# Geometry and Topology

# Geometry and Topology

Geometry and Topology at the University of Glasgow touches on a wide range of highly active subdisciplines, benefiting from and capitalizing on strong overlap with the Algebra, Analysis, and Integrable Systems and Mathematical Physics research groups within the School of Mathematics and Statistics.

Our interests lie in algebraic topology, geometric group theory, low-dimensional topology and quantum geometry, to name a few.

Broadly speaking, our research – performed by undergraduates, postgraduates, postdoctoral fellows, and academic staff – is concerned with the rich interaction and deep interconnections between algebra and geometry with a view to new applications and solutions to long-standing problems.

Some background on and context for our work in low-dimensional topology, homotopy theory and homological algebra, homological invariants and categorification, geometric group theory, quantum symplectic geometry, and noncommutative topology is given at areas of focus tab below.

## Staff

#### Dr Spiros Adams-Florou Lecturer

Geometric Topology, Algebraic Topology, Surgery theory, Controlled Topology

**Member of other research groups:** Algebra

#### Dr Chris Athorne Senior lecturer

Geometric representation theory; algebraic curves;soliton theory

**Member of other research groups:** Integrable Systems and Mathematical Physics

#### Dr Andrew J Baker Reader

Algebraic Topology, especially stable homotopy theory, operations in periodic cohomology theories, structured ring spectra including Galois theory and other applications of algebra, number theory and algebraic geometry.

**Member of other research groups:** Algebra

#### Dr Alex Bartel Senior lecturer

**Algebraic number theory:**- Galois module structures, e.g. the structure of the ring of integers of a number field as a Galois module, or of its unit group, or of the Mordell-Weil group of an abelian variety over a number field;
- Arithmetic statistics, especially the Cohenâ€•Lenstra heuristics on class groups of number fields and their generalisations;
- Arithmetic of elliptic curves over number fields.

**Representation theory of finite groups:**- Integral representations of finite groups;
- Connections between the Burnside ring and the representation ring of a finite group;
- Applications of the above to number theory and geometry.

**Geometry and topology:**- Actions of finite groups of low-dimensional manifolds.

**Member of other research groups:** Algebra

#### Dr Gwyn Bellamy Lecturer

My research interests are in geometric representation theory and its connections to algebraic geometry and algebraic combinatorics. In particular, I am interested in all aspects of symplectic representations, including symplectic reflection algebras, resolutions of symplectic singularties, D-modules and deformation-quantization algebras.

**Member of other research groups:** Algebra

**Research student:** Tomasz Przezdziecki

**Postgraduate opportunities:** Deformation-Quantization algebras on K3 surfaces, Homological properties of Deformation-Quantization algebras, Resolutions of symplectic singularities, Sheaves of Cherednik algebras, Rational Cherednik algebras and geometric Langlands

#### Dr Tara Brendle Professor of Mathematics

Geometric group theory; mapping class groups of surfaces; low-dimensional topology

**Member of other research groups:** Algebra

**Research students:** Alan McLeay, Luke Jeffreys

#### Dr Kenneth A Brown Professor of Mathematics

Noncommutative algebra; Hopf algebras; homological algebra

**Member of other research groups:** Algebra

**Research student:** Miguel Couto

#### Dr Ben Davison Lecturer

Algebraic geometry, geometric representation theory, cluster algebras, Higgs bundles, Nakajima quiver varieties

**Member of other research groups:** Algebra

#### Dr Mikhail Feigin Senior lecturer

Frobenius manifolds

**Member of other research groups:** Integrable Systems and Mathematical Physics, Algebra

**Research students:** Maali Alkadhem, Georgios Antoniou

#### Dr Vaibhav Gadre Lecturer

Teichmuller Dynamics, Mapping Class Groups.

**Member of other research groups:** Algebra

**Research student:** Luke Jeffreys

#### Dr Ciaran Meachan Lecturer

**Member of other research groups:** Algebra

#### Dr Brendan Owens Senior lecturer

Low-dimensional topology: knots, 3-manifolds, smooth 4-manifolds

**Research student:** Daniel Waite

#### Prof Ian A B Strachan Professor of Mathematical Physics

Geometry and integrable systems; Frobenius manifolds; Bi-Hamiltonian structures, twistor theory and self-duality

**Member of other research groups:** Integrable Systems and Mathematical Physics

**Research student:** Georgios Antoniou

#### Dr Christian Voigt Senior lecturer

Noncommutative geometry; K-theory; Quantum groups

**Member of other research groups:** Analysis, Algebra

**Research students:** Andrew Monk, Samuel Evington, Lucia Rotheray

#### Dr Andy Wand Lecturer

#### Dr Liam Watson Reader

Low-dimensional topology

**Research students:** Mel Chen, Michael Snape

#### Prof Michael Wemyss Professor of Mathematics

Algebraic geometry and its interactions, principally between noncommutative and homological algebra, resolutions of singularities, and the minimal model program. All related structures, including: deformation theory, derived categories, stability conditions, associated commutative and homological structures and their representation theory, curve invariants, McKay correspondence, Cohen--Macaulay modules, finite dimensional algebras and cluster-tilting theory.

**Member of other research groups:** Algebra

**Research staff:** Sam Dean

#### Dr Stuart White Professor of Mathematics

Non-commutative geometry

**Member of other research groups:** Analysis, Algebra

**Research student:** Samuel Evington

**Postgraduate opportunities:** Interactions between von Neumann and C*-algebras, Operator Algebras associated to groups

#### Dr Mike Whittaker Lecturer

Noncommutative geometry, topological dynamical systems, fractal geometry, and aperiodic substitution tilings.

**Member of other research groups:** Analysis, Algebra

**Research students:** Dimitrios Gerontogiannis , Mustafa Ozkaraca

**Postgraduate opportunities:** Aperiodic substitution tilings and their C*-algebras., Operator algebras associated to self-similar actions.

#### Dr Andrew Wilson University Teacher of Mathematics

#### Dr Joachim Zacharias Reader

C*-algebras, their classification and amenability properties; special examples of C*-algebras; K-theory and non commutative topology, noncommutative dynamical systems, geometric group theory with applications to C*-algebras.

**Member of other research groups:** Integrable Systems and Mathematical Physics, Analysis, Algebra

**Research staff:** Joan Bosa

**Research students:** Luke Hamblin, Dimitrios Gerontogiannis

## Postgraduates

#### Mel Chen PhD Student

**Supervisor:** Liam Watson

#### Alan McLeay PhD Student

**Research Topic:** Geometric group theory and mapping class groups of surfaces

**Member of other research groups:** Algebra

**Supervisor:** Tara Brendle

#### Michael Snape PhD Student

**Research Topic:** Homological invariants in low-dimensions

**Supervisor:** Liam Watson

#### Daniel Waite PhD Student

**Supervisor:** Brendan Owens

## Postgraduate opportunities

### Operator algebras associated to self-similar actions. (PhD)

**Supervisors:** Mike Whittaker

**Relevant research groups:** Geometry and Topology, Analysis, Algebra

This project will focus on self-similar groups and their operator algebras. The primary aim will be to examine a new class of groups that act self-similarly on the path space of a graph and to study the noncommutative geometry of a natural class of operator algebras associated to these self-similar groups.

### Aperiodic substitution tilings and their C*-algebras. (PhD)

**Supervisors:** Mike Whittaker

**Relevant research groups:** Geometry and Topology, Analysis

### Rational Cherednik algebras and geometric Langlands (PhD)

**Supervisors:** Gwyn Bellamy

**Relevant research groups:** Algebra, Geometry and Topology

The geometric Langlands program (google it!) is one of the most profound (conjectural) relations in mathematics, and is often viewed as the very pinnacle of the subject. It turns out thet there is a connection between rational Cherednik algebras and the affine Lie algebra glb n at the critical level (which is the key object of study in geometric Langlands). Very little is known about this relationship and virtually nothing written. The goal of this project would be to remedy this situation.

All projects are subject to the availability of funding.

### Sheaves of Cherednik algebras (PhD)

**Supervisors:** Gwyn Bellamy

**Relevant research groups:** Algebra, Geometry and Topology

Soon after Etingof and Giznburg introduced rational Cherednik algebras, Etingof showed that they belong to a much large family of sheaves of Cherednik algebras that can be defined for any smooth variety and finite group acting on this variety. In this generality, very little is know about the algebras. The goal of this project would be to develop tools, as exist in the theory of D-module, to study these algebras. Hopefully one can say something non-trivial about the representation theory of these algebras then.

All projects are subject to the availability of funding.

### Resolutions of symplectic singularities (PhD)

**Supervisors:** Gwyn Bellamy

**Relevant research groups:** Algebra, Geometry and Topology

Symplectic singularities appear naturally in representation theory of non-commutative algebras. As such their geometry tells us a lot about the representation theory of these algebras. Through work of Namikawa we now know a great deal about symplectic singularities and their resolutions. However, there are still many open problems. One of the most basic is to count, and explicitly construct, these resolutions. The goal of this project would be to do so for some concrete, but interesting examples.

All projects are subject to the availability of funding.

### Homological properties of Deformation-Quantization algebras (PhD)

**Supervisors:** Gwyn Bellamy

**Relevant research groups:** Algebra, Geometry and Topology

In the study of the representation theory of deformation-quantization algebras, several natural categories of sheaves play an important role. Most of these categories should be “smooth” in an appropriate sense. In particular, they should have finite global dimension. The goal of this project would be to understand the homological properties, in particular global dimension, of these categories. The idea here would be to generalize classical (but difficult) results in the theory of D-modules on the global dimension of sheaves of differential algebras.

All projects are subject to the availability of funding.

### Deformation-Quantization algebras on K3 surfaces (PhD)

**Supervisors:** Gwyn Bellamy

**Relevant research groups:** Algebra, Geometry and Topology

Smooth K3 surfaces are a natural sources of symplectic manifolds. They have been studied by algebraic geometers for over a century now. One can also study deformation-quantization algebras on them. The goal then would be to 1 relate the properties of these non-commutative algebras, and their representation theory, to the rich underlying geometric properties of the surfaces.

All projects are subject to the availability of funding.

## Research Areas of Focus

### Low-dimensional topology

Geometry and topology is particularly interesting and rich in low dimensions, namely, the dimensions of the universe we inhabit. This includes dimensions three and four as well as how knots and surfaces can inhabit these spaces. As a result, there is also a strong connection with mapping class groups of surfaces. Since the 1980s, gauge theory techniques from theoretical physics have been the leading tools for understanding smooth topology in four-dimensions. In the 21st century new approaches, in particular Heegaard Floer theory, have expanded the reach of these tools to three-dimensions, as well as to the study of knots and surfaces, and made fascinating connections with Khovanov homology — a theory that seems to stem from completely different origins.

**People:** Tara Brendle, Brendan Owens, Liam Watson

### Homotopy theory and homological algebra

Algebraic topology grew out of classical point-set topology giving rise to a theory of algebraic invariants of spaces (and maps between them) up to a natural notion of equivalence called homotopy. However, in recent decades these ideas have seeped into many other areas of mathematics and theoretical physics, often providing new frameworks for handling old problems. Abstract homotopy theory, then, provides a general algebraic framework for studying deformation; this has strong interaction with the general study of category theory. Stable homotopy theory involves the underlying structure of homology and cohomology theories and is usually pursued by working with a suitable generalization of spaces — called spectra — in which negative dimensions make sense. This is not unlike the birth of the complex numbers from considerations of √-1! There are rich algebraic structures available in modern versions of these categories and topics such as E∞ ring spectra lead to extensions of classical algebraic topics (Galois theory and Morita theory, for example).

**People:** Andy Baker, Gwyn Bellamy, Ken Brown, Uli Kraehmer, Richard Steiner, Liam Watson

### Homological invariants and categorification

How can you determine if two knots are different in an essential way? One good way is to produce an algebraic invariant to tell them apart. For example, Khovanov categorification of the Jones polynomial gives rise to an invariant of links in the three-sphere in the form of a bi-graded homology theory. This has seen a range of interesting applications in low-dimensional topology while providing a point of departure to many generalisations — now touching on homotopy theory, gauge theory and physics. But this seems to be just the tip of an iceberg: Categorification is now an essential tool in algebraic geometry and geometric representation theory. This, in turn, continues to feed back into low-dimensional topology by providing a range of new invariants stemming from diagrammatic algebras.

**People:** Gwyn Bellamy, Ghislain Fourier, Christian Korff Brendan Owens, Liam Watson

### Geometric group theory.

Geometric group theory studies groups by connecting their algebraic properties to the topological and geometric properties of spaces on which they act. Sometimes the group itself is treated as a geometric object; occasionally auxiliary structures on the group, such as orders, arise naturally. The field emerged as a distinct area in the late 1980s and has many interactions with other parts of mathematics, including computational group theory, low-dimensional topology, algebraic topology, hyperbolic geometry, the study of Lie groups and their discrete subgroups and K-theory.

**People:** Tara Brendle, Anne Thomas, Liam Watson

### Quantum symplectic geometry

Motivated by the key notion of quantization in quantum mechanics, quantum geometry (or, non-commutative geometry) aims to apply the tools and techniques of non-commuative algebra to study problems in geometry. In the opposite direction, it allows one to use powerful geometric tools to study the representation theory of non-commuative algebras, as epitomized by the famous Beilinson-Bernstein localization theorem. At Glasgow, we study quantum symplectic geometry from several different perspectives — via the theory of D-modules and deformation-quantization algebras on a symplectic manifold; via the deformation theory of Hopf algebras and their relation to operads; and via quantum integrable systems such as the quantum Calogero-Moser system. Taking such a broad approach to the subject allows one to see how truly interconnected these areas of mathematics really are.

**People:** Gwyn Bellamy, Ken Brown, Misha Feign, Uli Kraehmer

### Noncommutative Topology

This relatively young field grows out of the Gelfand-Naimark theorem, establishing a strong connection between compact Hausdorff spaces and commutative C*-algebras. This allows us to translate topology into algebra and functional analysis. Even more, once formulated algebraically, some of these concepts still make sense for noncommutative C*-algebras, opening the door to study these algebras using ideas from topology. The truly fascinating fact, however, is that the study of noncommutative C*-algebras in turn has deep applications to classical topology and geometry. For instance, the Baum-Connes conjecture, which is a central aspect of the noncommutative topology of groups, implies the Novikov conjecture on higher signatures and the stable Gromov-Lawson-Rosenberg conjecture on the existence of positive scalar curvature metrics. At Glasgow, various aspects of noncommutative topology are studied, ranging from the classification program for nuclear C*-algebras to quantum groups and bivariant K-theory, including links with geometric group theory.

**People:** Anne Thomas, Christian Voigt, Stuart White, Joachim Zacharias