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Abstract: Broadly speaking, geometric representation theory is a framework in which symmetries of geometric objects act on invariants of these objects such as cohomology theories and, more generally, derived categories associated to them. We then apply geometric machinery to study the structure of these invariants. Often the representation theoretic results obtained in this way are substantial and beyond the reach of purely algebraic methods. More specifically, in an algebro-geometric setting we can consider an algebraic group G with a subgroup H. The geometry of the space H\G/H gives rise to a number of interesting algebras and their representations, both classical and categorical. In this talk I will give several examples of this: 1) Historically, geometric representation theory was developed by Kazhdan, Lusztig, Kashiwara, Beilinson and Bernstein to prove Kazhdan-Lusztig conjectures.  Let G be a reductive  algebraic group e.g. GL(n) and let B be a Borel subgroup in G. I will discuss the Grotehndieck group of B x B-equivariant perverse sheaves on G, with the multiplication given by convolution, and its relation to Kazhdan-Lusztig theory for the finite Hecke algebra.  2) In the geometric setting of 1) Kostant and Kumar considered the Grothendieck group of B x B-equivariant coherent sheaves on G. The convolution product gives rise to an algebra structure on the group called the degenerate affine Hecke algebra. I will explain the recent work of Harada, Landveber and Sjamaar which relates this algebra to Demazure operators and its categorical version due to Arkhipov and Kanstrup. 3) If time permits, I will also discuss the geometric affine Hecke category of of Bezrukavnikov, Riche, Ben-Zvi and Nadler and its natural place in the framework of the geometric Langlands correspondence. Notes from Talk

22nd April 2015 Speaker: Prof. Alessio Corti (Imperial) Title: Lattice polygons, mirror symmetry and classification problems in algebraic geometry. Abstract: I state some elementary questions in the combinatorics of lattice polygons and explain some answers by Kasprzyk and others. Then I sketch how these questions have far-reaching implications in mirror symmetry and classification problems in algebraic geometry. If time permits I speculate about possible higher dimensional generalisations. 6th May 2015 Speaker: Prof. Mark Girolami (Warwick) Title: Differential Geometric Markov Chain Monte Carlo Methods Abstract: Monte Carlo methods are the dominant approach to perform inference over increasingly sophisticated statistical models used to describe complex phenomena. This presents a major challenge as issues surrounding correct and efficient MCMC -based statistical inference over such models are of growing importance. This talk will argue that differential geometry provides the tools required to develop MCMC sampling methods suitable for challenging statistical models. By defining appropriate Riemannian metric tensors and corresponding Levi-Civita manifold connections MCMC methods based on Langevin diffusions across the model manifold are developed. Furthermore proposal mechanisms which follow geodesic flows across the manifold will be presented. The optimality of these methods in terms of mixing time shall be discussed and the strengths (and weaknesses) of such methods will be experimentally assessed on a range of statistical models will also be considered. This talk is based on work that was presented as a Discussion Paper to the Royal Statistical Society and it remains the most downloaded article from the journal website. Details here.

13 May 2015 Speaker: Prof. Bernard Schutz (Cardiff) Title: Data Science Challenges at Cardiff University Abstract: The new Cardiff Data Innovation Institute (DII) has been established to do research in data science, defined rather broadly. The DII is just starting to build up staff and seek collaborations. The spectrum of research at the University that involves the exploitation of large and/or complex data sets is remarkably wide. I will describe the mission and organisation of the DII and then highlight some of the areas where there are challenges that the DII could hope to address, either by itself or in the role of a “matchmaker” between data producers and existing fundamental data science researchers at the University. 24th June 2015 Speaker: Prof. Jesus De Loera (UC Davis) Title: Helly's theorem: A jewel of 20th century geometry and its new 21st century applications. Abstract: The classical theorem of Eduard Helly (1913) is a masterpiece of geometry. It states that if a finite family $\Gamma$ of convex sets in $R^n$ has the property that every $n+1$ of the sets have a non-empty intersection, then all the convex sets must intersect. This theorem has since found applications in many areas, most particularly  the study of solvability of systems of linear inequalities and the theory of optimization. My lecture will be accessible to undergraduate students, it will begin explaining the basics of convex geometry and  proceed with a selection of lovely applications of Helly's theorem. The last part of my talk will deal with some surprising new generalizations, my favorite one is our brand new version when the intersection(s) contain(s) a lattice point. It originated in the 1970's  work of Doignon, Bell, and Scarf (arising in Economics theory). Along the way I will mention the history of the subject. All new results are based on joint work with I. Aliev, R. Basset, Q. Louveaux, R. La Haye, D. Oliveros and E. Roldan-Pensado.