# Seminars this semester

Series:

Oct 4 Wed Eoin Murphy ShEAF: postgraduate pure maths seminar
16:00 Hicks Room J11 t = 1 Limits of Hall Algebras and Quantum Groups

Abstract:
In this talk I hope to discuss how two isomorphic algebras degenerate on setting a parameter t=1. One of these is the quantum group Q associated to a simple Lie algebra. The other is a Hall algebra H of a certain category of complexes of quiver representations. The interesting thing is that for either algebra there are in fact different "ways to set t=1" resulting in different degenerate algebras. On the one hand one gets the universal enveloping algebra of a Lie algebra, on the other a Poisson algebra of functions on a Poisson-Lie group. We describe the t=1 theory of H and explain how it is related to that of Q. The story involves work for my PhD thesis and builds on results due to Ringel, Bridgeland, Deng, Chen and others.

Oct 11 Wed Jordan Williamson (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Morita Theory in Stable Homotopy Theory

Abstract:
Morita theory was developed in the 1950s as a tool for studying rings by studying their categories of modules. Since then, reincarnations of Morita theory for abelian categories, derived categories and stable model categories have been developed. We will outline the classical version of Morita theory, the extension to the world of stable homotopy theory, and then use this extension to show how this result can be powerful in the search for algebraic models of spectra.

Oct 18 Wed Angelo Rendina (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Ramanujan sums, the Casimir effect and the Riemann zeta function

Abstract:
In 1913 Ramanujan claimed in a letter to Hardy that $1+2+3+4+...=-1/12$, proving it with elementary methods. We also find examples of such divergent series appearing in some quantum physics phenomena, e.g. the Casimir effect, where a suitable renormalization allows to deal with converge problems; again, we find the same value of $-1/12$. The theory of analytic functions and meromorphic continuation makes sense of this absurd value: in particular, we will see how to extend the generalized harmonic series to the whole complex plane and find its functional equation.

Oct 25 Wed Giovanni Marchetti (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Perverse dimensions

Abstract:
Exactly 100 years ago, Felix Hausdorff came up with the original idea that some geometrical objects might have non-integer dimension. However, very few steps have been made in homological algebra to pursue this idea. We discuss how the formalism of Bridgeland's stability can be exploited to build homology objects indexed by sets more general than the integers. Finally, we borrow an example from the theory of perverse sheaves to show that dimensions could be even worse behaved: they could be uncomparable.

Nov 1 Wed Malte Heuer (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Generalised complex geometry

Abstract:
Generalised complex structures were introduced by Nigel Hitchin in 2003 and further developed by his student Marco Gualtieri. They give a unification of complex and symplectic geometry. We will see in which way these two seemingly very different structures can be thought of as extremal cases of generalised complex structures. The definition was motivated by phenomena in string theory, especially mirror symmetry where there is a link between symplectic and complex geometry.

Nov 15 Wed Daniel Graves (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks A Simplicial View of Algebraic Topology

Abstract:
Simplicial sets give a nice combinatorial model for topological spaces. In this talk I will introduce the foundations of the theory of simplicial sets and, time permitting, give some idea of how I use them in my own work

Nov 22 Wed Neil Hansford (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks A Whistle Stop Tour of C*-categories

Abstract:
C*-categories are categories in the usual 'objects and arrows' sense, equipped with additional structure comparable with the more analytic C*-algebras. Familiar concepts from either side of the fence include norms, completeness, involutions, the C*-identity, morphism sets, (C*-) functors, and much more. We will have a brief overview of C*-categories, taking in their abstract definition, some specific examples, functors, representations, ideals and quotients, as well as the generalisation of an important construction from the world of C*-algebras.

Nov 29 Wed Rudolf Chow (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks A Tale of Two Sieves

Abstract:
Over 300 years ago, the French mathematician Mersenne conjectured that $2^{251} − 1$ was a composite number. This was finally proved 120 years ago, but even 50 years ago the computational load to actually factor the number was considered insurmountable -- the technology and theory at that time would have taken roughly $10^{20}$ years to do so. But this all changed with the advent of accessible and fast computing power as the number was factorised in 1984, merely taking 32 hours.

In this talk we will first discuss the method that was used, the quadratic sieve by Pomerance in 1981, before moving on to a more complicated yet powerful version of it, the number field sieve by Pollard in 1996. There'll be plenty of actual numbers and examples!

Dec 6 Wed Di Zhang (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Reciprocity laws and $L$-functions

Abstract:
Hilbert's ninth problem was to prove the reciprocity law for $n$-th power residue for an arbitrary number field $K$ and for $n>2$, and it was partially solved by Emil Artin. How did Artin guess his reciprocity law? He was led to the law in trying to show that a new kind of L-function was a generalization of the usual L-function. In today's talk we will see why the factorization of some L-functions can be viewed as some form of ''reciprocity law''.

Dec 13 Wed Ed Pearce (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Jacobi's Identity and the Dirac Sea

Abstract:
The Jacobi Triple Product identity was first introduced in 1829 by Carl Jacobi in his works on the theory of elliptic functions. The identity also arises in the study of affine Kac-Moody Lie algebras. In this talk we will give a physical interpretation of the identity using the electron sea model introduced by Paul Dirac in 1930, which as a by-product theorised the antimatter particle, the positron, years before its experimental discovery. We further provide a combinatorial proof of the identity, where the techniques and concepts involved have applications in the theory of partitions, and interest beyond in the fields of algebraic geometry and number theory.

Feb 21 Wed Scott Balchin, Caitlin McAuley, Ariel Weiss ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks What is...?

Abstract:
Three fifteen minute introductory talks on some important themes from different areas of pure maths, namely Tiling Spaces, Mirror Symmetry, and the Langlands Program.

Feb 27 Tue Christos Aravanis (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Hopf algebras in categories of complexes

Abstract:
I will discuss about a generalization of the notion of a Hopf algebra in monoidal categories due to Brugieres and Virelizier. Of particular interest will be the derived category of coherent sheaves on a smooth complex projective variety.

Mar 14 Wed Igor Sikora (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Could the Philosophy of Mathematics be interesting for mathematicians

Abstract:
Philosophical reflections of mathematics are concerned with the fundamental problems about mathematics, such as existence of mathematical objects, subject of research of mathematics, how do we extend our mathematical knowledge, what are relations of mathematics with other sciences etc. In this talk I will attempt to describe several classical and modern problems in philosophy an approaches to solve them.

Mar 21 Wed Sam Morgan (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 F20 Lie groupoids and their application in symplectic geometry

Abstract:
The aim of this talk is to introduce the theory of Lie groupoids and Lie algebroids to a broad audience. It is hoped that the subject can be well motivated, without many prerequisites. In the second part of the talk, we will see why Lie groupoids were first introduced into symplectic and Poisson geometry, and what role they play here.

Apr 18 Wed David Spencer (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Visualising the values of a binary quadratic form

Abstract:
The solution of Diophantine equations is still a thriving area in number theory. In this talk I will consider binary quadratic forms, those of the form $ax^2 + bxy + cy^2$ with $a,b,c \in\mathbb Z$. I will show how to construct a graph which allows us to see the possible values such a quadratic form can take. This will allow us to determine when the Diophantine equation $ax^2 + bxy + cy^2 = k$ is solvable in integers $(x,y)$, and to find such integers when it is solvable.

Apr 25 Wed Caitlin McAuley (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Introducing stability conditions

Abstract:
The space of stability conditions is a complex manifold associated to a triangulated category. The definition of a stability condition was motivated by work in string theory and as such, an understanding of the stability manifold will have important consequences in mirror symmetry. I'll introduce stability conditions on an arbitrary triangulated category and discuss some of their most important features, as well as discussing some examples.

May 2 Wed *No talk* ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks

May 9 Wed Davide Spriano (ETH Zurich) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks What is geometric group theory (and why people care about it)

Abstract:
Geometric group theory is an area of mathematics that focuses on understanding a group by understanding its geometric properties, which are typically expressed in terms of actions on metric spaces. The advantage of the geometric group theory approach is that it provides a better understanding of groups of great interest on which more classical approach would fail. A prominent example is given by "automorphism groups of certain objects". They can be easily defined, but it is not at all clear when they are, for instance, finitely generated. The first part of the talk will be concerned in providing example of such groups motivating why are they interesting and which are the main difficulties that arises in the study of them. In the second part, we will introduce some of the fundamental concepts and tools of geometric group theory and coarse geometry.

May 16 Wed *No talk* ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks

May 30 Wed Nebojsa Pavic (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Intersection theory in algebraic geometry: History and motivation.

Abstract:
In this talk I'm going to motivate the notion of intersection theory in algebraic geometry by considering the example of Riemannian surfaces and only requiring a basic knowledge of complex analysis and a little bit of complex differential geometry. If time permits, I will give a rigorous definition of intersection groups, so called Chow groups, and relate them to the example.

Jun 6 Wed Ciaran Schembri (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Fuchsian group

Abstract:
Fuchsian groups are discrete subgroups of the special projective linear group. They act as isometries on the hyperbolic plane and are studied because of their role in generating Riemann surfaces among other things. In this talk I will outline their geometric properties and if time permits will discuss how they relate to modern number theory.

16:00 J11 Hicks

Abstract:

I will give an introduction to the main themes behind Cohen's forcing method in ZFC, as framed by Scott through the lens of Boolean algebras. Given time there will also be some discussion of why, contrary to Godel's belief, forcing shows that the inclusion of large cardinal axioms does not solve the continuum hypothesis

A Calculation of Some Group Cohomology (Brotherston)

I will give a very brief introduction to group cohomology and why we care in relation to algebraic K-theory. I will then move on to outline a mostly geometric calculation of the former in the case where $G = GL_n(\mathbb{F}_q)$ with coefficients in $\mathbb{Z}/l$ for $l$ coprime to the characteristic of the field $\mathbb{F}_q$. There will hopefully be a little of number theory, algebraic topology and algebraic geometry mentioned.

Tate Shafarevich Groups - The mysterious objects attached to Elliptic Curves (Zahoor)

One of the main unsolved mystery about elliptic curves is the size of the Tate Shafarevich Group (Sha), that is conjectured to be finite. The lack of proof of this particular fact is one of the main barriers in giving an algorithm to compute rank of an elliptic curve. Intuitively, Sha measures the obstruction to the Hasse (local-global) principle. This talk aims at understanding Tate Shafarevich Groups using torsors of elliptic curves and is deigned for anyone having a first look at the Sha.

Nov 14 Wed Jordan Williamson (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Equivariant Topology and Commutative Algebra

Abstract:
Equivariant topology is the study of spaces with a group action, and some invariants for studying these objects are equivariant cohomology theories. In this talk, we will explain how algebraic techniques can be used to study equivariant cohomology theories, and we will give a sketch-proof of a theorem of Greenlees-Shipley which classifies the equivariant cohomology theories on free G-spaces over the rational numbers. This will involve a discussion of Borel cohomology and its relation to representation theory, and the algebraicization theorem of Shipley, which provides a bridge between algebra and topology.

Nov 21 Wed Igor Sikora (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Eilenberg-Zilber map and Acyclic Models

Abstract:
When we are thinking about homology of a product of topological spaces, the first answer coming into mind is the Kunneth Theorem. Actually, it is only part of the truth. During the talk I will focus on the other part, which is chain level relation between singular complex of a product and a product of complexes - or more generally, between chain complexes associated to the simplicial abelian group and a product of complexes. This is done by very nice technique, called acyclic models. I assume basic knowledge of singular homology, i.e. definition of a chain complex and of a singular complex of a space.

Nov 28 Wed Daniel Graves (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Homology theories for algebras

Abstract:
Since the 40s people have been developing homology and cohomology theories to try to encode information about algebras over commutative rings. In this talk I will discuss three such: Hochschild homology, Cyclic homology and Symmetric homology. The first two are classical theories that have found applications in many diverse areas. Symmetric homology is a related theory that is less well studied and slightly mysterious in comparison.

Dec 12 Wed Luca Pol ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks The spectrum of a triangulated category

Abstract:
The spectrum of a commutative ring spec(R) is an interesting topological space that encodes lots of geometric and algebraic information. In 2004, Paul Balmer generalized this definition to any tensor triangulated category setting the scene for Tensor Triangular Geometry. One of the main achievement of the Balmer spectrum is to put in a unique framework three classification results: Devinats, Hopkins and Smith's theorem in Stable Homotopy Theory, Thomason's theorem in Algebraic Geometry and Benson, Carlson and Rickard's theorem in Modular Representation Theory. In this talk I will define the Balmer spectrum and show some concrete examples.

Feb 27 Wed Raven Waller (Nottingham) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Level structures - a crossroads between topology, representation theory, and number theory

Abstract:
The arithmetic, algebraic, and topological properties of local fields are intimately related. For higher dimensional local objects these relationships begin to break down, and this may cause considerable difficulty when studying them. The notion of a level structure allows us to work around some of these issues. We will discuss various applications of level structures, including the explicit study of representations of reductive groups over higher dimensional local fields, which is also related to the geometric Langlands program.

Mar 6 Wed Paolo Dolce (Nottingham) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Two dimensional adelic geometry

Abstract:
I will give an overview of a novel approach to the study of two dimensional algebraic and arithmetic geometry by means of adelic and idelic structures. Particular emphasis will be given to the case of arithmetic surfaces since the aim of the theory is to give a two dimensional version of Tate's thesis.

Mar 13 Wed Karoline Van Gemst (Birmingham) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Enumerative geometry in projective space and Kontsevich's formula

Abstract:
Enumerative geometers are interested in counting certain geometric objects given a set of conditions. One example of such a counting problem is how many degree d rational curves pass through 3d-1 generically placed given points in the projective plane. This particular problem proved extremely difficult using classical methods, even for low d. In the 1990s however, a revolution within this area took place, originating in the world of physics. This led to Kontsevich solving the counting problem by proving a recursive formula for calculating this number for any d. Kontsevich’s formula requires a single initial datum, the case of d=1, which translates to the fact that a single line passes through two given points in the plane. In this talk, I will present some of the crucial ingredients in setting up for and proving Kontsevich’s formula, and illustrate how it makes sense through a few examples. If time permits, I will also motivate how the formula can be viewed as expressing the associativity of the quantum product.

Mar 20 Wed Gianmarco Brocchi (Birmingham) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks What does extremise a Strichartz estimate?

Abstract:
This will be a blunt talk on sharp inequalities. Roughly speaking, these are inequalities which cannot be improved. In particular, I will introduce inequalities for the restriction of the Fourier transform, explaining why I got interested in them and how they are related to other inequalities in PDE, such as Strichartz estimates. These are a key tool in understanding the evolution of waves in dispersive PDE. If time allows, I will discuss a sharp Strichartz estimate for the fourth order Schrödinger equation from a joint work with Diogo Oliveira e Silva and René Quilodrán.

Mar 27 Wed Andreea Mocanu (Nottingham) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks On the connection between Jacobi forms and elliptic modular forms

Abstract:
Jacobi forms arise naturally in number theory, for example as functions of lattices or as Fourier-Jacobi coefficients of other types of modular forms. They have applications in algebraic geometry, string theory and the theory of vertex operator algebras, among other areas. We are interested in establishing a precise connection between Jacobi forms of lattice index and elliptic modular forms, in other to transfer information from one side to the other. In this talk, we illustrate this connection via an example, namely that of Jacobi forms whose indices are the root lattices of type $D_n$.

May 1 Wed Esmee te Winkel (Warwick) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks A combinatorial approach to surface homeomorphisms

Abstract:
In geometric group theory, it is common to try to study a group by finding a meaningful action on a metric space. This talk is about the mapping class group of a compact surface and its action on various graphs. The mapping class group is the group of homeomorphisms up to isotopy. I will define this group and state some of its properties and open questions. After this motivation, I will introduce the curve graph and the pants graph associated to a surface and explain how the mapping class group acts on them.

Oct 23 Wed Joseph Martin (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks An Introduction to the Periodic Table of n-Categories

Abstract:
The aim is to provide much of what is needed to understand the relationship between low-dimensional degenerate n-categories and their counterparts in the Periodic Table of n-categories. We first seek to establish a good understanding of equivalences between categories via a thorough study of adjunctions. Then we give an overview of the structures that can be found in the Periodic Table along with a useful result in each case. Finally, this is followed by an inspection of degenerate categories and bicategories, in particular we compare their totalities to that of monoids.

Oct 30 Wed Maram Alossaimi, Lewis Combes, & Yirui Xiong (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Poisson Algebra (Maram Alossaimi)
An Introduction to the Theory of Elliptic Curve Cryptography (Lewis Combes)
Calabi-Yau algebras and superpotentials (Yirui Xiong)

Abstract:
Poisson Algebra
The concept of a Poisson algebra comes from defining a bilinear product {·, ·} on a commuta- tive algebra over a field K to bring a new non-commutative algebra structure. I will give some definitions, examples and the main Lemma in our research. In the end, if there is enough time I will introduce our new Poisson algebra structure.

An Introduction to the Theory of Elliptic Curve Cryptography
An elliptic curve over a finite field can be endowed with the structure of an abelian group. Within this group there are computations that are easy to perform, but hard to reverse. These computations form the basis of elliptic curve cryptography, an encryption standard with advantages and disadvantages when compared to traditional RSA methods. The downsides are such that an intimate understanding of certain mathematical properties of the chosen elliptic curve is needed to keep the protocol secure. In this talk I will go through the theory behind using elliptic curves for encryption, as well as some of the mathematical considerations that should be made when designing such a system.

Calabi-Yau algebras and superpotentials
Calabi-Yau algebras arise from transporting the conception of Calabi-Yau manifolds to noncommutative geometry, and now have profound applications in algebraic geometry and representation theory. One of the central problems in the study of Calabi-Yau algebras is their structural problem: can Calabi-Yau algebras be derived from superpotentials? We will review the answers to the problem based on work in the past years. And if time is permitted, I will introduce some applications based on structural theorems of Calabi-Yau algebras.

Nov 6 Wed Eve Pound (Sheffield) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks The Structure of Chevalley Groups over Local Fields.

Abstract:
The Chevalley group is a subgroup of the automorphism group of a Lie algebra. In 1965, Iwahori and Matsumoto showed that, when the underlying field admits a nonarchimedean discrete valuation (for example, over Qp), these groups admit a double coset decomposition, or Bruhat decomposition. This decomposition allows lots of information about the group to be read off, and is intricately linked with the associated Bruhat-Tits building. In this talk, I'll start with the definition of a Lie algebra and try to motivate why we care about the Chevalley group, and give an overview of the geometric and combinatorial ideas in Iwahori and Matsumoto's work. If there is time, I will give some examples of how this links to buildings.

Nov 20 Wed Cesare Giulio Ardito (Manchester) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Classifying 2-blocks with an elementary abelian defect group

Abstract:
Donovan’s conjecture predicts that given a $p$-group $D$ there are only finitely many Morita equivalence classes of blocks of group algebras with defect group $D$. While the conjecture is still open for a generic $p$-group $D$, it has been proven in 2014 by Eaton, Kessar, Külshammer and Sambale when $D$ is an elementary abelian 2-group, and in 2018 by Eaton and Livesey when $D$ is any abelian 2-group. The proof, however, does not describe these equivalence classes explicitly. A classification up to Morita equivalence over a complete discrete valuation ring $\mathcal{O}$ has been achieved for $D$ with rank 3 or less, and for $D = (C_2)^4$. I have done $(C_2)^5$, and I have partial results on $(C_2)^6$. I will introduce the topic, give the relevant definitions and then describe the process of classifying this blocks, with a particular focus on the methodology and the individual tools needed to achieve a complete classification.

16:00 J11 Hicks TBA

Abstract:
TBA

Jan 15 Wed Igor Sikora (Warwick) ShEAF: postgraduate pure maths seminar
16:00 J11 Hicks Elmendorf's theorem

Abstract:
Elmendorf's theorem is an absolutely key result in the equivariant homotopy theory. It relates homotopy type of G-spaces with homotopy of its fixed points diagrams. During the talk I will state the theorem, discuss what actually a homotopy theory in some category is, discuss a little bit of model categories and eventually I may approach proving the theorem, but I cannot promise the latter right now.