## 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.

# Information about

## Staff

#### 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 Cohen―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, Samuel (Sam) Lewis, Kellan Steele, Simone Castellan, Ross Paterson

#### Dr Vitalijs Brejevs : Research Associate

**Supervisors:** Andy Wand, Brendan Owens

#### 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:** Tudur Lewis, 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, Leo Kaminski, Georgios Antoniou, Johan Wright

#### Dr Vaibhav Gadre : Lecturer

Teichmuller Dynamics, Mapping Class Groups.

**Member of other research groups:** Algebra

**Research students:** Luke Jeffreys, Tudur Lewis

#### Dr Alan Lazarus : Research Associate

**Supervisor:** Dirk Husmeier

#### Dr Ana Lecuona : Lecturer

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

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

#### Dr Brendan Owens : Senior lecturer

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

**Research staff:** Vitalijs Brejevs

**Research students:** Weizhe Niu, Daniel Waite, Miguel Rodriguez, Tanushree Shah

#### Dr Matthew Pressland : EPSRC Postdoctoral Fellow

representation theory, homological algebra, cluster algebras, algebraic geometry

**Member of other research groups:** Algebra

#### Dr Franco Rota : Research Associate

Algebraic geometry. In particular, derived categories, moduli spaces of sheaves, the McKay correspondence, Bridgeland stability conditions, the stability manifold and its relation with mirror symmetric questions.

**Member of other research groups:** Algebra

#### Dr Greg Stevenson : Lecturer

**Member of other research groups:** Algebra

**Research staff:** James Rowe

**Research students:** David Murphy, Hao Zhang

#### 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

#### Prof Christian Voigt : Professor

Noncommutative geometry; K-theory; Quantum groups

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

**Research student:** Owen Tanner

#### Dr Andy Wand : Senior lecturer

**Research staff:** Vitalijs Brejevs

**Research students:** 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), Hao Zhang, Samuel (Sam) Lewis

#### Prof Mike Whittaker : Professor of Mathematics

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

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

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

#### Dr Andrew Wilson : Lecturer

,

#### Prof Joachim Zacharias : Professor

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

## Postgraduates

#### Simone Castellan : PhD Student

**Research Topic:** Short star products in (Poisson) vertex algebras

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

**Supervisors:** Daniele Valeri, Gwyn Bellamy

#### 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

#### Leo Kaminski : PhD Student

**Research Topic:** Special solutions of WDVV equations

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

**Supervisor:** Mikhail Feigin

#### Samuel (Sam) Lewis : PhD Student

**Research Topic:** The derived geometry of moduli spaces

**Supervisors:** Gwyn Bellamy, Michael Wemyss

#### Tudur Lewis : PhD Student

**Research Topic:** Mapping class groups and related structures.

**Member of other research groups:** Algebra

**Supervisors:** Tara Brendle, Vaibhav Gadre

#### Elizaveta Lokteva : PhD Student

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

**Supervisors:** Ana Lecuona, Andy Wand

#### Weizhe Niu : PhD Student

**Research Topic:** Embedding problems in low-dimensional topology

**Supervisors:** Brendan Owens, Ana Lecuona

#### Tanushree Shah : PhD Student

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

#### Kellan Steele : PhD Student

**Member of other research groups:** Algebra

**Supervisor:** Gwyn Bellamy

#### Owen Tanner : PhD Student

**Research Topic:** Topological full groups and continuous orbit equivalence.

**Member of other research groups:** Algebra

**Supervisors:** Xin Li, Christian Voigt

#### Daniel Waite : PhD Student

**Supervisor:** Brendan Owens

#### Hao Zhang : PhD Student

**Research Topic:** Bridgeland stability manifolds of Divisor to Curve Contractions

**Member of other research groups:** Algebra

**Supervisors:** Michael Wemyss, Greg Stevenson

## Geometry and Topology example research projects

#### 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