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# University of York Geometry, Analysis and Mathematical Physics seminar 2019-20

## Autumn Term 2019

 Date Speaker Title 8 October 15 October 22 October 29 October 5 November Andrea Appel (Edinburgh) Quantisation via differential twists 12 November 19 November Yuuji Tanaka (Oxford) On two generalisations of Hitchin's equations in four dimensions from Theoretical Physics 26 November Andrew Dancer (Oxford) Symplectic duality and implosion 3 December Johan Martens (Edinburgh) A general approach to the Hitchin connection

## Spring Term 2020

 Date Speaker Title 14 January 21 January Alina Vdovina (Newcastle) $$C^*$$-algebras coming from cube complexes and buildings 28 January Nuno Romao (Augsburg) Pochhammer states on Riemann surfaces 4 February 11 February 18 February Matthew Young (MPIM Bonn) Unoriented Dijkgraaf-Witten theory 25 February Mathew Bullimore (Durham) Symplectic Vortices and Symplectic Duality 3 March Aron Wall (Cambridge) The Maximin Method: How to Prove Holographic Entropy Inequalities 10 March Samuel Borza (Durham) Measure Contraction Property of generalised Grushin planes

## Summer Term 2020

 Date Speaker Title 14 April 21 April 28 April Ben Davison (Edinburgh) Strong positivity for quantum cluster algebras 5 May 12 May Yue Fan (Maryland) Construction of the moduli space of Higgs bundles using analytic methods (video) 26 May Jan Swoboda (Heidelberg) Moduli spaces of parabolic Higgs bundles: their ends structure and asymptotic geometry (video) 2 June Victoria Hoskins (Nijmegen) On the motive of the moduli space of Higgs bundles 9 June 16 June 23 June Szilard Szabo (Budapest) Geometry of Hodge moduli spaces in the Painlevé cases 30 June

## Abstracts

5 November. Andrea Appel (Edinburgh)
Title. Quantisation via differential twists
Abstract. A universal Drinfeld twist is an inner automorphism which relate the coproduct of a quantum enveloping algebra to the standard coproduct of the undeformed enveloping algebra. In this talk I will outline the construction of a family of Drinfeld twists, obtained from the dynamical Knizhnick-Zamolodchikov equations of a Kac-Moody algebra by exploiting their singularity at infinity. This leads to a "transcendental" construction of quantum groups, gives a new direct proof of the Drinfeld-Kohno theorem, and produces in finite type a quantum dual exponential in the sense of Ginzburg-Weinstein.

19 November. Yuuji Tanaka (Oxford)
Title. On two generalisations of Hitchin's equations in four dimensions from Theoretical Physics
Abstract. This talk is about two sets of gauge-theoretic equations called Vafa-Witten and Kapustin-Witten ones, both of which have the origin in $$N=4$$ super Yang-Mills theory in Theoretical Physics. We present them as two kinds of generalisations of Hitchin's prominent equations on Riemann surfaces to ones in dimension four. We then discuss issues on how to solve gauge-theoretic moduli problems for them and convey recent progress on the problems from both analytic and algebraic aspects.

26 November. Andrew Dancer (Oxford)
Title. Symplectic duality and implosion
Abstract. Symplectic duality is a conjectured duality, arising from physics, between certain kinds of holomorphic-symplectic or hyperkahler manifolds. We discuss some of the issues around this duality and present candidates for duals of implosion spaces.
This is joint work with Frances Kirwan and Amihay Hanany.

3 December. Johan Martens (Edinburgh)
Title. A general approach to the Hitchin connection
Abstract. The Hitchin connection is a flat projective connection on bundles of non-abelian theta-functions over the moduli space of curves, originally introduced by Hitchin in a Kähler context. It can be understood through a heat operator, generalising the heat equation that (abelian) theta functions have classically been known to satisfy, and is also equivalent to the WZW/KZB connection in conformal field theory. We will describe a general construction of this connection that also works in (most) positive characteristics. A key ingredient is an alternative to the Narasimhan-Atiyah-Bott Kähler form on the moduli space of bundles on a curve. We will comment on the connection with some related topics, such as the Grothendieck-Katz p-curvature conjecture.
This is joint work with Baier, Bolognesi and Pauly.

21 January. Alina Vdovina (Newcastle)
Title. $$C^*$$-algebras coming from cube complexes and buildings
Abstract. We will give an elementary introduction to the theory of cube complexes and buildings. Then we show how explicit geometric structures help in studying higher-rank graph $$C^*$$-algebras and their K-theory.

28 January. Nuno Romao (Augsburg)
Title. Pochhammer states on Riemann surfaces
Abstract. For a compact Riemann surface of genus $$g>1$$, the $$L^2$$-Betti numbers (of universal or maximal abelian covers, say) concentrate in middle degree 1, with first $$L^2$$-Betti number $$2g-2$$. This reflects the fact that, on such surfaces (equipped with a fixed riemannian metric), nonzero harmonic $$p$$-forms valued in families of local systems, and of finite $$L^2$$-norm, only exist for $$p=1$$; in fact, there is an infinite-dimensional Hilbert space of such 1-forms, with finite von Neumann dimension $$2g-2$$. In my talk, I shall construct a complete basis for this space depending on the choice of a particular kind of pair-of-pants decomposition of the surface. We have baptised the elements of such a basis "Pochhammer states", since they localise to 1-cycles lifting Pochhammer curves immersed on each pair of pants, and moreover they span all the ground states (waveforms) in supersymmetric quantum mechanics incorporating Aharonov-Bohm phases. This construction provides useful geometric intuition on such a space of ground states.
This is joint work with Marcel Bökstedt.

18 February. Matthew Young (MPIM Bonn)
Title. Unoriented Dijkgraaf-Witten theory
Abstract. I will begin the talk with a basic introduction to unoriented topological field theory in low dimensions, explaining the importance of such theories and highlighting some open problems. I will then describe a new class of topological gauge theories, namely, unoriented twisted Dijkgraaf-Witten theory, and explain how computations in these theories can be understood in terms of a categorified real representation theory of finite groups.

25 February. Mathew Bullimore (Durham)
Title. Symplectic vortices and symplectic duality
Abstract. I will discuss moduli spaces of symplectic vortices on Riemann surfaces and associated enumerative problems arising in (supersymmetric) gauge theory. I will also explain how symplectic duality, a conjectural relationship between pairs of holomorphic symplectic manifolds, is reflected in highly non-trivial relationships between different enumerative invariants.

3 March. Aron Wall (Cambridge)
Title. The Maximin Method: How to Prove Holographic Entropy Inequalities
Abstract. There are a number of interesting inequalities satisfied by minimal-area surfaces in asymptotically hyperbolic geometries. In the static version of AdS/CFT, these surfaces are interpreted as the entropy of the corresponding boundary regions. This relates quantum information inequalities (e.g. Strong Subadditivity) to geometrical inequalities which can be given simple picture-proofs.
Since the dynamical version of AdS/CFT involves Lorentzian geometries, it is important to be able to prove the corresponding inequalities for asymptotically AdS geometries. In this case, we have to use extremal surfaces instead of globally minimal ones, making it more difficult to prove global inequalities. But by reformulating extremal surfaces in terms of an equivalent "maximin" variational principle, elegant proofs can be constructed. (Unlike the static case, these results require use of the null curvature condition.)

10 March. Samuel Borza (Durham)
Title. Measure Contraction Property of generalised Grushin planes
Abstract. Synthetic notions of curvature play an important role in the study of the singular spaces where a smooth curvature bound is not available. The Grushin plane, a sub-Riemannian manifold, is one of these. We will look at a generalisation of the standard Grushin plane and investigate its geometry. A careful analysis will help us to derive distortion coefficients of the geodesic flow. Appropriate estimates will then tell us whether these spaces satisfy a curvature-dimension property. As a consequence, we can establish new geometric and functional inequalities, such as a Brunn-Minkowski inequality

28 April. Ben Davison (Edinburgh)
Title. Strong positivity for quantum cluster algebras
Abstract. Quantum cluster algebras are quantizations of cluster algebras, which are a class of algebras interpolating between integrable systems and combinatorics. These algebras were originally introduced to study positivity phenomena arising in the study of quantum groups, and so one of the key questions regarding them (and their quantum analogues) is whether they admit a basis for which the structure constants are positive.
The classical version of this question was settled in the affirmative by Gross, Hacking, Keel and Kontsevich. I will present a proof of the quantum version of this positivity, due to joint work with Travis Mandel, based on results in categorified Donaldson-Thomas theory obtained in joint work with Sven Meinhardt.

12 May. Yue Fan (Maryland)
Title. Construction of the moduli space of Higgs bundles using analytic methods
Abstract. Introduced by Hitchin, a Higgs bundle $$(E,\Phi)$$ on a complex manifold $$X$$ is a holomorphic vector bundle $$E$$ together with an $$\mathrm{End} (E)$$-valued holomorphic 1-form $$\Phi$$. The moduli space of Higgs bundles was constructed by Nitsure where $$X$$ is a smooth projective curve and by Simpson where $$X$$ is a smooth projective variety. They both used Geometric Invariant Theory, and the moduli space is a quasi-projective variety. It is a folklore theorem that the Kuranishi slice method can be used to construct this moduli space as a complex space where $$X$$ is a closed Riemann surface. I will present a proof of this folklore theorem and show that the resulting complex space is biholomorphic to the one in the category of schemes. Moreover, I will briefly talk about some applications of this new construction.

26 May. Jan Swoboda (Heidelberg)
Title. Moduli spaces of parabolic Higgs bundles: their ends structure and asymptotic geometry
Abstract. Moduli spaces of Higgs bundles are mathematical objects which are of interest from various points of view: as holomorphic objects, generalizing the concept of holomorphic structures on vector bundles, as topological objects, relating to surface group representations, and as analytic objects which admit a description through a nonlinear PDE.
In my talk, I will mostly take up this latter point of view and give an introduction to Higgs bundles on Riemann surfaces both in the smooth and the parabolic setting. In the parabolic case, i.e. in the situation where the Higgs bundles are permitted to have poles in a discrete set of points, I will discuss recent joint work with L. Fredrickson, R. Mazzeo and H. Weiß concerning the asymptotic geometric structure of their moduli spaces. Here the focus lies on the hyperkähler metric these spaces are naturally equipped with. One implication of a recent conjectural picture due to Gaiotto-Moore-Neitzke suggests that this metric is asymptotic to a so-called semiflat model metric which comes from the description of the moduli space as a completely integrable system.
We shall also discuss several open questions in the case where the Riemann surface is a four-punctured sphere and these moduli spaces turn out to be gravitational instantons of type ALG.

2 June. Victoria Hoskins (Nijmegen)
Title. On the motive of the moduli space of Higgs bundles
Abstract. In joint work with Simon Pepin Lehalleur, we study the motive of the moduli space of Higgs bundles on a curve in Voevodsky's category of motives. After introducing moduli spaces of Higgs bundles and highlighting some key geometric properties we will need, I will give a brief introduction to motives, which were envisaged by Grothendieck as providing universal cohomological invariants. I will then present two results concerning the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve. First we give a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic. Second, when working with rational coefficients, we show that the motive of the Higgs moduli space is a direct factor of the motive of a sufficiently large power of the curve.

23 June. Szilard Szabo (Budapest)
Title. Geometry of Hodge moduli spaces in the Painlevé cases
Abstract. We will describe the Geometry and Topology of the real 4-dimensional Hodge moduli spaces associated to the Painlevé cases. We will analyze the possible configurations of singular fibers of the Hitchin fibration (joint work with P. Ivanics and A. Stipsicz). We will give a smooth compactification of the associated character varieties, and prove the Geometric P = W conjecture for these spaces (joint work with A. Némethi).

Last updated 27 May, 2020