## Spring 2016 Seminars and Abstracts

Seminars take place on Monday afternoons 15:00 - 16:00. Everyone is welcome.

### Talks

- January 25, (JSC 1.02),
**Man Wai Cheung (UC San Diego and Cambridge University),**Stratification of Quiver Grassmannians via Theta Bases. - February 3,
**Wednesday, 16:00**, (LT3), School Colloquium,**Michael Berry (University of Bristol),**Divergent Series: From Thomas Bayes' Bewilderment to Today's Resurgence Via The Rainbow - February 8, (JSC 1.03),
**Martina Balagovic (Newcastle University)**, Universal K-Matrices Via Quantum Symmetric Pairs. - February 15, (SCI 3.05),
**John Britnell (Imperial College London),**Classes of Commuting Matrices. - February 22, (JSC 1.03),
**Richard Elwes (University of Leeds),**Infinite Random Networks and (di-)(multi-)graphs. - February 29, (JSC 1.03),
**Gwyneth Harrison-Shermoen (University of Leeds),**Independence, Via Limits. - March 7,
**14:00**, (Queens 1.03),**Tanmay Inamdar (UEA)**and**Awatif Alhowaity (UEA)**, UEA Joint Pure and Applied PhD Student Presentations. - April 11,
**14:00**, (LT4),**Markus Linckelmann (City University London),**On Integrable - April 18, (ARTS 2.05),
**Sibylle Schroll (University of Leicester),**Special Algebras - April 25, (JSC 2.03),
**Jyun-Ao Lin (Academia Sinica, Taipei),**Spherical Hall Algebra of a Weighted Projective Curve. - May 9, (JSC 1.02),
**Jeffrey Adler (American University, USA),**Transferring Representations Between Finite Reductive Groups - May 16,
**14:00**, (SCI 0.31),**Carola-Bibiane Schoenlieb (University of Cambridge),**Joint Pure and Applied Seminar: A Generalised Model for Optimal Transport of Images Including Dissipation and Density Modulation - June 6, (EFRY 01.08),
**Igor Dolinka (University of Novi Sad, Serbia),**Representing Groups by Endomorphisms of the Random Graph

### Abstracts

**Man Wai Cheung: **Scattering diagrams were first developed by Kontsevich and Soibelman, later by Gross and Siebert, to solve problems in mirror symmetry. Later on, it is found that the diagrams encodes information about cluster translation. While the canonical basis for cluster algebra is still unclear, Gross-Hacking-Keel-Kontservich proposed one, the theta basis, for cluster algebra constructed from scattering diagram. And the construction is able to solve one of the major conjecture in cluster algebra. In the talk, we are going to describe this construction. Furthermore, we will discuss how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.

**Michael Berry: **Following the discovery by Bayes in 1747 that Stirling’s series for the factorial is divergent, the study of asymptotic series has today reached the stage of enabling summation of the divergent tails of many series with an accuracy far beyond that of the smallest term. Several of these advances sprang from developments of Airy’s theory of waves near optical caustics such as the rainbow. Key understandings by Euler, Stokes, Dingle and Écalle unify the different series corresponding to different parameter domains, culminating in the concept of resurgence: quantifying the way in which the low orders of such series reappear in the high orders.

**Martina Balagovic:** The construction of the universal R-matrix for quantum groups produces solutions of the Yang-Baxter equation on tensor products of representations of that quantum group. This gives an action of the braid group of type A, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category.

I will explain how the theory of quantum symmetric pairs allows an analogous construction of a universal K- matrix, which produces solutions of the reflection equation on tensor products of representations of that quantum group. This gives a representation of the braid group of type B, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category with a cylinder twist.

**John Britnell:** (Joint work with Mark Wildon.) Let C and D be similarity classes of matrices. We say that C and D commute if there exists an element of C which commutes with an element of D. The commuting relation on similarity classes is in fact best understood as a relation on type classes, a notion introduced (in the case of matrices over a finite field) in work of Steinberg and Green from the 1950s. I will explain this theory, and introduce an equivalent formulation of type classes which allows a sensible extension to matrices over arbitrary fields. I will also describe how the question of which classes of matrices commute reduces to the case of nilpotent classes over an extension field, and I will discuss what is known about this case.

**Richard Elwes:** Random graphs and networks have been studied by mathematicians at least since some famous work of Erdős-Rényi around 1959. Their model has been studied from two main angles. Firstly, people have analysed the resulting finite structures, looking at their degree distributions, giant components and such like. Secondly, a wonderful theorem establishes that there is a unique infinite structure which any Erdős-Rényi process approaches with probability one. This object is known variously as “the random graph”, “the Erdős-Rényi graph”, and “the Rado graph”.

Recently, network scientists hoping to model real-world structures such as the worldwide web have turned away from Erdős-Rényi processes and looked to other models. In particular, the “preferential attachment” paradigm of Barabasi and Albert in 1999 has been intensively studied in recent years. However, very little work has been done on the infinite limits of such processes. I will try to say something about this.

**Gwyneth Harrison-Shermoen**: The concept of an independence relation (a ternary relation among sets, satisfying several properties) generalises that of linear independence in vector spaces and algebraic independence in fields, and gives us a way to determine what behaviour is "generic" in a structure. Certain model-theoretically tame mathematical structures are known to have independence relations, while others are known not to have any such relations satisfying certain (extra) properties. In this talk, I will consider structures that might not be so very tame, but can be viewed as the limit of their more well-behaved substructures. I will describe a method of lifting the independence relations of the tame substructures to a relation on the larger structure. While some properties might be lost along the way, this gives us a relatively nice notion of independence in the large structure. As a bonus, I may explain what I mean by "tame," "well-behaved," or "nice" in some of these contexts.

**Awatif Alhowaity:** "Solidification Caused by Under-Cooling". Many crude oils contain dissolved waxes that can precipitate out of solution and become deposited on the internal walls oil pipes. The waxy oils are transported through very long pipelines from warm walls to cooler conditions in the pipe. An important phenomenon occurring during the under-cooling of the pipeline is the formation of solid matter inside the pipe. The wax deposition is one of the most serious problems, potentially restricting flow and plugging the pipe. However, the wax deposits begin to form when the temperature is below the wax appearance temperature (WAT). We model the particle’s growth in the oil pipe once the temperature falls below the WAT. We determine the temperature distribution, formulate and solve the self-similar problem of wax particle growth from a single point. A numerical method is used to compute the solution of initial value problem of diffusion and transport of wax towards the particle. The numerical solution is compared to the self-similar solution.

**Tanmay Inamdar:** "Measures and Slaloms". Any compact subset of the reals has the following properties:.

1) It is compact,

2) It has a linear order which generates its topology,

3) It has a countable dense set, i.e., it is separable.

A classical result of G. Cantor implies that these properties in fact characterise compact subsets of the reals. This led M. Suslin to ask if separability above can be somewhat weakened to the property of having the countable chain condition, i.e., that every disjoint family of open sets is countable. This question is now known to be independent of ZFC.

General topologists have, however, tried to strengthen Suslin's hypothesis in various ways in order to obtain another characterisation of the compact subsets of the reals. A promising candidate until recently consisted of a strengthening of 'has a linear order' with 'does not map continuously onto an uncountable power of [0,1]', until S. Todorcevic constructed a counterexample in ZFC. We consider a further strengthening of this statement, where 'ccc' is replaced by 'supports a measure'. We show that under an extra axiom, Todorcevic's space can be constructed so as to have a measure, whereas under another (necessarily contradictory) extra axiom, no such space can carry a measure. We also show, without assuming any extra axiom, that such a space can be constructed so as to not support a measure. Various other non-separable compactifications of the natural numbers are constructed.

All of this proceeds by an analysis of the effect of forcing with a measure algebra on certain families of subsets of the natural numbers known as 'slaloms' which were used by Todorcevic in his construction. Joint work with Piotr Borodulin-Nadzieja (Wroclaw).

**Markus Linckelmann:** Derivations on an algebra over a commutative ring $k$ are $k$ -linear endomorphisms satisfying the product rule of differentiation. Integrable derivations arise from automorphisms of an extension of the algebra. This notion, due to Gerstenhaber, plays an important role in the context of deformations of algebras. We consider integrable derivations in the context of modular representations of finite groups. We investigate structural connections between block algebras of finite groups, their local structure, their character theory, and the subgroups of the first Hochschild cohomology groups determined by integrable derivations. This is motivated by the question to which extent the local structure of a block algebra is invariant under Morita equivalences or stable equivalences of Morita type.

**Sibylle Schroll**: We will start by motivating the study of special algebras by considering finite dimensional algebras defined by quiver and relations. The class of special algebras contains many well-studied examples of algebras such as special biserial algebras which in turn contain many of the tame algebras connected to modular representation theory of finite groups such as many tame blocks of finite groups and all tame blocks of Hecke algebras. However, unlike special biserial algebras, special (multiserial) algebras are in general of wild representation type. We will show that nevertheless there is some control over their representation theory.

**Jyun-Ao Lin: **In this talk, we will introduce the Hall algebra (and its spherical subalgebra) associated to the category of (parabolic) coherent sheaves on a smooth projective curve defined over a finite field and describe their structures in terms of shuffle algebras. We will then provide several applications in the context of quantum groups as well as on counting the Poincare polynomials of the moduli spaces of the stable (parabolic) Higgs bundles. If the time allows, we will discuss the position that Hall algebras play in the geometric Langlands program.** **

**Jeffrey Adler: **Suppose $\stackrel{~}{G}$ is a connected reductive group over a finite field $k$ , and $\mathrm{\Gamma}$ is a finite group acting on $\stackrel{~}{G}$ , preserving a Borel-torus pair. Then the connected part $G$ of the group of $\mathrm{\Gamma}$ -fixed points of $\stackrel{~}{G}$ is reductive, and there is a natural map from (packets of) representations of $G(k)$ to those of $\stackrel{~}{G}(k)$ . I will discuss this map, its motivation in the study of $p$ -adic base change, prospects for refining it, and a generalization: the pair of groups $(\stackrel{~}{G},G)$ must satisfy some axioms, but $G$ need not be a fixed-point subgroup of $\stackrel{~}{G}$ , nor even a subgroup at all.

**Carola-Bibiane Schoenlieb:** In this talk I will present a new model in which the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouv'e and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects. This is joint work with Jan Maas, Martin Rumpf and Stefan Simon.

**Igor Dolinka**: I will begin by introducing the so-called *random graph R*, a prime example of a countable ultrahomogeneous first-order structure. This will lead us to the outline of a beautiful subject in model theory, the *Fraïssè theory*, centred around the notion of an amalgamation class of structures.

The other basic ingredient of the talk will be semigroups, especially in the light of their links to groups via maximal subgroups of a semigroup and ‘hidden’ group structures called Schützenberger groups (of a D-class). Hence, I will provide a gentle crash-course (excuse the pun) into the basics of semigroup theory. These fundamentals will be then applied to endomorphism monoids of structures, naturally leading to the concept of an algebraically closed structure.

Our main results establish links between countable algebraically closed graphs and the endomorphisms of the random graph *R*. In particular, we show that, for any countable graph Γ there are uncountable many maximal subgroups of the endomorphism monoid of *R* isomorphic to the automorphism group of Γ. Further structural information about End(*R*) is established including that Aut(Γ) arises in uncountably many ways as a Schützenberger group. Similar results hold for the countable universal directed graph and the countable universal bipartite graph.

The presented original results are obtained jointly with R.D.Gray (UEA, Norwich), J.D.McPhee, J.D.Mitchell and M.Quick (St Andrews).

## Autumn 2015 Seminars and Abstracts

### Seminars take place on Monday afternoons 15:00 - 16:00. Everyone is welcome.

### Talks

- September 2, (SCI 3.05),
**Liron Speyer (UEA)**, The graded representation theory of the symmetric group and dominated homomorphisms. - October 5, (JSC 3.02),
**Oliver King (Leeds),**Blobs, Paths and Tiles. - October 12, (SCI 0.31),
**David Stewart (Cambridge University),**On the Jacobson-Morozov Theorem for Lie Algebras. - October 19, (SCI 0.31),
**David Pauksztello (University of Manchester),**The Combinatorics of Silting Mutation - October 26, (SCI 0.31),
**Emily Cliff (University of Oxford),**Factorisation Spaces in Algebraic Geometry, and Examples from Hilbert Schemes of Points. - November 9, (ARTS 2.01),
**Ilke Canakci (Durham University),**Ptolemy Relations, Cluster Algebras and Their Application to Gentle Algebras. - November 16, (EFRY 1.34),
**Robert Laugwitz (UEA)**, Categorical Modules over the Monoidal Centre. - November 23, (C.Hall 01.12),
**David Evans (Imperial College London),**Extremely Amenable Subgroups of Infinite Permutation Groups. - November 30, (SCI 0.31),
**Chris Bowman (City University, London),**The Co- Pieri Rule for Kronecker Coefficients - December 7, (SCI 3.05),
**Peter Latham (UEA),**The Unicity of Types for SL(N).

### Abstracts

**Liron Speyer**: I will give a brief overview of the cyclotomic KLR algebra and its use in studying the (graded) representation theory of the symmetric group. I will then talk about dominated homomorphisms between Specht modules and give examples to illustrate why they are so useful. If there is time at the end, I will briefly discuss a generalisation of the results and their application to Ariki-Koike algebras. This is joint work with Matthew Fayers.

**Oliver King**: The symplectic blob algebra is a quotient of the 2-boundary Temperley-Lieb algebra, which is itself an extension of the Temperley-Lieb algebra. In this talk I will define these objects, construct their cell modules and compute the corresponding Gram determinants. This will involve a discussion on finding good parameterisations and a path basis that lends itself well to tackling this problem. This is joint work with Paul Martin and Alison Parker.

**David Stewart: **The original Jacobson-Morozov theorem states that any nilpotent element e of a simple complex Lie algebra \g can be embedded into an \sl_2 subalgebra of \g. Kostant showed that this could be done uniquely up to conjugacy by the centraliser G_e in the associated algebraic group. Much work has been done on extending these important results to the modular case, that is where \g=\Lie(G) for G a reductive algebraic group over an algebraically closed field k of characteristic p>0: Pommerening showed that under the mild restriction that p is a good prime, one can always find an \sl_2-subalgebra containing a given nilpotent element, but this may not be unique up to conjugacy.

If h=h(G) denotes the Coxeter number of G, then Springer and Steinberg proved that the uniqueness holds whenever p\geq 4h-1 and in his book, Carter establishes the result under the weaker condition p>3h-3; both proofs go essentially by an exponentiation argument. We establish precisely when there is a bijection [ sl_2-subalgebras } -> { nilpotent orbits }.

Roughly speaking one only needs p>rk(G).

**David Pauksztello**: In this talk I will discuss silting objects in a triangulated category, which are a generalisation of tilting objects. Whilst tilting objects do not necessarily admit a good mutation theory, passing to silting objects gives rise to a mutation theory analogous to that in cluster-tilting theory. In this talk, I will discuss how to build a regular CW complex from silting mutation, generalising the "silting exchange graph" and, time permitting, explaining its connection to the Bridgeland stability manifold. This will be a report on joint work with Nathan Broomhead (Bielefeld) and David Ploog (Hannover).

**Emily Cliff**: Following Beilinson and Drinfeld, we introduce the Ran space of a complex variety X, and define factorisation spaces and factorisation algebras over X. The archetypal factorisation space is the Beilinson-Drinfeld Grassmannian associated to a reductive group G, but it is only interesting in the case that X is a curve. We briefly recall (or perhaps introduce) the Hilbert scheme of points of the variety X, and show how we can use it to produce a factorisation space over X; in this way we obtain factorisation spaces and algebras over varieties of arbitrary dimension.

**Ilke Canakci:** The well-known Ptolemy relations relate the four sides and two diagonals of a quadrilateral. In the context of triangulated marked surfaces, the notion of 'generalized Ptolemy relations' give rise to identities in the corresponding cluster algebra. In joint work with Schroll, we interpret these relations in the cluster category associated to marked surfaces and use this result to address a long-standing problem for the module categories of gentle algebras. This talk does not require any background on cluster algebras or quiver representations and will lay emphasis on the key ideas involved.

**Robert Laugwitz:** The centre of a monoidal category is a fundamental construction in the field of Quantum Algebra and has applications for example in topological quantum field theory. In the case where the monoidal category is given by modules over a bialgebra, it can be described as modules over the Drinfeld (or quantum) double of the bialgebra.

In this talk I will describe how the monoidal centre naturally acts of a class of categories, which can be described using the Majid–Radford biproduct, or Majid's construction of the dual of a monoidal category with respect to a fiber functor.

**David Evans:** A topological group is extremely amenable if whenever it acts continuously on a compact space there is a fixed point. For infinite permutation groups (with the topology of pointwise convergence), work of Kechris, Pestov and Todorcevic gives a beautiful and surprising connection between extreme amenability and Ramsey classes of finite structures (as studied in work of Nesetril, Rodl and others). In this talk, I will give an overview of these results and some of their applications; I will also discuss recent work showing that certain model-theoretic constructions provide counterexamples to some conjectures in the area.

**Chris Bowman:** A central problem in algebraic combinatorics is to provide an algorithm for calculating the coefficients arising in the decomposition of a tensor product of two simple representations of the symmetric group. The coefficients in such a decomposition are known as the “Kronecker coefficients”; these coefficients include the Littlewood—Richardson coefficients as a special case. In this subcase, the solution to the problem takes the form of a tableaux counting algorithm known as the Littlewood—Richardson rule.

The ultimate goal in this area is to generalise the Littlewood—Richardson rule to the general case. We shall discuss recent work with Maud De Visscher and John Enyang in which this problem is solved for Kronecker coefficients labelled by “co-Pieri triples" of partitions.

**Peter Latham**: The local Langlands conjectures predict a parameterization of representations of the Galois groups of p-adic fields in terms of the irreducible representations of reductive groups over these fields. The latter class of representations admits some extremely explicit constructions via the theory of types; it's then natural to ask how this transfers to Galois representations. This then leads to the idea of the inertial Langlands correspondence, which in the case of general linear groups has found numerous applications around the Langlands programme. I'll explain the theory for GL(N), which relies on a technical property known as the "unicity of types", and how to transfer these results to SL(N) in order to obtain an analogue of the inertial correspondence.