# Seminars 2016-2017

## 2016-2017 Leicester Pure Mathematics Seminar (Semester 2)

**31/01/2017** Jelena Grbic (Southampton) "*Homotopy theory of toric objects*"

**Abstract:** At the beginning of this millennium, Toric Topology has been recognised as a new branch of Topology closely related to Algebraic Geometry, Combinatorics and Algebra. Initially problems of Toric Topology were motivated by the study of toric geometry. The approach I take departs from geometry and brings in the tools and techniques of homotopy theory. That allows one to generalise the fundamental concepts of Toric Topology to new ones which will further have applications to geometric group theory, robotics and applied mathematics.

**07/02/2017** Melanie Rupflin (Oxford) "*Minimal surfaces and geometric flows*"

**Abstract: **The classical Plateau problems has been one of the most influential problems in the development of modern analysis. Posed initially by Lagrange, it asks whether a closed curve in Euclidean space always spans a surfaces with minimal possible area, a question that was answered positively by Douglas and Rado around 1930. In this talk I want to consider some aspects of the classical Plateau Problem and its generalisations and discuss furthermore how one can "flow" to such minimal surfaces by following a suitably defined gradient flow of the Dirichlet energy, i.e. of the integral of gradient squared.

**14/02/2017** Muneerah Saad Al Nuwairan (King Faisal University) *"The additivity problem"*

**Abstract: **In our talk, we present the well- known problem in the field of quantum information theory known as “The additivity problem”. It concerns transferring classical information using quantum channels. We will build a mathematical model for the problem, and show that the field of Representation theory provides rich source for solutions. We will also introduce EPOSIC channels, a class of SU(2)-covariant quantum channels that form the extreme points of all SU(2)-irreducibly covariant channels. These channels provide us with a potential solution for the problem.

**22/02/2017** Sebastian del Baño (Queen Mary London) *"Gaussian distributions on affine spaces"*

**Abstract:** We present intrinsic variants of some classical results on real and p-adic Gaussian distributions. Results such as the Isserles-Wick formula for higher moments are considerably simplified by using an intrinsic tensorial approach.

**28/02/2017** Ilke Canakci (Durham) "*Infinite rank surface cluster algebras*"

**Abstract:** Cluster algebras were introduced by Fomin and Zelevinsky in the context of Lie theory in 2002, however they received much attention since many applications in diverse areas of mathematics have been discovered including quiver representations, Teichmuller theory, integrable systems and string theory. An important class of cluster algebras, called surface cluster algebras, were introduced by Fomin, Shapiro and Thurston by associating cluster algebra structure to triangulated marked surfaces. I will review the state of the art in research associated to surface cluster algebras and report on joint work with Anna Felikson where we introduce a generalisation of surface cluster algebras to infinite rank by associating cluster algebras to surfaces with finitely many accumulation points of boundary marked points.

**07/03/2017** Nathan Broomhead (Plymouth) "*Thick subcategories, arc-collections and mutation*"

**Abstract:** I will explain some work, in which I describe the lattices of thick subcategories of discrete derived categories. This is done using certain collections of exceptional and sphere-like objects related to non-crossing configurations of arcs in a geometric model.

**14/03/2017** Tom Bridgeland (Sheffield) "*Wall-crossing and Riemann-Hilbert problems"*

**Abstract:** The subject of the talk will be wall-crossing phenomena for Donaldson-Thomas invariants, but I will only discuss the simplest examples, which are completely concrete. In the first half I will recall the representation theory of the A2 quiver, and explain how the Kontsevich-Soibelman wall-crossing formula works in this case. In the second half I will discuss a natural Riemann-Hilbert problem suggested by the form of the wall-crossing formula, and show how to solve it in the case of the A1 quiver.

**21/03/2017** Justin Lynd (Aberdeen) "*Fusion systems and classifying spaces"*

**Abstract:** The Martino-Priddy conjecture asserts that two finite groups have equivalent p-completed classifying spaces if and only if their associated fusion systems at the prime p are isomorphic. This was first proved by Bob Oliver in 2004 and 2006. Andrew Chermak generalized this in 2013 by showing that each saturated fusion system over a finite p-group has a unique classifying space attached to it. The proofs of these results depend on the classification of the finite simple groups (CFSG). The focus for this talk will be on the group theoretic aspects of joint work with George Glauberman that helped us remove the dependence of these results on the CFSG. These results concern the question: given a finite group G acting on a finite abelian p-group, when can one find a p-local subgroup H (i.e., a normalizer of a nonidentity p-subgroup) having the same fixed points on the module as does G itself?

**02/05/2017** Joao Faria Martins (Leeds) "*The fundamental crossed module of the complement of a knotted surface in the 4-sphere"*

**Abstract:** After a review on homotopy 2-types and crossed modules, I will show a method to calculate the homotopy 2-type of the complement of a knotted surface in the 4-sphere given a movie presentation of it. We therefore generalise Wirtinger relations for the fundamental group of a knot complement.

**09/05/2017** Sarah Zerbes (UCL) "*Elliptic curves and the conjecture of Birch--Swinnerton-Dyer"*

**Abstract:** An important problem in number theory is to understand the rational solutions to algebraic equations. One of the first non-trivial examples, cubics in two variables, leads to the theory of so-called elliptic curves. The famous Birch—Swinnerton-Dyer conjecture, one of the Clay Millennium Problems, predicts a relation between the rational points on an elliptic curve and a certain complex-analytic function, the L-function on an elliptic curve. In my talk, I will give an overview of the conjecture and of some new results establishing the conjecture in certain cases.

**22/05/2017** Yann Palu (Picardie) "*Non-kissing complex and tau-tilting over gentle algebras"*

**Abstract:** Gentle algebras form a class of algebras, described in terms of quivers and relations, whose representations are well understood and can be described combinatorially. In this talk, I will introduce a combinatorial notion, called "non-kissing". I will then explain how the non-kissing condition can be interpreted in terms of the representation theory of gentle algebras

## 2016-2017 Leicester Pure Mathematics Seminar (Semester 1)

**04/10/2016** Giuseppe Tinaglia (King's College, London) *"The geometry of constant mean curvature surfaces in Euclidean space"*

**Abstract:** In this talk I will begin by reviewing classical geometric properties of constant mean curvature surfaces, H>0, in R^3. I will then talk about several more recent results for surfaces embedded in R^3 with constant mean curvature, such as curvature and radius estimates for simply-connected surfaces embedded in R^3 with constant mean curvature. Finally I will show applications of such estimates including a characterisation of the round sphere as the only simply-connected surface embedded in R^3 with constant mean curvature and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Meeks.

**11/10/2016** Artie Prendergast-Smith (Loughborough) "*Cones of positive classes"*

**Abstract:** I will explain why convex cones of "positive" cohomology classes arise naturally in algebraic geometry, why they are important, and some things we know about them.

**18/10/2016** Simon Willerton (Sheffield) "*The magnitude of odd balls (From category theory to potential theory)*"

**Abstract:** Tom Leinster defined the notion of the magnitude of a finite metric space, which is some notion of size, using his idea of Euler characteristic of a finite category. You can extend this notion to nice infinite metric spaces such as subsets of Euclidean space. Tom and I conjectured a formula for the magnitude of a convex body involving classical invariants such as volume and surface area, but it turned out to be difficult to calculate precisely for non-trivial bodies. I will explain all of this and how Tony Carbery and Juan-Antonio Barceló recently used potential theory to calculate the magnitude of odd dimensional balls.

**20/10/2016** Neil Ghani (Strathclyde) "*Compositional Game Theory*"

**Abstract:** The central concept of economic game theory is that of Nash Equilibria consisting of a collection of strategies for each player in a game such that no player has a reason to change their strategy assuming all players keep their strategies the same. This talk reports on a new categorical foundation for economic game theory - compositional game theory (CGT). At its core, CGT involves a new representation of games where large games are built from smaller subcomponents of the game. However, while natural from a categorical perspective, developing CGT is no simple matter, eg all current models of game theory are inherently non-compositional! More fundamentally, not all reasoning can be put in the compositional form, especially if there is significant emergent behaviour present in a system which is not present in its subsystems. And, this is certainly the norm in game theory, eg an optimal strategy for a game may not remain optimal when that game is part of a larger network of games.

**8/11/2016** Bob Coecke (Oxford) "*From quantum foundations to natural language meaning via diagrams*"

**Abstract:** This talk concerns how mathematical structures immersing in one discipline can be relevant in an entirely different discipline. The conceptual focus is on structures that enable one to describe interaction. Earlier work on an entirely diagrammatic formulation of quantum theory, which is soon to appear in the form of a textbook [1], has somewhat surprisingly guided us towards an answer for the following question [2, 3]: how do we produce the meaning of a sentence given that we understand the meaning of its words? This work has practical applications in the area of natural language processing, and the resulting tools have meanwhile outperformed existing methods. Recent developments involve the use of several more concepts from quantum theory, for example, density matrices for modelling ambiguity [4] and lexical entailment [5], and convex state spaces to model interaction of psychological concepts [6].

NB: this talk doesn’t require any background in quantum theory, nor in linguistics, nor in category theory

[1] B. Coecke & A. Kissinger (2016, 850 pages) Picturing Quantum Processes. A first course on quantum theory and diagrammatic reasoning. Cambridge University Press.

[2] B. Coecke, M. Sadrzadeh & S. Clark (2010) Mathematical foundations for a compositional distributional model of meaning. arXiv:1003.4394

[3] S. Clark, S., B. Coecke, E. Grefenstette, S. Pulman & M. Sadrzadeh (2013) A quantum teleportation inspired algorithm produces sentence meaning from word meaning and grammatical structure. arXiv:1305.0556.

[4] R. Piedeleu, D. Kartsaklis, B. Coecke & M. Sadrzadeh (2015) Open System Categorical Quantum Semantics in Natural Language Processing. arXiv:1502.00831

[5] D. Bankova, B. Coecke, M. Lewis & D. Marsden (2015): Graded Entailment for Compositional Distributional Semantics. arXiv:1601.04908

[6] J. Bolt, B. Coecke, F. Genovese, M. Lewis, D. Marsden & R. Piedeleu (2016) Interacting Conceptual Spaces. SLPCS. arXiv:1608.01402

**15/11/2016** Giuseppe Tinaglia (King's College, London) "*The geometry of constant mean curvature surfaces in Euclidean space*"

(This talk was postponed from 4 October, see above)

**22/11/2016** Brita Nucinkis (Royal Holloway) "*Classifying spaces for families and their finiteness conditions*"

**Abstract:** I will give a survey on cohomological finiteness conditions for classifying spaces for families of subgroups, such as the dimension or the type and will discuss some old and new questions.

**29/11/2016** Markus Linckelmann (City) "*When do derivations on an algebra yield a simple Lie algebra?*"

**Abstract:** A derivation on an algebra A is a linear endomorphism which satisfies the product rule. The quotient of the space of derivations by the subspace of inner derivations is the first Hochschild cohomology of the algebra A. This is a Lie algebra - and if A is defined over a field of prime characteristic p, then this is a p-restricted Lie algebra. Motivated by questions arising in modular representation theory, we investigate connections between the algebra structure of A and the Lie algebra structure of its first Hochschild cohomology. What are the implications for A if its first Hochschild cohomology is a simple Lie algebra? In joint work with Lleonard Rubio y Degrassi, we answer this question for block algebras of finite groups with one simple module.

**13/12/2016** Peter Jorgensen (Newcastle) "*Thick subcategories of d-abelian categories*" (report on joint work with Martin Herschend and Laertis Vaso)

**Abstract:**Let d be a positive integer. The notion of d-abelian categories was introduced by Jasso. Such a category does not have kernels and cokernels, but rather d-kernels and d-cokernels which are longer complexes with weaker universal properties. Canonical examples of d-abelian categories are d-cluster tilting subcategories of abelian categories. We introduce the notion of thick subcategories of d-abelian categories and show a classification of the thick subcategories of a family of d-abelian categories associated to quivers of type A_n. If time permits, we show how thick subcategories are in bijective correspondence with a particularly nice class of algebra epimorphisms. This generalises a classic result by Geigle and Lenzing.