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Differential geometry
In differential geometry one studies geometric structures with analytical methods. Undergraduate students encounter this topic normally for the first time in a course on curves and surfaces, where one investigates one- and two-dimensional smooth objects in three-dimensional Euclidean space. Differential geometry is some kind of generalisation to finite or infinite dimensions. However, there are different ways to generalise Euclidean geometry, leading to different branches of differential geometry. Of particular relevance for research at King's College are Riemannian geometry, symplectic geometry and complex geometry. Riemannian geometry is a generalisation of Euclidean geometry to curved spaces such as spheres, projective spaces and hyperbolic spaces. Symplectic geometry is a generalisation of the interpretation of Euclidean space as phase space in Hamiltonian mechanics. Complex geometry is a generalisation of the Euclidean (or Hermitian) geometry of a complex vector space.
Geometric analysis
In geometric analysis one studies differential equations with geometric methods. This topic is closely related to differential geometry. For example, a minimal surface in three-dimensional Euclidean space is a locally area minimising surface. It can be realised for instance by dipping a wire frame into a soap solution. Such surfaces can be described analytically by a system of partial differential equations. The investigation of this system of differential equations with geometric methods leads to significant insight into the structure of minimal surfaces.
Algebraic geometry
In algebraic geometry one studies the interplay of algebraic and geometric objects. For example, the solutions of an algebraic equation form a geometric object known as an algebraic variety. The investigation of algebraic varieties provides important insight into the structure of solutions of algebraic equations. The underlying algebraic structures in algebraic geometry normally satisfy commutative laws. Noncommutative geometry is concerned with objects arising from noncommutative algebraic structures.
Symmetries and Lie theory
A central theme of the research in geometry at King's College is that of symmetries and Lie theory. Lie theory is an area of mathematics originally conceived by the 19th century mathematician Sophus Lie around the notion of continuous symmetries of a geometric object, such as for example the group of rotations preserving a sphere. This so-called 'Lie group' illustrates a basic feature of the theory, involving a group structure (the composition of rotations) on a set that can be understood as a geometric object in its own right (in this case the three-dimensional real projective space). This point of view gave rise to the important notion of a 'Lie algebra' of infinitesimal symmetries, and its enveloping algebra, and was originally associated with differential geometry.
Quantum groups
A modern variation of this theme are the so-called quantum groups, which first arose in the physics literature in the late 1970's, particularly in that dealing with 'integrable' models in statistical mechanics. Mathematically they became to be understood to be certain naturally occurring Hopf algebras depending on a parameter q, such that as q approaches 1 they become the enveloping algebra associated to a corresponding Lie group. This Lie group is called the 'classical limit' of the quantum group, since the way in which quantum groups give rise to Lie groups when q approaches 1 is analogous to the way in which quantum mechanics 'becomes' classical mechanics when Planck's constant approaches zero. Quantum groups have attracted much attention from mathematicians, due largely to their surprising connections with areas of 'classical' mathematics, such as the theory of knots
Algebraic groups
A different variation on this theme of geometry and groups of symmetries is related to algebraic geometry. Most groups of interest in Lie theory are also 'algebraic groups' which can be geometrically described by coordinates with polynomial relations, such as for example 2x2-matrices with determinant 1. Such a description opens the way to associating a group, in this case SL2(K), to any field K. In other words the matrices considered could be complex, real, with entries in a nonarchimedian local field or indeed a finite field (giving rise to a "finite group of Lie type"), with a different but related theory in each case.
Geometry
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DESCRIPTION
Members of the group conduct research in differential geometry, geometric analysis, algebraic geometry and noncommutative geometry. Differential geometry
In differential geometry one studies geometric structures with analytical methods. Undergraduate students encounter this topic normally for the first time in a course on curves and surfaces, where one investigates one- and two-dimensional smooth objects in three-dimensional Euclidean space. Differential geometry is some kind of generalisation to finite or infinite dimensions. However, there are different ways to generalise Euclidean geometry, leading to different branches of differential geometry. Of particular relevance for research at King's College are Riemannian geometry, symplectic geometry and complex geometry. Riemannian geometry is a generalisation of Euclidean geometry to curved spaces such as spheres, projective spaces and hyperbolic spaces. Symplectic geometry is a generalisation of the interpretation of Euclidean space as phase space in Hamiltonian mechanics. Complex geometry is a generalisation of the Euclidean (or Hermitian) geometry of a complex vector space.
Geometric analysis
In geometric analysis one studies differential equations with geometric methods. This topic is closely related to differential geometry. For example, a minimal surface in three-dimensional Euclidean space is a locally area minimising surface. It can be realised for instance by dipping a wire frame into a soap solution. Such surfaces can be described analytically by a system of partial differential equations. The investigation of this system of differential equations with geometric methods leads to significant insight into the structure of minimal surfaces.
Algebraic geometry
In algebraic geometry one studies the interplay of algebraic and geometric objects. For example, the solutions of an algebraic equation form a geometric object known as an algebraic variety. The investigation of algebraic varieties provides important insight into the structure of solutions of algebraic equations. The underlying algebraic structures in algebraic geometry normally satisfy commutative laws. Noncommutative geometry is concerned with objects arising from noncommutative algebraic structures.
Symmetries and Lie theory
A central theme of the research in geometry at King's College is that of symmetries and Lie theory. Lie theory is an area of mathematics originally conceived by the 19th century mathematician Sophus Lie around the notion of continuous symmetries of a geometric object, such as for example the group of rotations preserving a sphere. This so-called 'Lie group' illustrates a basic feature of the theory, involving a group structure (the composition of rotations) on a set that can be understood as a geometric object in its own right (in this case the three-dimensional real projective space). This point of view gave rise to the important notion of a 'Lie algebra' of infinitesimal symmetries, and its enveloping algebra, and was originally associated with differential geometry.
Quantum groups
A modern variation of this theme are the so-called quantum groups, which first arose in the physics literature in the late 1970's, particularly in that dealing with 'integrable' models in statistical mechanics. Mathematically they became to be understood to be certain naturally occurring Hopf algebras depending on a parameter q, such that as q approaches 1 they become the enveloping algebra associated to a corresponding Lie group. This Lie group is called the 'classical limit' of the quantum group, since the way in which quantum groups give rise to Lie groups when q approaches 1 is analogous to the way in which quantum mechanics 'becomes' classical mechanics when Planck's constant approaches zero. Quantum groups have attracted much attention from mathematicians, due largely to their surprising connections with areas of 'classical' mathematics, such as the theory of knots
Algebraic groups
A different variation on this theme of geometry and groups of symmetries is related to algebraic geometry. Most groups of interest in Lie theory are also 'algebraic groups' which can be geometrically described by coordinates with polynomial relations, such as for example 2x2-matrices with determinant 1. Such a description opens the way to associating a group, in this case SL2(K), to any field K. In other words the matrices considered could be complex, real, with entries in a nonarchimedian local field or indeed a finite field (giving rise to a "finite group of Lie type"), with a different but related theory in each case.
Associated research programmes
Mathematics Research MPhil/PhD (Applied Mathematics, Pure Mathematics), option of joint PhD with HKU
Associated staff research interests
Interests:
Infinite dimensional Lie groups and algebras; quantum groups.
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Interests:
Complex, differential, and symplectic geometry.
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My main research interest is geometric analysis with emphasis currently on the theory of minimal and constant mean curvature surfaces.
Minimal surfaces are defined as surfaces which are critical points for the area functional. The mean curvature of a minimal surface, the average of the two principal curvatures, is identically zero. Surfaces which are critical points for the area functional under a volume constraint are instead called constant mean curvature (CMC) surfaces and in fact the average of the two principal curvatures is constant. Not only are there plenty of mathematical examples for both of these surfaces (for instance planes, helicoids and catenoids are minimal surfaces while spheres, cylinders and Delaunay surfaces are non zero CMC surfaces), but they overtly present themselves in the real world. In nature, the shape of a soap film approximates with great accuracy that of a minimal surface while soap bubbles provide the analogous approximation for CMC surfaces. In other words, a soap bubble is the least-area surface that encloses the fixed volume inside.
The results in my papers and my recent interests can be divided into the following areas:
Minimal surfaces are defined as surfaces which are critical points for the area functional. The mean curvature of a minimal surface, the average of the two principal curvatures, is identically zero. Surfaces which are critical points for the area functional under a volume constraint are instead called constant mean curvature (CMC) surfaces and in fact the average of the two principal curvatures is constant. Not only are there plenty of mathematical examples for both of these surfaces (for instance planes, helicoids and catenoids are minimal surfaces while spheres, cylinders and Delaunay surfaces are non zero CMC surfaces), but they overtly present themselves in the real world. In nature, the shape of a soap film approximates with great accuracy that of a minimal surface while soap bubbles provide the analogous approximation for CMC surfaces. In other words, a soap bubble is the least-area surface that encloses the fixed volume inside.
The results in my papers and my recent interests can be divided into the following areas:
- Curvature estimates for nonzero CMC disks embedded in locally homogeneous manifolds.
- The asymptotic geometric structure of nonzero CMC surfaces properly embedded in locally homogeneous manifolds.
- The rigidity of CMC surfaces immersed in locally homogeneous manifolds.
- The geometry of surfaces embedded in Euclidean space with integral bounds for the mean curvature.
- Problems and results on the number, genus, shape and other geometric properties of compact CMC surfaces from the geometry of their boundary.
Tel:
020 7848 2981
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My own research is in geometry, but not restricted to it, and I often deal with problems and methods that are related to other areas. Geometry is one of the traditional areas of mathematics. In ancient civilizations geometry was used for solving practical problems. This changed drastically with the ancient Greeks, where geometry became the center of mathematics. Its axiomatic foundation and the theory based on propositions that were logically deducted from it influenced enormously our way of thinking and lead over the last 2000 years to many fundamental developments and discoveries in mathematics and the sciences. For instance, during the Renaissance, geometry influenced strongly the development of astronomy, geodesy, cartography, mechanics, optics and arts. Geometry nowadays has great impact on our lives through many applications, for example in medicine (diagnosis of cancer, brain imaging), building and construction industry (computer aided design), and manufacturing (robotics). This shows the potential for geometry to get involved in collaborative projects with partners from industry. On the theoretical side, there is a fundamental interaction between geometry and physics, for instance between Euclidean geometry and Newtonian mechanics, or Riemannian geometry and relativity theory, which represent remarkable scientific achievements. Currently there is an intense interaction between geometry and modern physical theories (string theories, supergravity theories, M-theory). Because of all these applications and its particular feature of visualization, geometry is also an extremely attractive area of mathematics for teaching at all levels in higher education.
Tel:
020 7848 2814
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My main work is on geometry of flag varieties, particularly in relation to Gromov-Witten theory and mirror symmetry. I also have a related interest in total positivity, a theory that allows many geometric objects occuring in Lie theory to be defined, in some sense, over the positive real numbers.
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I am interested in understanding modular forms and their generalizations. I take a particular interest in studying congruences between these objects modulo primes numbers. The general way to approach this problem is to study various cohomology groups that naturally arise, and I mostly study those cohomolgy groups which arise over certain algebraic varieties called Shimura varieties.
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These are mainly in the field of differential geometry and centred around the study of manifolds (typically Riemannian or complex ones) defined with reference to the action of a Lie group.
Tel:
020 7848 1222
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Interests:
My research work utilizes concepts from pseudodifferential operators, index theory and non-commutative geometry in the construction of representations of geometric semigroups and categories whose characters define invariants of the background manifold. The character of the index bundle associated to a family of Dirac operators over a Riemann suraface, for example, can be used to enumerate holomorphic maps from the surface into complex projective space, while for manifolds with boundary the characters are connected to gauge groups reoresentations constructed using Grassmannians of elliptic boundary conditions for Dirac operators; this is closely tied in with ideas from quantum and conformal field theory.
Tel:
020 7848 2778
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CONTACTS FOR FURTHER INFORMATION
Postgraduate Administrator, 020 7848 2107, fax 020 7848 2017
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