University of York Geometry, Analysis and Mathematical Physics seminar 2021-22

All are welcome to attend the seminar, which usually takes place at 2pm on Tuesdays in the Topos during term time.

Autumn Term 2021

Date	Speaker	Title
5 October
12 October
19 October
29 October at 2pm in the Dusa McDuff Room Note the date, location and time!	Derek Harland (Leeds)	Cyclic monopole chains and parabolic Higgs bundles
2 November
9 November	Corina Keller (Montpellier)	Generalized Character Varieties and Quantization via Factorization Homology
16 November	Sergey Cherkis (Arizona)	Bow Construction of Yang-Mills Instantons on Taub-NUT space
23 November	Lorenzo Foscolo (UCL)	Anti-self-dual instantons and codimension-1 collapse
30 November
7 December

Spring Term 2022

Date	Speaker	Title
18 January
1 February
15 February
1 March
15 March

Summer Term 2022

Date	Speaker	Title
19 April	Markus Upmeier (Aberdeen)	Cobordism categories with applications to enumerative invariants
26 April	Stuart Hall (Newcastle)	Kähler Quantisation and the geometry of molecular surfaces
3 May	John Parker (Durham)	Fenchel-Nielsen coordinates for SL(3,C) representations of surface groups
10 May	Lashi Bandara (Brunel)	Index theory and boundary value problems for general first-order elliptic differential operators

Previous years' seminars

2020-21
2019-20

Abstracts

29 October. Derek Harland (Leeds)
Title. Cyclic monopole chains and parabolic Higgs bundles
Abstract. Monopoles are solutions of a natural gauge theoretical equation on a three-manifold. Under favourable conditions, they are an integrable system and form hyperkaehler moduli spaces. Much of the work on monopoles to date has focussed on the simplest possible base manifold, namely \(\mathbb{R}^3\). In this talk we instead discuss monopoles on \(\mathbb{R}^3 \times S^1\), aka "monopole chains". By a result of Cherkis and Kapustin, these are naturally dual to Higgs bundles on a cylinder. A problem that has received a lot of attention in both the monopole and Higgs bundle literature is the construction of symmetric solutions; in this talk I will describe one such result, namely a classification of monopole chains invariant under actions of cyclic symmetries.

9 November. Corina Keller (Montpellier)
Title. Generalized Character Varieties and Quantization via Factorization Homology
Abstract. Factorization homology is a local-to-global invariant which "integrates" disk algebras in symmetric monoidal higher categories over manifolds. In this talk we will focus on a particular instance of factorization homology on surfaces where the input algebraic data is a braided monoidal category. If one takes the representation category of a quantum group as an input, it was shown by Ben-Zvi, Brochier and Jordan (BZBJ) that categorical factorization homology quantizes the category of quasi-coherent sheaves on the moduli space of G-local systems. In this talk I will present two applications of the factorization homology approach for quantizing (generalized) character varieties. In a first part I will explain how to compute categorical factorization homology on surfaces with principal D-bundles decorations, for D a finite group. The main example for us comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We will see that in this case factorization homology gives rise to a quantization of Out(G)-twisted G-character varieties (This is based on joint work with Lukas Müller). In a second part we will allow for surfaces with certain stratifications, namely marked points. It was shown by BZBJ that the algebraic data governing marked points are braided module categories and I will discuss an example related the theory of dynamical quantum groups.

16 November. Sergey Cherkis (Arizona)
Title. Bow Construction of Yang-Mills Instantons on Taub-NUT space
Abstract. We describe the relationship between bows and instantons. The bows give a complete construction of all instantons (with assumption of regular asymptotic holonomy). We illustrate how it can be used to study both instantons and their moduli spaces.

23 November. Lorenzo Foscolo (UCL)
Title. Anti-self-dual instantons and codimension-1 collapse
Abstract. We study the behaviour of anti-self-dual instantons on \(\mathbb{R}^3 \times S^1\) (also known as calorons) under codimension-1 collapse, i.e. when the circle factor shrinks to zero length. In this limit, the instanton equation reduces to the well-known Bogomolny equation of magnetic monopoles on \(\mathbb{R}^3\). However, inspired by work of Kraan and van Baal in the mathematical physics literature, we show how \(SU(2)\) instantons can be realised as superpositions of monopoles and "rotated monopoles" glued into a singular background abelian configuration consisting of Dirac monopoles of positive and negative charges. If time permits, I will also discuss potential applications of this construction to moduli spaces of instantons on more general ALF spaces.
This is joint work with Calum Ross.

19 April. Markus Upmeier (Aberdeen)
Title. Cobordism categories with applications to enumerative invariants
Abstract. The construction of enumerative invariants requires a good understanding of differential-topological properties of moduli spaces, which will be the focus of my talk. Important examples include instanton counting problems in gauge theory and algebraic geometry.
I will discuss a new approach to these index-theoretic questions based on cobordism categories, of which I shall construct a new categorical group representation. The main application produces canonical orientations for Donaldson-Thomas invariants for Calabi-Yau 4-folds and sheaves with \(c_2=0\). Finally, I will comment on how the general case can be solved using flag structures, a new concept that arises naturally from the point of view cobordism categories.

26 April. Stuart Hall (Newcastle)
Title. Kähler Quantisation and the geometry of molecular surfaces
Abstract. I will report on a joint 'applied geometry' project with colleagues in the school of Chemistry at Newcastle. A fruitful method for drug development and discovery is to find a compound with a known activity or effect and then search for molecules with a similar 'shape'. Advances in computing mean that it is feasible to screen enormous databases of potential compounds; to do this one needs to develop fast yet discriminating ways of describing a molecule's shape.
One such description comes from Kähler quantisation - a technique developed around 20 years ago to provide numerical approximations to Calabi-Yau and other Kähler-Einstein metrics. I'll explain the technique and how it can be applied to the shapes in question as well as give some idea how effective it is. If I have time I might touch on other related descriptors one could consider based on ideas from Riemannian geometry.
This is joint work with Daniel Cole, Thomas Murphy and Rachael Pirie.

3 May. John Parker (Durham)
Title. Fenchel-Nielsen coordinates for \(SL(3,\mathbb{C})\) representations of surface groups
Abstract. Teichmüller space is the space of marked hyperbolic structures on a surface of genus at least two. Fenchel and Nielsen gave this space beautiful geometrical coordinates. We can also view Teichmüller space as the space of conjugacy classes of representations of the fundamental group of the surface to \(SL(2,\mathbb{R})\). We can then interpret Fenchel-Nielsen coordinates in terms of traces and eigenvalues. In the last few years, there has been a programme studying the action of discrete subgroups of \(SL(3,\mathbb{C})\) on \(\mathbb{C}P^2\). In this talk, I will describe how to generalise Fenchel-Nielsen coordinated to \(SL(3,\mathbb{C})\) representations of surface groups. I will also explain how this includes as special cases the earlier generalisations of these coordinates to \(SL(2,\mathbb{C})\), \(SL(3,\mathbb{R})\) and \(SU(2,1)\) representations.
This material is joint work with my student Rodrigo Davila.

10 May. Lashi Bandara (Brunel)
Title. Index theory and boundary value problems for general first-order elliptic differential operators
Abstract. Connections between index theory and boundary value problems are an old topic, dating back to the seminal work of Atiyah-Patodi-Singer in the mid-70s where they proved the famed APS Index Theorem for Dirac-type operators. From relative index theory arising in the study of positive scalar curvature metrics to a rigorous understanding of the chiral anomaly for the electron in particle physics, this index theorem has been a central tool to many aspects of modern mathematics.

APS showed that local boundary conditions are topologically obstructed for index theory. Therefore, a central theme emerging from the work of APS is the significance of non-local boundary conditions for first-order elliptic differential operators. An important contribution from APS was to demonstrate how their crucial non-local boundary condition for the index theorem could be obtained by a spectral projection associated to a so-called adapted boundary operator. In their application, this was a self-adjoint first-order elliptic differential operator.

The work of APS generated tremendous amount of activity in the topic from the mid-70s onwards, culminating with the Bär-Ballmann framework in 2010. This is a comprehensive machine useful to study elliptic boundary value problems for first-order elliptic operators on measured manifolds with compact and smooth boundary. It also featured an alternative and conceptual reformulation of the famous relative index theorem from the point of view of boundary value problems. However, as with other generalisations, a fundamental assumption in their work was that an adapted boundary operator can always be chosen self-adjoint.

Many operators, including all Dirac-type operators, satisfy this requirement. In particular, this includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general first-order elliptic operators, with the quintessential example being the Rarita-Schwinger operator on 3/2-spinors. This operator has physical significance, arising in the study of the delta baryon, analogous to the way in which the Atiyah-Singer Dirac operator arises in the study of the electron. However, not only does the Rarita-Schwinger operator fail to be of Dirac-type, it can be shown that outside of trivial geometric situations, this operator can never admit a self-adjoint adapted boundary operator.

In this talk, I will present work with Bär where we extend the theory for first-order elliptic differential operators to full generality. That is, we make no assumptions on the spectral theory of the adapted boundary operator. The ellipticity of the original operator allows us to show that, modulo a lower order additive perturbation, the adapted boundary operator is in fact bi-sectorial. Identifying the spectral theory makes the problem tractable, although departure from self-adjointness significantly complicates the analysis. Therefore, we employ a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory, and maximal regularity to extend the Bär-Ballman framework to the fully general situation.

Last updated 23 April, 2022