One of the oldest disciplines in mathematics, geometry goes back to the work of Pythagoras and Euclid. Geometry in modern mathematics takes on a multitude of forms, many of which are studied here at UNSW.
The interests of the Geometry research group are very broad, including algebraic geometry, differential geometry, hyperbolic geometry, Banach space geometry and noncommutative geometry.
Algebraic geometry studies solutions to polynomial equations using techniques from algebra, geometry, topology and analysis. This rich subject is intimately connected to number theory. Differential geometry studies manifolds, a key concept used to formulate many of the ideas in physics, from relativity to string theory. Hyperbolic geometry was developed in the nineteenth century and is a geometry in which there are many lines parallel to a given line through a given point.
Noncommutative geometry is inspired by the tantalising prospect of extending the natural duality between commutative algebra and geometry to the noncommutative setting. It has revolutionised the theory of operator algebras and noncommutative noetherian rings.
The group also performs research into problems related to physics, including operator algebras, path integrals and quantization.
Works in noncommutative algebraic geometry, a branch of mathematics which explores various connections between non-commutative algebra and algebraic geometry. His main interest is in noncommutative surfaces, the simplest examples of which are sheaves of algebras on projective surfaces called orders. His work includes noncommutative versions of Mori's minimal model program and the study of moduli spaces for noncommutative algebras.
Interested in sub-Finsler geometry, the differential geometry associated to systems of differential equations, and in the geometry of Lie groups, particularly in the question of when algebraic conditions ensure smoothness and so allow the application of differential geometric methods.
Now semi-retired, is currently working in trace formulas in Hochschild cohomology of algebraic varieties. He has long-standing interests in trace formulas in adelic geometry and geometric models associated to modular representations of finite groups.
His work in functional analysis has had an increasingly geometric aspect. In operator theory his research investigates the relationship between the geometry of compact sets in the plane and the algebraic properties of certain algebras of functions defined on such sets. In a quite different direction he has recently been investigating geometric properties (such as generalised roundness) in the metric space setting.
He's a computational number theorist, whose interest in geometry arises because many problems in number theory can now be turned into algebraic geometric questions. For example, Fermat's Last Theorem which was proved via a very difficult study of elliptic curves, which are the curves that arise as the zero sets of polynomials of degree 3 in two variables.
Works on classical and quantum superintegrable systems. These are natural Hamiltonian systems having the maximum number of independent symmetries and as a result possess many useful and interesting properties. The most celebrated examples are the harmonic oscillator and Kepler-Coulomb system, but recently many more examples have been found.
Interested in scattering theory of mathematical physics. Her particular interest is in the area of spectral analysis of first-order differential operator, specifically Dirac operators, and spectral shift function associated with them.
Interested in sub-Riemannian geometry, which is the metric geometry for non-holonomic mechanical systems, and in Lie groups and Lie algebras theory. Lie groups provide a case study for sub-Riemannian spaces, in which the metric is left-invariant. In particular, he is interested in the mappings between such spaces that preserve certain properties of the metric (e.g., isometries, conformal and quasiconformal maps).
His interests are in the area of General Relativity, particularly in exact solutions of the Einstein Field Equations, their symmetries and interpretation. He is also interested in geometric aspects of mathematical physics and the history of mathematical physics.
Noncommutative geometry is partly motivated by the study of solutions of integral and differential equations. Fedor is interested in the functional-analytical aspects of such a study and its applications to noncommutative geometry.
Works in algebraic geometry, in particular in the theory of cohomological invariants associated to algebraic varieties. Cohomological invariants like motivic cohomology, Lawson homology or etale cohomology can be used in problems of classification of algebraic varieties. For example, Artin and Mumford give example of a unirational variety that is not rational by using etale cohomology.
The discoverer of Rational Trigonometry, a new completely algebraic framework for trigonometry and Euclidean geometry. This is developed in his 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry. He has been extending this approach to Hyperbolic Geometry, to a new threefold symmetry in planar geometry called Chromogeometry, to Spherical Geometry and recently to Triangle Geometry. He is particularly interested in Pascal's Hexagrammum Mysticum. His video series WildTrig and UnivHypGeom at his YouTube channel are bringing geometry to a wide audience.
Studies functions and the functions of functions, with applications ranging from quantum physics to signal processing.