Current members

Staff

Neil Strickland

I work in stable homotopy theory, a branch of topology in which one studies phenomena that occur uniformly in all sufficiently high dimensions. 

On the one hand, the subject involves many direct geometrical constructions with interesting spaces such as complex algebraic varieties, coset spaces of Lie groups, spaces of subsets of Euclidean space, and so on. On the other hand, one can use generalised cohomology theories to translate problems in stable homotopy theory into questions in pure algebra, in a strikingly rich and beautiful way. 

The algebra involved centres around the theory of formal groups, which is essentially a branch of algebraic geometry, although not one of the most familiar branches. It has connections with commutative algebra, Galois theory, the study of elliptic curves, finite and profinite groups, modular representation theory, and many other areas. 

To translate efficiently between algebra and topology we need to make heavy use of category theory, and this also has applications both on the purely algebraic and the purely topological side.

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Markus Szymik

My research revolves around symmetries in topology, geometry, and algebra. I use homotopy theory and homological algebra, representation theory and K-theory to understand groups and their generalisations. 

These methods have applications far beyond the study of symmetry: they have been used in data and computer science (topological data analysis, homotopy type theory, and topological quantum computing). 

On the theoretical side, I am currently thinking about knot theory and analogies with number theory.

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Sarah Whitehouse

In algebraic topology, we use algebra to extract topological or geometric information about spaces. Among the standard tools used to carry out the translation from topology to algebra are generalised cohomology theories. Given such a theory there is an associated algebra of operations, carrying a wealth of structure and information. 

I am particularly interested in studying the structure and properties of these algebras. A simple, yet still interesting example, is given by complex K-theory. This theory has a strong geometric flavour and yet there is a rich interplay with other areas, such as number theory. 

The interplay between topology and algebra leads to "up to homotopy" versions of algebraic structures. I am interested in A-infinity and E-infinity algebras, arising when one weakens associativity and commutativity conditions.

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Simon Willerton

I am interested in various ideas in low-dimensional topology coming from quantum physics, and in their relationship to geometry and algebraic topology. In particular, methods from quantum field theory give rise to new invariants of knots and three-manifolds – these are the so-called quantum and Vassiliev (or finite-type) invariants. 

A large part of the motivation for my work is to understand these invariants from a topological or geometric point of view. For instance, the Kontsevich integral is a construction which takes a knot and gives back a sort of Feynman diagram expansion: this embodies a rich algebraic structure that is reminiscent of certain objects from algebraic topology, but it is not clear at the moment how to relate these. 

Well-studied examples of quantum invariants arise when one fixes a Lie group. Motivated in part by trying to understand the Kontsevich integral, I have considered (with collaborators in San Diego and Oxford) the less well-studied invariants which arise when one fixes a hyper-Kahler manifold. This work has revealed unexpected algebraic structures in the derived category of coherent sheaves on a complex manifold. 

The theory of gerbes is a related interest of mine. Gerbes can be thought of as the next step beyond line bundles. Ideas from this area feed into K-theory, string theory and the quantum invariants mentioned above. In recent times I have been interested in the connections between metric spaces and category theory. This has lead in particular to me studying measures of biodiversity.

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Ieke Moerdijk

My focus of research is on the interface of category theory and algebraic topology. Currently, I concentrate mainly on developing the theory of dendroidal sets and dendroidal spaces. This is an extension of the simplicial theory, and can be used to model the homotopy theory of operads and their algebras, of infinite loop spaces and of several related structures. 

Earlier on, I also did work on Lie groupoids and foliations, and on applications of topology to mathematical logic. My main affiliation is with Utrecht University. I also work on improving cross fertilisation between the topologists in Utrecht and in Sheffield.

Ieke's University of Sheffield staff profile

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James Cranch

I'm a member of teaching staff. When I get a chance to do research, I think about topology and higher category theory, two increasingly closely-related areas. 

Recently I've been working with Sheffield's computer scientists, trying to explain why similar algebraic structures occur both in topology and in the study of concurrent programs.

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Cheuk Yu Mak

I study symplectic topology and its interactions with low-dimensional topology, algebraic geometry, and geometric representation theory. 

Khovanov homology is an invariant of links in S3 that categorifies the Jones polynomial. It is a powerful invariant that, in particular, can be used to show the existence of exotic R4 spaces that are homeomorphic but not diffeomorphic to R4. Despite its representation-theoretic origin, Seidel and Smith found a symplectic analog. The symplectic Khovanov homology is shown to be isomorphic to the ordinary Khovanov homology over a characteristic 0 field, and it is better suited for studying symmetries. Part of my work involves gaining a better understanding of its categorification, its relation to other categorifications (in geometric representation theory and algebraic geometry), and its generalizations.

By definition, symplectic Khovanov homology is a Lagrangian Floer cohomology, which is a type of Morse homology of an action functional on a path space. Homotopy theory enters in two ways. On one hand, the Lagrangian Floer cochain inherits the structure of an A∞ algebra, similar to the chain on a based loop space. Its closed-string analog, the symplectic cohomology, inherits the structure of a BV∞ algebra, similar to the chain on a free loop space. Floer theory and homotopical algebra are essential to my study of symplectic topology. On the other hand, instead of taking the homology of the action functional, one can consider the stable homotopy type with respect to the action functional. This is a new branch of research termed Floer homotopy theory. I am learning this new technique and trying to incorporate it into my research.

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