# 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.

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

#### 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 Senior 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 students:** Niall Hird, Tomasz Przezdziecki, Kellan Steele, Kellan Steele

#### 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

**Research student:** Ogier Van Garderen

#### 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 Jamie Gabe Honorary Research Fellow

**Member of other research groups:** Mathematical Biology, Algebra

#### Dr Vaibhav Gadre Lecturer

Teichmuller Dynamics, Mapping Class Groups.

**Member of other research groups:** Algebra

**Research student:** Luke Jeffreys

#### Dr Dimitra Kosta LKAS Fellowship

Birational geometry; toric geometry; log canonical thersholds; resolution of singularities; Fano varieties.

**Member of other research groups:** Statistical Methodology, Environmental Statistics, Biostatistics and Statistical Genetics, Algebra

#### 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 students:** Daniel Waite, Vitalijs Brejevs

#### Dr Greg Stevenson Lecturer

**Member of other research groups:** Algebra

#### 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:** Jamie Antoun, Andrew Monk, Samuel Evington

#### Dr Andy Wand Senior lecturer

**Research student:** Vitalijs Brejevs

#### 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:** Theo Raedschelders

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

#### 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, Jamie Antoun

**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 students:** Luke Hamblin, 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

#### Alan Lazarus PhD Student

**Supervisors:** Dirk Husmeier, Theo Papamarkou

#### 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

#### Kellan Steele PhD Student

**Member of other research groups:** Algebra

**Supervisor:** Gwyn Bellamy

#### Ogier Van Garderen PhD Student

**Member of other research groups:** Algebra

**Supervisors:** Michael Wemyss, Ben Davison

#### Daniel Waite PhD Student

**Supervisor:** Brendan Owens

## Postgraduate opportunities

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

**Supervisors:** Mike Whittaker

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

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.

Self-similar groups are an important and active new area of group theory. The most famous example is the Grigorchuk group, which was the first known example of a group with intermediate growth. This makes investigating C*-algebras associated to them particularly interesting. In particular, these groups are often defined by their action on a graph, and the associated C*-algebra encodes both the group and path space of the graph in a single algebraic object, as well as the interaction between them.

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

**Supervisors:** Mike Whittaker

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

A tiling is a collection of subsets of the plane, called tiles, for which any intersection of the interiors of two distinct tiles is empty and whose union is all of the plane. A tiling said to be aperiodic if it lacks translational periodicity. The most common method of producing aperiodic tilings is to use a substitution rule; a method for breaking each tile into smaller pieces, each of which is a scaled down copy of one of the original tiles, and then expanding so that each tile is congruent to one of the original tiles.

This project will focus on a natural class of operator algebras associated with an aperiodic substitution tiling. These algebras were first considered by Kellendonk and reflect the symmetries of a tiling in an algebraic object that allows up to consider invariants in a noncommutative framework. A key area of study are spectral triples associated with aperiodic tilings, which allow us to think of tilings as noncommutative geometric objects.

## 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