-Nikshych-Ostrik study generalizations of finite groups called fusion categories. Then following Deligne we will use this new perspective to answer the following question: `What should the symmetric group on $\pi$ letters be?' Note we indeed mean $\pi = 3.1415...$. | ||

March 9, 1:45pm | Trang Nguyen | The Hitchin fibration |

Abstract: In this talk, we give an introduction to the theory of Higgs bundles and the Hitchin fibration. We discuss spectral covers, cameral covers, and how to describe the generic fibers of the Hitchin fibration. | ||

April 6, 1:45pm | Graham Manuell | A resolution of the Banach-Tarski paradox via pointfree measure theory |

Abstract: Pointfree topology is a generalisation of topology in which spaces are identified with their lattices of open sets. The resulting theory is better behaved in various ways. I will give a brief introduction to pointfree topology and describe the pointfree approach to measure theory. An advantage of this approach is that every subset of the reals can be assigned a reasonable measure and so, in particular, the Banach-Tarski paradox fails, even in the presence of choice. |

##### Semester 1

October 6, 1:45pm | Matt Booth | DGAs and A-infinity algebras |

Abstract: A differential graded algebra (dga) over a ring R is a chain complex of R-modules equipped with a multiplication. Examples include the tensor algebra or the endomorphism algebra of a dg-R-module, the de Rham complex of a smooth manifold, and any graded R-algebra. Over a field of characteristic 0, commutative dgas are the affine derived schemes. Unfortunately, dgas can behave badly with respect to homotopy: if R is a field, any chain complex C is quasi-isomorphic to its homology HC, and if C is a dga then so is HC, but they will generally not be quasi-isomorphic as dgas. A-infinity algebras extend dgas by providing a notion of 'dgas up to homotopy', and have better behaviour. In particular, Kadeishvili's Theorem says that if A is an A-infinity algebra, then HA admits an A-infinity algebra structure such that the natural map from A to HA is an A-infinity quasi-isomorphism. | ||

October 13, 1:45pm | Graham Manuell | Is the closed interval compact? |

Abstract: The closed real interval is one of the most familiar examples of a compact topological space. However, there are systems of mathematics in which it isn't compact at all. In particular, compactness fails in situations where every function is Turing-computable. We will explore the conditions under which compactness holds and fails to hold and consider a way to salvage compactness in the bad case. | ||

October 20, 1:45pm | Carlos Zapata-Carratala | Generalisations of Symplectic Geometry and Symplectic Reduction |

Abstract: In this talk I will present a collection of geometric structures that generalise symplectic structures (classical phase spaces) and we will discuss the notion of morphisms and reductions between them. The material will be presented from an introductory level and we will cover popular topics like Lie Algebroids, Poisson manifolds and discussions around the symplectic "category". | ||

October 27, 1:45pm | Ruth Reynolds | The Noetherianity of Idealizer Subrings |

Abstract: We define the idealizer of a right ideal to be the largest subring such that the right ideal becomes a two-sided ideal in this subring. As you can tell by this definition, idealizers are a purely non-commutative construction. In this talk we will introduce the idealizer and describe some interesting results about the noetherianity of these subrings. In particular, we will see how the noetherianity of idealizers played an essential role in the proof by Sierra and Walton that the universal enveloping algebra of the positive Witt algebra is not noetherian. | ||

November 3, 1:45pm | Jenny August | Cluster Algebras |

Abstract: Cluster algebras were developed as a tool to study a particular property of matrices but the interesting combinatorics contained in these algebras has led to connections to other parts of maths such as number theory and representation theory. In this talk, I will try to give a gentle introduction to these algebras and give examples of their applications in these different areas. | ||

November 10, 1:45pm | Juliet Cooke | Kauffman Bracket Skein Algebras |

Abstract: In this talk I will introduce a graphical calculus for working with Kauffman bracket Skein algebras of handlebodies, and use this calculus to compute the action of a loop on the Kauffman bracket Skein algebra of the 2-torus. | ||

November 17, 1:45pm | Simon Crawford | Polynomial Identity Rings and Azumaya Algebras |

Abstract: Polynomial identity (PI) rings are rings which are "close to commutative" in some appropriate sense. They are of interest to noncommutative ring theorists because a number of open conjectures which are true for commutative rings are also true for PI rings. I will give many examples of PI rings, and talk about how Azumaya algebras arise from PI rings once we impose a form of regularity on their prime ideals. I'll close the talk with a result of mine concerning Azumaya skew group rings, before working through an example. | ||

November 24, 1:45pm | Fatemeh Rezaee | From classical number theory to modern algebraic geometry : the story of the Weil conjectures |

Abstract: The Weil conjectures show the unity of mathematics and bridge some areas in mathematics like number theory, algebraic topology, algebraic geometry, arithmetic geometry, and so on. The story of the Weil conjectures started from Gauss and number theory . I will give a historical review of relevant conjectures which were given before AndrĂ© Weil, and one of them (the Riemann Hypothesis) is still open. Then I will give the statement of the Weil conjectures, and if time permits, mention the tools and ideas of the proof and give some examples to clarify the connection between arithmetic and topology obtained from the conjectures. |

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