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Christine O'Keefe and Mathai Varghese share the Australian Mathematical Society 2000 Medal
The Medal of the Australian Mathematical Society for 2000 is shared between Dr Christine O'Keefe and Dr Mathai Varghese. Both work in geometry, although their specialties within geometry are very different, and the dual medal highlights Australia's strength in this area of mathematics.
Dr O'Keefe's specialty is finite geometry, which involves generalisation of smooth or continuous features like curves and surfaces from the ordinary geometry of Euclid and Pythagoras to discrete or non-continuous objects. She has proved a number of important results in finite geometry, involving structures such as hyper-ovals and generalised quadrangles. Finite geometry, and O'Keefe's work in particular, has potential applications in information security, an area of vital importance in the development of e-commerce and the internet. (Research citation)
Dr Varghese works in the area of continuous or differentiable geometry. Many of his results involve the geometry of manifolds, familiar examples of which are again curves and surfaces. However, mathematicians are not restricted to three dimensions, and much of Varghese's work is in spaces of higher dimension, and sometimes even in infinitely many dimensions. Varghese has proved a number of important results related to classifying manifolds with different geometric structures on them. His work finds applications in physics, in particular to string theory and the pursuit of what physicists like to call `the theory of everything'. (Research citation)
Finite geometry is the study of finite analogues of the classical geometric structures such as projective and polar spaces, curves and surfaces. The major problems are the construction, characterisation and eventual classification of structures, and the investigation of the network of connections between them, with the fundamental question in mind: which properties transfer from the classical to the finite case, and which new properties arise? The most important projective and polar spaces are those coordinatised by finite fields, and the basic geometric objects are the finite analogues of the conics and quadrics, namely arcs and ovoids. These geometric objects have been the subject of intense study since the 1950s and the remarkable work of the Italian geometer Beniamino Segre. They have been used in many ways, for example, in constructing new designs, understanding finite simple groups, studying maximum distance-separable and algebraic-geometric codes and constructing efficient solutions to many problems in information security, such as secret sharing, authentication and key distribution.
During her PhD candidature, O'Keefe began to work on spreads in finite projective spaces, resulting in several published articles. Since then, she has carried this work on spreads in two different directions; the first being to investigate the construction of designs from spreads and the second being to generalise spreads to covers. In 1989 O'Keefe began work on ovals, hyperovals and ovoids in projective spaces. These are incredibly important structures for several reasons, but mostly because they can be thought of as sort of building blocks for many other geometric structures. Despite its clear importance, the classification problem has defied attack since the first non-classical examples were found in the 1950s. Motivated by a remarkable result of Penttila and Praeger, O'Keefe embarked on a highly successful partnership with Penttila.
Among many other contributions in this area, they attracted the attention of the international community by obtaining classification results in small order fields and strong characterisations. The so-called O'Keefe-Penttila hyperoval, constructed in 1992, has exploded the conjecture that hyperovals could be classified by their connection with certain flocks and hence is the seed of a new theory of hyperovals, currently under investigation by an international team of experts.
Ovoids are the building blocks for one of the few classes of generalized quadrangles with extremal parameters Thus O'Keefe's results on ovoids have direct consequences for these generalized quadrangles. Another class is the flock quadrangles, important because of the wealth of associated structures and the relative paucity of examples. In joint work with Penttila, O'Keefe has made significant progress on the highly-sought classification problem for these generalized quadrangles, showing that a solution of the problem might in fact be feasible. In important fundamental work, O'Keefe and Penttila introduced a new, simpler method for the calculation of automorphism groups of flock quadrangles in characteristic 2, and illustrated the power or this perspective by obtaining new characterisations of three of the four known families of flock quadrangles.
Ovoids are also the building blocks for the Buekenhout-Metz unitals, and a major problem is to prove either that all unitals are Buekenhout-Metz or to construct a new example which is not. This problem has been attracting considerable interest, including O'Keefee's characterisations and other results including a geometric link with the underlying ovoid.
There are five classes of finite classical polar spaces, one of which is the family Q(2n,q) of parabolic quadrics. Ovoids are absolutely fundamental objects in polar spaces, since they are extremal objects and arise in many constructions and other contexts. The basic question of the existence or otherwise of ovoids in Q(2n,q) was settled for q even and n>2 by Thas in 1981 but the case of q odd and n>3 is much more difficult and remained open. O'Keefe, together with Thas, proved nonexistence under a further simple hypothesis, and completely settled the cases q = 5, 7, 11 and 13.
In further contributions in finite geometry and design theory, O'Keefe has obtained results in relation to blocking sets, (k,r)-sets, block-transitive point-imprimitive and other designs, arcs, flocks, defining sets, permutation polynomials and quasi-quadrics.
In the early 90's, O'Keefe became interested in the application of her geometric expertise in the area of information security. She has a very productive collaboration with W. A. Jackson and K. Martin in the area of information security, with particular success in the construction of optimal geometric secret sharing schemes with various capabilities. In a ground-breaking paper O'Keefe and her collaborators completely remove the traditional use of a mutually trusted authority to generate and distribute shares in the setup phase of a secret sharing scheme. In their innovative protocol the participants generate their own shares and communicate sub-shares to the other participants. They proposed new efficiency measures, proved bounds on these and provided optimal constructions for schemes in the important case of thre shold schemes.
1. Superconnections, Thom classes and equivariant differential forms (with D. G. Quillen).
This is a major paper published in Topology shortly after Varghese completed his PhD at MIT. Using the superconnection formalism, he and Quillen obtained a refinement of the Riemann-Roch formula linking the Thom classes in K-theory and cohomology, as an equality on the level of differential forms. This has an interpretation in physics as computing the classical and quantum (super) partition functions for the Fermion analogue of a harmonic oscillator with source term. In particular, they obtain a nice Gaussian shaped representative of the Thom class in cohomology, which is peaked along the zero section. Its universal representative is obtained using the machinery of equivariant differential forms. This formalism has been extraordinarily influential, being widely used in index theory and in topological field theories in mathematical physics.
2. Determinant lines, Von Neumann algebras and torsion (with A. Carey and M. Farber).
Mathai Varghese is one of the inventors of the notion of torsion for operators on covering spaces of manifolds. He has worked extensively on invariants which arise from generalising the work of Atiyah and Singer on the index theorem. This work has culminated in this paper in Crelle's Journal which studies both combinatorial and analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds. These are viewed as volume forms on the (infinite dimensional) reduced homology and cohomology. These torsion invariants specialize to the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional case. Under the assumption that the homology vanishes, the determinant line can be canonically identified with the reals, and these torsion invariants specialize to the torsion invariants previously constructed by Carey, Mathai and Lott. The main theorem in this game due to Burghelea et al can be reformulated as stating equality between two volume forms (the combinatorial and the analytic) on the reduced cohomology. torsion is that it was shown by V. Mathai and J. Lott to be proportional to hyperbolic volume for hyperbolic manifolds.
3. Equivariant holomorphic Morse inequalities I : a heat-kernel proof. (with S. Wu).
This important contribution appeared in the Journal of Differential Geometry. It gives a heat-kernel proof of the equivariant holomorphic Morse inequalities, using some techniques developed by Bismut and Lebeau. These inequalities, first obtained by Witten using a different argument, produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomologies in terms of the data of the fixed points. This work was subsequently extended by Wu to different general contexts.
4. Approximating invariants on amenable covering spaces: a combinatorial approach (with J. Dodziuk).
In this paper in the Journal of Functional Analysis, the Betti numbers of an amenable covering space are shown to be approximated by the average Betti numbers of a regular regular exhaustion, proving earlier conjectures of the authors. These results establish special cases of some general conjectures for spectral theory of combinatorial Laplacians for covering spaces.
5. Quantum Hall effect on the hyperbolic plane (with A. Carey, K. Hannabus and P. McCann).
This paper in Communications in Mathematical Physics extends some ideas of A. Connes and others concerning analogues of the quantum Hall effect which are defined on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. The novel technique is a twisted analogue of the Kasparov map, which enables the use of the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case. Mathai in a subsequent series of papers has obtained some remarkable generalisations of these results including a method to compute the K-theory of twisted group -algebras in terms of the (twisted) K-theory of certain classifying spaces. This can be viewed as a first step towards understanding the structure theory of these algebras and also a first step towards their classification. A main achievement is to use this method to study obstructions to the existence of Riemannian metrics of positive scalar curvature on a compact spin manifold (the Gromov-Lawson-Rosenberg conjecture), and to establish a special case of this conjecture using these methods, putting in particular a result due to Gromov into a -algebraic context. In a recent preprint he has generalized these results using higher dimensional group cocycles.
6. Noncommutative Bloch theory (with M. Marcolli).
Long ago, Novikov and Shubin initiated a mathematical study of Schrodinger operators which involve almost periodic potentials or magnetic potentials. This study continues today and in this paper an important new step is taken by studying the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. The authors obtain qualitative results on real and complex hyperbolic spaces in 2 and 4 dimensions, related to generalizations of the Bethe-Sommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompact group. This generalises a previous application of the Baum-Connes conjecture by Varghese which obtained complete results for the torsion-free case.
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