Seminars this semester
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 PoissonLie 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 noninteger 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 Lfunction was a generalization of the usual Lfunction. In today's talk we will see why the factorization of some Lfunctions 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 KacMoody 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 byproduct 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. 



Oct 24  Wed  James Brotherston, Callum Reader, Sadiah Zahoor  ShEAF: postgraduate pure maths seminar  
16:00  J11 Hicks  
Abstract: What is Forcing? (Reader) 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 Ktheory. 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 (localglobal) 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 sketchproof of a theorem of GreenleesShipley which classifies the equivariant cohomology theories on free Gspaces 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  EilenbergZilber 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 3d1 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 FourierJacobi 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 nCategories  
Abstract: The aim is to provide much of what is needed to understand the relationship between lowdimensional degenerate ncategories and their counterparts in the Periodic Table of ncategories. 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) CalabiYau 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 noncommutative 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. CalabiYau algebras and superpotentials CalabiYau algebras arise from transporting the conception of CalabiYau manifolds to noncommutative geometry, and now have profound applications in algebraic geometry and representation theory. One of the central problems in the study of CalabiYau algebras is their structural problem: can CalabiYau 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 CalabiYau 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 BruhatTits 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 2blocks 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 2group, and in 2018 by Eaton and Livesey when $D$ is any abelian 2group. 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. 



Dec 11  Wed  Callum Reader  ShEAF: postgraduate pure maths seminar  
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 Gspaces 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. 



Feb 19  Wed  Sadiah Zahoor (Sheffield)  ShEAF: postgraduate pure maths seminar  
16:00  J11 Hicks  Modular forms and their congruences  
Abstract: Starting with a unit disc embedded inside a complex plane, we act on it by a group of symmetries. Our primary interest lies in holomorphic functions defined over this disc which are invariant under this action. These functions are called modular forms. Modular forms show bizarre symmetries due to the remarkable way they transform. I shall begin with an informal introduction to modular forms building up an insight to my current research project dealing with congruences between modular forms and similar objects. 



May 25  Wed  Dan Graves  ShEAF: postgraduate pure maths seminar  
16:00  J11 Hicks  From Algebraic Topology to Terrapin Station in three easy steps  

