## 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.

All information about our group, our members, our activities, and a full list of our expertise, can be found at our Core Structures webpage.

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

#### Prof Alex Bartel : Professor of Mathematics

**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

**Research student:** Ross Paterson

#### Prof Gwyn Bellamy : Professor of Mathematics

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 students:** Niall Hird, Kellan Steele, Ross Paterson

#### Dr Tara Brendle : Professor of Mathematics

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

**Member of other research groups:** Algebra

**Research student:** 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 Mikhail Feigin : Senior lecturer

Frobenius manifolds

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

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

#### Dr Maxime Fortier-Bourque : Lecturer

#### Dr Vaibhav Gadre : Lecturer

Teichmuller Dynamics, Mapping Class Groups.

**Member of other research groups:** Algebra

**Research student:** Luke Jeffreys

#### Dr Ana Lecuona : Lecturer

Low dimensional topology; knot theory; interactions between 3 and 4 dimensional manifolds.

**Research students:** Elizaveta Lokteva, Tanushree Shah

#### Dr Brendan Owens : Senior lecturer

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

**Research students:** Cheng Chen, Daniel Waite, Vitalijs Brejevs, Miguel Rodriguez, Tanushree Shah

#### Dr Greg Stevenson : Lecturer

**Member of other research groups:** Algebra

**Research students:** David Murphy, James Rowe

#### 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 student:** Jamie Antoun

#### Dr James Walton : Research Assistant

**Member of other research groups:** Analysis

#### Dr Andy Wand : Senior lecturer

**Research students:** Vitalijs Brejevs, Miguel Rodriguez, Elizaveta Lokteva, Tanushree Shah

#### 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 students:** Sarah Kelleher (Mackie), Ogier Van Garderen

#### 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 , Kate Gibbins, Mustafa Ozkaraca, Jamie Antoun, Cheng Chen

#### Dr Andrew Wilson : Lecturer

,

#### 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 student:** Dimitrios Gerontogiannis

## Postgraduates

#### Jamie Antoun : PhD Student

**Supervisors:** Christian Voigt, Mike Whittaker

#### Vitalijs Brejevs : PhD Student

**Supervisors:** Andy Wand, Brendan Owens

6880

#### Mel Chen : PhD Student

**Supervisor:** Liam Watson

#### Luke Jeffreys : PhD Student

**Research Topic:** Teichmüller dynamics

**Member of other research groups:** Algebra

**Supervisors:** Vaibhav Gadre, Tara Brendle

#### Alan Lazarus : PhD Student

**Supervisor:** Dirk Husmeier

#### Elizaveta Lokteva : PhD Student

**Research Topic:** Slice disks and interactions between 3 and 4 dimensional
topology.

**Supervisors:** Ana Lecuona, Andy Wand

#### Tanushree Shah : PhD Student

**Supervisors:** Ana Lecuona, Brendan Owens, Andy Wand

#### Kellan Steele : PhD Student

**Member of other research groups:** Algebra

**Supervisor:** Gwyn Bellamy

#### Ogier Van Garderen : PhD Student

**Member of other research groups:** Algebra

**Supervisor:** Michael Wemyss

#### Daniel Waite : PhD Student

**Supervisor:** Brendan Owens

## Postgraduate opportunities

### Topological full groups and continuous orbit equivalence (PhD)

**Supervisors:** Xin Li

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

This proposed PhD project is part of a research programme whose aim is to develop connections between C*-algebras, topological dynamics and geometric group theory which emerged recently.

More specifically, the main goal of this project is to study topological full groups, which are in many cases complete invariants for topological dynamical systems up to continuous orbit equivalence. Topological full groups have been the basis for spectacular developments recently since they led to first examples of groups with certain approximation properties, solving long-standing open questions in group theory. The goal of this project would be to systematically study algebraic and analytic properties of topological full groups. This is related to algebraic and analytic properties of topological groupoids, the latter being a unifying theme in topological dynamics and operator algebras. A better understanding of the general construction of topological full group -- which has the potential of solving deep open questions in group theory and dynamics -- goes hand in hand with the study of concrete examples, which arise from a rich variety of sources, for instance from symbolic dynamics, group theory, semigroup theory or number theory.

Another goal of this project is to develop a better understanding of the closely related concept of continuous orbit equivalence for topological dynamical systems. This new notion has not been studied in detail before, and there are many interesting and important questions which are not well-understood, for instance rigidity phenomena. Apart from being interesting on its own right from the point of view of dynamics, the concept of continuous orbit equivalence is also closely related to Cartan subalgebras in C*-algebras and the notion of quasi-isometry in geometric group theory. Hence we expect that progress made in the context of this project will have an important impact on establishing a fruitful interplay between C*-algebras, topological dynamics and the geometry of groups.

The theme of this research project has the potential of shedding some light on long-standing open problems. At the same time, it leads to many interesting and feasible research problems.

## 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

### 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:** Gwyn Bellamy, Ken Brown

### 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, Christian Korff Brendan Owens

### 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

### 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,

### 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:** Christian Voigt, Stuart White, Joachim Zacharias