My interests include: perturbative structures of gauge and gravitational theories, novel presentations of physical observables, aspects of twistor theory and string theory, and asymptotic properties of general relativity.

I am particularly interested in studying the interplay of perturbative and non-perturbative structures using resurgence techniques in the context of topological string theory, mirror symmetry and BPS structures of supersymmetric theories.

Algebraic geometry of generalised $\wp$-functions – geometric structure of partial differential equations on Jacobians of hyperelliptic curves, applications of Lie Algebras in differential equations and Hamiltonian theory – use of representation theory to describe state space for quantum dimer, intertwining relations for differential operators – generalised Laplace maps and Toda lattices, and Gauss-Manin connections – formulation in terms of algebraic invariants.

At the moment, my main interest is in the area of Algebraic Number Theory called Arithmetic Statistics, in particular in the Cohen-Lenstra heuristics on ideal class groups and its many generalisations and variants. I also think about the arithmetic of elliptic curves over number fields, and about the Galois module structure of arithmetic objects, such as rings of integers of number fields, their units, and their higher K-groups. In Representation Theory I think about rational and integral representations of finite groups. I particularly enjoy exploiting connections between representation theory and number theory. Recently, I have also done some research on group actions on the cohomology of low-dimensional manifolds.

I am interested in connections between homotopy theory, physics, and arithmetic geometry. In the past I have done work in pure higher category theory, homotopical algebra, algebraic K-theory, equivariant homotopy theory, stratified homotopy theory, and étale homotopy theory. My latest project is develop forms of quantum field theory in arithmetic. Currently, this involves using higher categorical structures to reimagine the mathematical foundations of quantum field theory in ways that make it applicable to the sorts of geometries that arise in arithmetic.

I have been working on questions in algebraic geometry motivated by string theory and mirror symmetry, in the last few years mostly derived categories and stability conditions. I am particular intrigued when these modern methods connect to classical questions within algebraic geometry, in particular to the classification of algebraic varieties.

I am interested in applications of pure mathematics and mathematical statistics to causal questions in population biomedicine and public health policy. This includes targeted learning, causality, representation learning, and topological data analysis. I’m not recruiting PhD students at this time.

I am interested in supervising PhD students in geometric group theory or geometric topology, and my research has significant connections to complex dynamics, fractal geometry, automata theory, quasiconformal geometry, and symbolic dynamics. Please see my web page or contact me for more information.

I am interested in representation theory and its interaction with geometry. In particular, I am interested in anything that is even remotely related to symplectic reflection algebras (and especially rational Cherednik algebras). To date, “anything” includes symplectic algebraic geometry, D-modules, Calogero-Moser systems, algebraic combinatorics and the representation theory of certain objects of Lie type such as Hecke algebras and q-Schur algebras.

I am interested in developing new topological methods to study groups, algebras, and low dimensional manifolds. Some topics I’m interested in are Artin and Coxeter groups and monoids, topological invariants of diagram algebras and moduli spaces of low dimensional manifolds (specifically links and 3-manifolds). My work lies in the intersection of algebraic topology, geometric topology, and geometric group theory.

I am interested in a variety of different topics including cosmology, particle physics, gravitational waves, black holes, and mathematical physics.

Possible PhD projects include a variety of topics in geometric group theory and low-dimensional topology, with a particular focus on structures related to surfaces, such as mapping class groups and braid groups.

I am birational geometer interested in K-stability and rationality of Fano varieties, subgroups of Cremona groups, polarized cylinders in rational varieties, and relevant classification type problems.

I research equations, decision problems, complexity of algorithms, fixed subgroups, etc. in free, hyperbolic, rel. hyp., Artin, Coxeter groups; Group actions on trees, Λ-trees; Formal languages and growth in groups; Analytic properties of groups; Computation in solvable and nilpotent groups. I’m not recruiting PhD students at this time.

I work in the field of geometric (approximate) group theory. My research interests include groups and spaces with some aspect of negative curvature and their boundaries.

I work on applications of refined and categorified Donaldson-Thomas theory. This includes work in cluster algebras, quantum groups, nonabelian Hodge theory and enumerative geometry of 3-Calabi-Yau varieties and noncommutative varieties.

I am interested in the collective dynamics of quantum field theories with strong interactions. For example, understanding how QFTs thermalize and what is the effective theory that applies once this happens. I am particularly interested in a class of theories called holographic theories, where these questions can be answered by studying the dynamics of black holes.

I research complex geometry, which is roughly the intersection of algebraic and differential geometry. I’m interested in both the analytic and algebro-geometric sides of complex geometry, and at the moment most of my work lies somewhere between the two. Topics I’m currently interested in include Kähler geometry, K-stability, geometric invariant theory and moduli theory, amongst others.

I am a mathematical physicist and my research focuses on quantum and classical integrable models. Specifically, I have been studying integrable quantum spin chains and the Bethe ansatz formulation, quantum groups or quasi-triangular Hopf algebras, braid groups, Hecke and Temperley-Lieb algebras, solutions of the Yang-Baxter and reflection equations, and their set-theoretic versions. I am also interested in classical discrete integrable systems & integrable field theories, in particular the Hamiltonian formulation, dressing methods, solitons and integrable PDEs & ODEs.

My research lies at the interface of theoretical physics and pure mathematics, inspired by the dual questions: What algebraic and geometric structures govern the behavior of elementary quantum particles and fields? Conversely, how can physics intuition be used to derive new predictions and results in algebra, geometry, and other branches of mathematics? Recently I’ve been interested in how quantum groups (and their cousins) arise in representing extended operators in topological or holomorphic QFT’s, and how they govern representation theory of logarithmic vertex operator algebras.

The area of my research is theory of integrable systems in relations with algebra, geometry and mathematical physics. More specifically, I am working on quantum integrable systems of Calogero-Moser type and their relativistic analogues known as Ruijsenaars-Macdonald operators, Baker-Akhiezer functions and Darboux transformations in many dimensions, Hadamard’s problem in the theory of Huygens’ Principle, Coxeter and other hyperplane arrangements, rings of quasi-invariants, representations of Cherednik algebras, Frobenius manifolds, WDVV equations.

I am broadly interested in geometry, topology and dynamics with particular focus on Teichmüller dynamics.

My research interests are in perturbative scattering amplitudes in Quantum Chromodynamics and related gauge theories, motivated by collider physics as well as fundamental questions. The mathematical structure of amplitudes encodes key principles such as unitarity. locality and causality. I am particularly interested in long-distance singularities of amplitudes, and in their behaviour in special kinematic limits where new factorization properties emerge (e.g. Regge or collinear limits). I also study the mathematical structure of Feynman integrals, their asymptotic expansions, their cuts and discontinuities, as well as the differential equations they satisfy.

My research interests centre around the area of non-commutative geometry, quantum gravity and algebraic quantum field theory.

Continuous and discrete integrable systems with a particular interest in noncommutative systems.

I am interested in Lie theoretic representation theory and its connections with combinatorics, geometry and noncommutative things.

I work at the intersection of algebraic geometry and number theory on the behaviour of rational points on algebraic varieties. My past research interests have included K3 surfaces, zero-cycles, Brauer-Manin obstruction, density of rational points, the Hilbert Property and p-adic geometry.

My research concerns holographic dualities in quantum gravity and string theory. In particular I am interested in limits of the AdS/CFT correspondence in which the duality becomes more tractable. In relation to this, I have worked on the foundations and applications of non-Lorentzian geometries, such as Newton-Cartan and Carroll geometries, that arise in limits of string theory and general relativity. For example, I am interested in asymptotically flat spacetimes and the Carroll geometry and symmetries that appear on the boundaries of such spacetimes.

I am working on algebraic geometry, commutative algebra, and combinatorics. My interests include positivity properties of line bundles and syzygies, toric varieties, and tropical varieties.

I am interested in applications of mirror symmetry (an equivalence between algebraic and symplectic geometry) towards problems in tropical geometry and toric varieties. I also think about problems in pure symplectic topology.

My research is on the interface of quantum physics and mathematics. On the physics side I am inspired by fundamental questions in supersymmetric gauge theory and string theory. On the mathematics side this often prompts deep relations with geometry, topology and algebra.

I am interested in strong field effects in both gauge theory and gravity, in particular all-orders and non-perturbative effects. This includes the definition and properties of scattering amplitudes in strong fields and curved spacetime, double copy in background fields, and applications of quantum field theoretic methods to the phenomenology of intense laser-matter interactions.

The problems that interest me the most concern the interactions between low dimensional topology, non-commutative algebra, and representation theory, especially as these interactions arise in mathematical physics and gauge theory.

My research is mostly in arithmetic geometry, the study of structures that have a combination of arithmetic and geometric natures, Recently, I’ve been mostly interested in the interface between arithmetic geometry and quantum field theory. You can see some of my papers listed on my webpage. My main contribution to mathematics is the discovery of the non-abelian method of Chabauty, a theoretical framework for applying ideas of topology, especially homotopy theory, to the algorithmic resolution of Diophantine equations.

I am interested in areas where algebra and representation theory meet problems arising in physical systems. My research focusses on quantum integrable models connected with solutions of the Yang-Baxter equation. The latter include exactly solvable lattice models in statistical mechanics, quantum many body systems and lower dimensional quantum field theories.

My research interests have expanded to include algebraic statistics, phylogenetic algebraic geometry, computational commutative algebra and applied algebraic geometry.

My research interests are in two-dimensional conformal field theory (2D CFT) and its applications. More specifically, I am interested in applying 2D CFT methods to the study of renormalization group (RG) flows in low-dimensional quantum systems. I am working on developing new non-perturbative methods such as constraints from topological defects, renormalisation interfaces, variational methods and numerical Hamiltonian truncation methods that would allow one to chart the topology and geometry of spaces of RG flows.

I am interested in representation theory with a view towards quantum physics. I like affine Hecke algebras and quantum groups, along with their Schur–Weyl dualities and elliptic generalisations. My research focusses on the interplay between such algebraic structures and quantum many-body systems of Calogero–Sutherland and Ruijsenaars–Macdonald type, along with their connections to quantum spin chains.

I am currently working on the connections between partial symmetries, topological groupoids and C*-algebras. These diverse topics are linked by a non-commutative generalization of Stone duality.

My research interests are in low dimensional topology. I particularly like knots and questions that arise from them. My research tends to be more on the algebraic/combinatorial side. I’m not recruiting PhD students at this time.

I am interested in developing tools at the interface of probability, geometry, and category theory for applications in machine learning. This involves establishing new theoretical results, deriving efficient computational algorithms, and working with scientists/engineers to apply to real-world problems. My recent work uses tools from higher geometry to build composable features for surface-valued data (ie. images).

Most of what I do is about category theory and its many applications, including some nonstandard ones such as metric geometry and aspects of both analysis and theoretical ecology. They are loosely unified by the themes of size and measurement. My papers, talks and blog entries can all be found through my home page – https://www.maths.ed.ac.uk/~tl.

I am interested in taking new PhD students. I am working on C*-algebras and their connections to other mathematical disciplines such as topological dynamics or group theory.

I work on general relativity and gravitational theories inspired by string theory and holography. Much of my research focuses on black hole solutions and related geometries in these contexts, with an emphasis on their construction and classification. I have a particular interest in higher-dimensional black holes, supersymmetric black holes, extremal black holes and near-horizon geometries.

My primary interests are in moduli spaces of objects in the derived category of sheaves on low dimensional algebraic varieties, especially Calabi-Yau varieties and more especially abelian and K3 surfaces. I am also interested in triangulated categories, higher triangles and hyperkahler geometry more generally. I am especially fond of the Fourier-Mukai transform as a means to understand the detailed structure of varieties and their moduli spaces of sheaves.

I study algebro-geometric and symplectic structures of various moduli spaces, mostly with links to physics (gauge theory, conformal field theory, integrable systems).

I am interested in understanding groups through their actions on nonpositively curved spaces. Groups that admit such nice actions include Coxeter groups, Artin groups, mapping class groups, and many more. Recently, I have used this geometric approach to understand the algebraic structure (subgroups, automorphisms, etc.) of Artin groups, as well as their geometry.

I am interested in topological problems in soft matter and biology.

My interests lie in Representation Theory and Algebraic Geometry. Specifically, I am interested in loop groups and double loop groups. This involves many aspects: affine Grassmannians, p-adic groups, Hecke algebras, and recently quiver varieties and Coulomb branches to name a few. Through p-adic groups, many of these topics make contact with number theory and the Langlands program.

My research is in low-dimensional topology. I am interested in smooth 3-manifolds, 4-manifolds and their submanifolds, and in the use of gauge-theoretic invariants of manifolds, especially Floer homology groups.

My work on degenerate symplectic structures — origami manifolds and b-symplectic or log-symplectic manifolds — led me to connections with toric topology, contact geometry, and Poisson geometry. More recently I have become interested in quantitative symplectic geometry by working on symplectic embedding problems.

Topology of manifolds, in particular smooth and topological 4-manifolds. Symmetries, classification, and knotted surfaces.

My research has two main strands: derived algebraic geometry, and homotopy theory of algebraic varieties (including arithmetic and motivic aspects). You can find my papers on the arXiv or my homepage. I’m not recruiting PhD students at this time.

My interests lie at the intersection of representation theory, number theory, and algebraic combinatorics. Generally, I like thinking about objects coming from representation theory or number theory in terms of algebraic combinatorics.

My research interests include twistor geometry and integrability, noncommutative geometry and geometric quantization as well as string and M theory.

I am interested in problems lying at the intersection of representation theory (of quantum groups), algebraic geometry (in particular, derived algebraic and derived symplectic geometry) and mathematical physics (supersymmetric field theories, topological field theories, deformation quantization). Currently I am thinking about derived microlocal geometry.

My research interests include topological solitons (particularly magnetic monopoles, vortices and skyrmions), quantum groups and quantum gravity in (2+1) dimensions.

I am interested in representation theory and Poisson geometry. My work concerns quantum groups, cluster algebra, character varieties, and integrable systems. Most of my work concerns applying cluster algebraic methods to problems in quantum groups, integrable systems, higher Teichmüller theory, and low-dimensional topology.

I work on homological mirror symmetry and symplectic topology.

I am a noncommutative ring theorist, specialising in interactions between ring theory and algebraic geometry, particularly the use of geometric techniques to solve problems in algebra. I am particularly interested in birationally commutative algebras (graded algebras that embed in a skew polynomial extension over a commutative ring) and in graded domains of low GK-dimension. Much of my research is motivated by the programme to classify connected graded domains of GK-dimension 3. I am also interested in moduli problems for graded rings and in algebraic geometry, and recently I have begun to work on problems coming from Lie theory, particularly chain conditions in enveloping algebras of infinite-dimensional Lie algebras. I’m not recruiting PhD students at this time.

My research interests lie in the understanding of the emergence of space and time, using the study of black holes and cosmological singularities in either the framework provided by string theory or more modern directions based on AdS/CFT, in which one explores the relation between gravitational physics, quantum information and quantum computation, building on the lessons that quantum entanglement has brought in our understanding of the fabric of spacetime.

I am interested in noncommutative algebra. Polynomial rings, matrix rings. Algebras with finite Gelfand-Kirillov dimension. Domains with finite and infinite Gelfand-Kirillov dimension. Algebras with exponential growth.

I work in geometric group theory, and in particular I specialise in Gromov-hyperbolicity and generalisations, and I especially like to study groups coming from topology such as mapping class groups.

My whole research is about the number 1729. It is the sum of two integer cubes in two interesting ways. I use tools from analysis, geometry and probability to find similar representations for other similar equations-the area goes by the name Manin’s conjecture : https://en.wikipedia.org/wiki/Manin_conjecture

I am interested in the interaction between complex differential geometry and the theory of integrable nonlinear systems. This include the study of equations which encode Gromov-Witten invariants (via Frobenius manifolds) and Donaldson-Thomas invariants (via Joyce structures and hyperKahler geometry).

My research centres around the applications of geometrical and topological methods to problems in quantum field theory and string theory, focusing in particular on techniques drawn from K-theory, noncommutative geometry, algebraic geometry and higher structures.

I am interested in fundamental questions of cosmology

My research area is noncommutative geometry, a modern part of mathematics with links to classical disciplines like topology, differential geometry, and mathematical physics. I am particularly interested in operator algebras, quantum groups, and cyclic homology.

I am interested in various nonperturbative aspects of quantum field theory, including scattering, thermalization, phase transitions, and confinement, often from the perspective of the conformal bootstrap and holography.

I currently supervise/co-supervise several students in various topics across low dimensional topology, focusing mainly on contact topology, aspects of knot theory, and mapping class groups of surfaces. I am happy to discuss potential supervision

My main research interests are in algebraic geometry and its interactions, principally between noncommutative and homological algebra, resolutions of singularities, and the minimal model program. In the process of doing this, I have research interests in 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.

His research interests are primarily in integrable quantum systems, both statistical-mechanical models and quantum field theories. My recent work focuses on solvable lattice models and in particular their analysis in terms of infinite-dimensional symmetry algebras.

My primary research interest is the connection between topological dynamical systems and operator algebras. Starting with a dynamical system we construct C*-algebras that are used to ascertain dynamical invariants in a noncommutative framework. Some of my specific interests in this area include hyperbolic dynamical systems called Smale spaces, self-similar group actions, graph and k-graph algebras, and aperiodic substitution tilings. Using the noncommutative geometry program, developed by Alain Connes, we have used these algebras to construct K-theoretic invariants including Poincaré duality classes, KMS equilibrium states, and spectral triples. I have been particularly interested in dynamical systems that are fractal in nature, with comprehensible local structure but chaotic global behaviour. Aperiodic tilings are extremely useful for further understanding this type of phenomenon and serve as a geometric model for self-similar dynamical systems.

I have wide interests in topics related to interactions between operator algebras and harmonic analysis. My recent research focus is on Calderón–Zygmund operators/Fourier multipliers on non-commutative Lp spaces associated with von Neumann algebras (non-commutative counterparts of measure spaces).

I am interested in supervising further PhD students in the field of operator algebras and related topics. Please contact me by e-mail if you would like to do a PhD under my supervision/co-supervision. I am happy to suggest topics in C*-algebra theory and dynamical systems, classification of simple C*-algebras, approximation properties of groups and operator algebras, K-theory and related topics.