The Hodge Club is the seminar for Hodge Institute graduate students and postdocs. That means we're interested in Algebra, Geometry, Topology, Number Theory, and all possible combinations and derivations of the four. Before the 2016/17 academic year, the Hodge Club was known as the Geometry Club.
We meet every Friday at 1:45pm in JCMB 4312 where we take it in turn to present a topic of interest to the rest of the group. Talks tend to be fairly informal and provide excellent practice for conference talks in front of a friendly audience. You can find our current schedule and a historical list of talks below.
Future events are circulated on our mailing list and advertised on the Graduate School calendar. See instructions below on how to join our mailing list.
Current schedule of talks for 2017/18:
|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 André 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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