## Spring 2019 Seminars and Abstracts Spring 2019 Seminars and Abstracts

Seminars take place on Monday afternoons 14:00–15:00, unless specified otherwise. Everyone is welcome.

### Talks

January 28, (LT4), Alison Parker (University of Leeds), Some central idempotents for the Brauer algebra

February 4, (LT4), Isobel Webster (University of Leeds), A lattice isomorphism theorem for cluster groups of type A

February 11, (SCI 0.31), Thomas Lanard (IMJ Paris Rive Gauche), On the l-blocks of p-adic groups

February 25, (Queens 1.03), Joel David Hamkins (University of Oxford), Must there be numbers we cannot describe or define? Definability in mathematics and the Math Tea argument

March 4, (Queens 1.03), Sven-Ake Wegner (Teeside University), The heart of the LB-spaces

March 11, (EFRY 01.08), Asaf Karagila (UEA), On the power of power sets and unions, and how choice makes sense of them

March 18, (EFRY 01.08) Omar Leon Sanchez (University of Manchester), On differentially large fields

March 25, (C.HALL 0.13), Jessica Fintzen (University of Cambridge and University of Michigan), Representations of p-adic groups

April 29, (EFRY 01.10), Christos Aravanis (University of Sheffield), Hopf monads on derived categories of sheaves and Hopf algebras

### Abstracts

Alison Parker (January 28):

The Brauer algebra is an important algebra in representation theory, partly because it includes the representation theory of the symmetric group as a special case.I will introduce this algebra and give some background as well as explain why it is so important. I will then describe a method for constructing central idempotents in the Brauer algebra that is more efficient and computational tractable than previous methods. This is joint work with Oliver King and Paul Martin.

Isobel Webster (February 4):

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers that are mutationally equivalent to oriented simply-laced Dynkin diagrams, the associated cluster groups are isomorphic to finite reflection groups and thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and we are interested in whether the cluster group presentations possess comparable properties.

I will define a cluster group associated to a cluster quiver and explain how the theory of cluster algebras forms the basis of research into cluster groups. As for Coxeter groups, we can consider parabolic subgroups of cluster groups. I will outline a proof which shows that, in the mutation-Dynkin type A case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups.

Thomas Lanard (February 11):

We will talk about the category of smooth representations of a p-adic group. Our main focus will be to decompose it into a product of subcategories. When the field of coefficients is $\mathbb{C}$, it is well known thanks to Bernstein decomposition theorem. But when we are over $\bar{\mathbb{Z}}_l$ it is more mysterious. We will see what can be done and make some links with the local Langlands correspondence.

Joel David Hamkins (February 25):

An old argument, heard perhaps at a good math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many real numbers, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable structures in mathematics, structures in which every object has a property that only it exhibits. A mathematical structure is Leibnizian, in contrast, if any pair of distinct objects in it exhibit different properties. Is there a Leibnizian structure with no definable elements? Must indiscernible elements in a mathematical structure be automorphic images of one another? We shall discuss many elementary yet interesting examples, eventually working up to the proof that every countable model of set theory has a pointwise definable extension, in which every mathematical object is definable.

Discussion and commentary can be made at: http://jdh.hamkins.org/must-there-be-number-we-cannot-define-norwich-february-2019/

Sven-Ake Wegner (March 4):

Let X be a Banach space and let Y be a linear subspace. If Y is closed in X then X/Y is a Banach space in the quotient norm. If Y is not closed then this wrong---even if Y is a Banach space in a norm stronger than those induced by X. The unpleasant fact, that there is no reasonable Banach space X/Y in this case, motivated Waelbroeck in 1982 to consider formal quotients of Banach spaces. Amazingly, in the same year, Beilinson, Bernstein, Deligne published their famous theory about hearts of t-structures on triangulated categories. It turnes out that in their terminology, and for the special case of Banach spaces, the heart is precisely the category of formal quotients considered by Waelbroeck. In the talk we show that a heart-like category can be defined also for categories that are even more non-abelian than the Banach spaces. The LB-spaces are a typical example from functional analysis in this context.

Asaf Karagila (March 11):

What is the power of a power set? It turns out that assuming the standard axioms of set theory, or ZFC, this is something which is undetermined. Cantor proved that it has to be larger, and later König proved an additional technical requirement. But other than this, the axioms will not decide anything specific. But what about upper bounds? We know that ZFC proves that for sets of certain size, their power set can only be so large. But without choice? It turns out that we cannot say much, even in relatively basic cases.

We will review the definitions and theorems necessary for the discussion in the talk, so no actual background in set theory is necessary.

Omar Leon Sanchez (March 18):

Recall that a field K is large if it is existentially closed in the field of  Laurent series K((t)). Examples of such fields are the complex, the real, and the p-adic numbers. This class of fields has been exploited significantly by F. Pop and others in inverse Galois-theoretic problems. In recent work with M. Tressl we introduced and explored a differential analogue of largeness, that we conveniently call “differentially large”. I will present some properties of such fields and characterise them using formal Laurent series and to even construct “natural” examples (which ultimately yield examples of DCFs and CODFs... acronyms that will be explained in the talk).

Jessica Fintzen (March 25):

In the 1990s Moy and Prasad revolutionized the representation theory of p-adic groups by showing how to use Bruhat-Tits theory to assign invariants to representations of p-adic groups. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results also beyond representation theory is an explicit construction of (types for) representations of p-adic groups.

In this talk I will introduce some of the basic objects and concepts, survey what is known about the construction of (the building blocks of) representations of p-adic groups and mention recent developments.

Christos Aravanis (April 29):

The notion of a Hopf monad on a monoidal category consists of a generalisation of the notion of a Hopf algebra in a braided monoidal category to a non braided setup. In this talk, we will discuss the construction of a Hopf monad on a category of orbits of the bounded derived category of coherent sheaves on a nonsingular complex projective variety. We will explain how this Hopf monad determines the Hopf algebra object in this category.

## Autumn 2018 Seminars and Abstracts Autumn 2018 Seminars and Abstracts

Seminars take place on Monday afternoons 14:00–15:00, unless specified otherwise. Everyone is welcome.

### Talks

September 14, (SCI 3.05), Fenjin Liu (Jiao Tong University and Chang'an University, Xi'an, P.R. China), The Spectrum of a Graph and its Walk Matrix

October 8, (LT4), Francesca Fedele (University of Newcastle), A (d+2)-Angulated Generalisation of a Theorem by Brüning

October 15, (EFRY 01.02), Jason Semeraro (University of Leicester), Weight Conjectures for Fusion Systems

October 22, (LT4), Amit Shah (University of Leeds), Cluster Categories and Partial Cluster-Tilted Algebras

October 29, (LT4), Amit Hazi (University of Leeds), Indecomposable Tilting Modules for the Blob Algebra

November 7, 1pm, (SCI 3.05), Laure Daviaud (University of Warwick), Tropical Algebra and Applications

November 12, (LT4), Jesper Grodel (University of Copenhagen; Visiting INI), Modular Representation Theory via Homotopy Theory

November 19, (ARTS 2.02), Robert Kurinczuk (Imperial College, London), Local Langlands in Families

November 26, (LT4), Sam Morley (UEA), Extensions of Uniform Algebras

December 3, (LT4), Peter Latham (Kings College London), Towards the Unicity of Types for Essentially Tame Cuspidals

### Abstracts

Fenjin Liu (September 14):

Francesca Fedele (October 8):

Let d be a ﬁxed positive integer, k an algebraically closed ﬁeld and Φ a ﬁnite dimensional k-algebra with gldim Φ ≤ d. When d = 1, then mod Φ is hereditary and it follows from Br¨uning’s result [1, theorem 1.1] that there is a bijection between wide subcategories of mod Φ and wide subcategories of the bounded derived category Db(mod Φ).

For d ≥ 2, assume that there is a d-cluster tilting subcategory F ⊆ mod Φ and consider F := add{ΣidF | i ∈ Z} as a subcategory of Db(mod Φ). The d-abelian category F plays the role of a higher mod Φ and the (d + 2)-angulated category F of its higher derived category. In this context, Br¨uning’s classic result generalises as follows.

Theorem. There is a bijection between functorially ﬁnite wide subcategories of F and functorially ﬁnite wide subcategories of F, sending a wide subcategory W of F to W.

For m and l positive integers such that (m − 1)/l = d/2, consider the C-algebra Φ = CAm/(radCAm)l from [2, section 4]. We use the above theorem to describe all the wide subcategories of F, where F is the unique d-cluster tilting subcategory of mod Φ.

Jasam Semeraro (October 15):

Many of the conjectures of current interest in the representation theory of finite groups in characteristic p are local-to-global statements, in that they predict consequences for the representations of a finite group G given data about the representations of the p-local subgroups of G. The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of Külshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples (S,F,alpha) consisting of a saturated fusion system F on a finite p-group S and a compatible family alpha.

Amit Shah (October 22):

In this talk, I will first try to indicate why some people care about cluster algebras and cluster categories. Then I will focus on a specific cluster category, namely the cluster category coming from a Dynkin graph of type A. Although the formal definition for a cluster category can look a bit intimidating, we will see that the category has a nice Auslander-Reiten quiver — pictorial description of the category — (at least for type A) and so can be easily understood. Hopefully there will be time to give the definition of a partial cluster-tilted algebra, give an idea of how I’ve been trying to study them.

Amit Hazi (October 29):

The blob algebra is an example of a diagram algebra, that is to say, an algebra whose generators and relations can be described pictorially.  In this talk I will describe the representation theory of this algebra and give a description of its indecomposable tilting modules.  This new result is related to KLR algebras, categorification, and (diagrammatic) Soergel bimodules.

Laure Daviaud (November 7):

Tropical algebra is the counterpart of linear algebra when the addition and multiplication operators are replaced by the minimum (or maximum) and addition operators. Tropical algebra is thus particularly suitable to deal with optimisation problems, such as public transport scheduling. In this talk, I will give an introduction to tropical algebra and present some applications, for example in computer science (with the analysis of running time of programs) and in biology (phylogenetic).

Jesper Grodel (November 12):

The modular representation theory of finite groups is a wild, and sometimes bewildering, place: Indecomposable representations may not be simple, and the indecomposable modules can be shown to be, in general, "unclassifiable", in a precise sense. In this harsh and unforgiving climate there are however islands of structure and beauty: One such is the class of so-called endo-trivial modules, modules M such that End(M) is isomorphic to a trivial module plus a projective module. These modules occur in many parts of representation theory as "almost 1-dimensional modules". I'll describe the quest to classify such modules spanning the last 40 years, starting from the work of Dade in the 70s and leading into the present, where methods from homotopy theory come to play a role.

Robert Kurinczuk (November 19):

The local Langlands correspondence provides a remarkable connection between the representation theory of Galois groups and general linear groups over p-adic fields.  For a prime \ell different to p, this correspondence was interpolated in \ell-adic families in the recent pioneering work of Emerton-Helm-Moss.  I will explain what all of this means, and then at the end describe current joint work with Jean-François Dat, David Helm, and Gil Moss on a conjectural generalisation to a quite general class of reductive p-adic groups.

Sam Morley (November 26):

The theory of algebraic extensions of Banach algebras is well established, particularly for commutative Banach algebras, and have been used to solve many problems. In his thesis, Cole constructed algebraic extensions of a certain uniform algebra to give a counterexample to the peak point conjecture. Cole’s method for extending uniform algebras ensures that certain properties of the original algebra are preserved by the extension. In this talk, we discuss the general theory of uniform algebra extensions and a certain class of uniform algebra extensions which generalise Cole’s construction.

Peter Latham (December 3):

Let R be a cuspidal representation of a semisimple, simply connected p-adic group G. It is possible that the restriction of R to a compact subgroup K of G contains a type: a component which can only be contained in R-isotypic representations of G. Due to work by Yu, Kim and Fintzen, under mild assumptions on p, given such an R one may always find a maximal compact K such that the restriction of R to K contains a type. Conjecturally, these are the only possible types; if so then we say that the unicity of types holds for R. I will present joint work with Monica Nevins (University of Ottawa), which reduces the unicity of types for essentially tame cuspidals to an open question regarding the representation theory of finite groups of Lie type. When the split rank of G is small, this question is often manageable; this allows us to obtain many new examples of groups for which the unicity of types holds.