Department of Mathematics

Spectral theory of PDEs; Schrodinger operators; scattering theory.

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Analysis: Partial differential and integral equations, microlocal analysis, global analysis, complex variables, spectral theory, functional analysis. Applications: Fluid dynamics, elasticity theory.

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Geometric index theory of elliptic operators; applications to quantum field theory.

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My mathematical interests lie between spectral theory and geometry. I have studied spectral and other properties of differential operators on manifolds in connection with geometric properties of the manifolds. Some of my recent work is connected with convex analysis, the abstract theory of operators on a Hilbert space and their applications to problems of spectral theory.

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Theory of disordered systems; processes on complex networks; non-equilibrium statistical mechanics; econophysics

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Applications of Statistical Mechanics in a broad range of fields including Soft Condensed Matter (fracture, friction), Packing Problems, Random Matrix Theory and methods in Statistical Mechanics (the Self-Consistent Expansion).

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Statistical mechanics of disordered systems; theory of minority games; metabolic networks; quantam integrable models.

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Statistical mechanics of disordered systems (soft materials; rheology, polydispersity effects on phase behaviour; glassy dynamics), statistical inference and learning processes including Gaussian processes, support vector machines, non-parametric Bayesian inference

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Physics of glassy systems, neural networks and risk modelling.

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Portfolio management in markets with "friction factors" (taxes and transaction costs), utility maximisation, optimal stopping and stochastic control problems, numerical methods for free-boundary problems.

Pricing, risk measurement, credit, counterparty risk, and stochastic models for commodities and inflation

Asymptotics for stochastic volatility models and Lévy processes with an emphasis on large deviations theory, and diffusion-type processes.

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Financial risk management, financial econometrics, mathematical finance and the development of computational techniques for risk management.

• Econophysics
• Application of methods from Statistical Physics to Finance
• Complex Systems
• Science of Networks

Others:
• Granular materials
• Numerical simulations of diffusive processes for the analysis of the magnetic properties in new materials
• Strongly Correlated Electronic Systems and High-Tc Superconductivity

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Infinite dimensional Lie groups and algebras; quantum groups.

Complex, differential, and symplectic geometry.

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:

• 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.

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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.

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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.

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.

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.

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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.

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+44 020 7848 2488

Number theory: Galois module structure, arithmetic algebraic geometry.

Number theory, especially modular forms and Galois representations.

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Number theory and Arithmetic geometry with a special interest in Iwasawa theory.

mahesh.kakde@kcl.ac.uk

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.

Supermanifold theory; applications to supersymmetry; supergravity, BRST theory and geometry.

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Conformal field theory; strings; branes.

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My research interest is in quantum field theory, in particular the integrable or conformal kind. I'm quite interested in looking at QFT as a powerful theory for emergent fluctuations (collective behaviours) in many-body systems. This point of view connects it to condensed matter and statistical systems, but also provides a fundamental understanding of renormalisation group and an alternative view on the fundamental particles of physics. One of my present research paths is to develop the connection between mathematical measures (conformal loop ensembles) describing emergent fluctuations for a large class of statistical models, and the algebraic construction of conformal field theory based on the usual tools and ideas of QFT.

Other research paths include: the physics and mathematical formulation of quantum impurities out of equilibrium; the entanglement entropy in extended quantum systems; the relation between classical integrability and twist fields in quantum field theory models of free particles; as well as various fundamental aspects of relativistic integrable quantum field theory and conformal field theory, including the study of form factors and of vertex operator algebras.

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String Theory, Supergravity, Supersymmetric Gauge Theories, AdS/CFT Correspondence, Geometry.

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My main interests are in string theory, supergravity and M-theory. I currently work on the classification of supersymmetric supergravity backgrounds.

Two-dimensional quantum field theory, in particular conformal field theory and integrable field theories.

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String theory, supersymmetric field theories and exactly soluble models.

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Supersymmetry; string theory and M-theory.

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Field theories, supersymmetry, string theories and the underlying symmetries of nature

Supersymmetric Gauge theory, String theory, String/SUSY phenomenology, N=4 Super-Yang Mills, Integrability.