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# Geometry

**Differential geometry**

In differential geometry one studies geometric structures with analytical methods. Undergraduates usually encounter this topic for the first time in a course on curves and surfaces, in which one investigates smooth objects in ordinary space. Differential geometry is a kind of generalisation to finite or infinite dimensions, in which the key concept is that of a *differentiable manifold*. There are different ways to generalise Euclidean geometry, leading to different branches of the subject. Of particular relevance for research at King's is Riemannian, symplectic and twistor geometry. Riemannian geometry allows one to study curved spaces such as spheres and their analogues with negative curvature, namely hyperbolic spaces. Symplectic geometry extends the notion of phase space in Hamiltonian mechanics. Twistor geometry is based on the use of *complex manifolds *to describe objects in 4 dimensions, and reflects the essential role that complex numbers play in quantum theory.

**Geometric analysis**

In geometric analysis, one studies differential equations with geometric methods. This topic is closely related to differential geometry, and in particular to the study of length, area, volume and other measures. For example, a*minimal surface* in 3-dimensional Euclidean space is a surface small parts of which minimise surface area. Minimal surfaces can be realised by dipping a wire frame into a soap solution, and are described analytically by a system of partial differential equations. This investigation of this system of differential equations with geometric methods leads to significant insight into the structure of minimal surfaces. Similar techniques can be applied to surfaces with *constant mean curvature* (that arises from soap bubbles rather than films).

**Algebraic geometry**

In algebraic geometry, one studies the interplay of algebraic and geometric objects, often in complex projective space. The solutions of one or more polynomial equations form a geometric object known as an*algebraic variety*. The investigation of algebraic varieties provides important insight into the structure of solutions of very complicated equations. The underlying structures in algebraic geometry normally satisfy commutative laws, though there is an expanding field of non-commutative geometry, including such topics as quantum groups, which have attracted much attention due to their surprising connections with areas of mathematics such as knot theory.

**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 in space. This simple 'Lie group' illustrates a basic feature of the theory, involving a multiplication (the composition of rotations) on a set that can be understood as a geometric object in its own right (in this case the 3-dimensional real projective space). There is also an algebraic counterpart, a so-called*Lie algebra* that consists of infinitesimal symmetries. Most groups of interest in Lie theory are also 'algebraic groups' which can be described by coordinates with polynomial relations, as is the case for 2x2-matrices with determinant 1. Such a description allows one to work in any field, and to study *representations* of the corresponding symmetries.

DESCRIPTION

Members of the group conduct research in differential geometry, geometric analysis, algebraic geometry and Lie groups. In geometric analysis, one studies differential equations with geometric methods. This topic is closely related to differential geometry, and in particular to the study of length, area, volume and other measures. For example, a

In algebraic geometry, one studies the interplay of algebraic and geometric objects, often in complex projective space. The solutions of one or more polynomial equations form a geometric object known as an

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 in space. This simple 'Lie group' illustrates a basic feature of the theory, involving a multiplication (the composition of rotations) on a set that can be understood as a geometric object in its own right (in this case the 3-dimensional real projective space). There is also an algebraic counterpart, a so-called

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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Interests:

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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Interests:

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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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 2217

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