Twareque Ali's research during the year 1996-97 was centred around the theory of square integrable group representations and their applications to wavelet analysis. Specifically, using representations of the full Poincaré group, a class of wavelet-like transforms has been constructed, which were subsequently used to develop remarkably fast algorithms for analysing two and three-dimensional images. On the theoretical side, a generalization of the classical theorem on the square-integrability of group representations has been obtained which incorporates vector coherent states and square integrability modulo a subgroup. The extended theorem has applications to the theory of matrix-kernels appearing in Hilbert spaces of holomorphic functions of several variables.
P. Arminjon's main research interest lies in the domain of numerical methods for nonlinear hyperbolic systems, with applications to engineering problems in gas dynamics and electrostatics/electrodynamics. For transonic/supersonic compressible flows, P. Arminjon studies, with his collaborators, A. Dervieux and M.C. Viallon, the design and numerical analysis of high accuracy finite difference, finite element or finite volume methods, and their application to typical flows arising in aerodynamics and aerospace engineering. Recently, they have obtained a family of non-oscillatory 2nd-order accurate schemes based on:
In joint work with M.C. Viallon, they have recently proved the convergence of this latter scheme for a linear conservation equation, and they are presently extending the proof to the nonlinear case.
Nonlinear dynamics gives an interpretation of complex changes in physiological rhythms (as bifurcations) when the values of the control parameters are modified. The theory leads to predictions for the possible behaviours in experimental settings and gives a unified explanation for the various regimes. Bélair's work has concentrated on nonlinear delayed feedback in control and in hormonal and neuromuscular system oscillations, stressing the role of the delay, the multiple feedback loops and the variable delays in the generation of periodic (oscillatory) or irregular behaviours.
Systems of neural networks of small size were analysed, with an emphasis on the combined role of time delays incorporated into the model to take into account processing time and the architecture of the network, with a view to establishing the deleterious effects of the delays.
In collaboration with J. Mahaffy and M.Mackey, a model for erythropoiesiswas developed which incorporates a constant rate destruction mechanism. This work is ongoing, with an attempt to incorporate recent discoveries on thrombopoietin.
Together with researchers in pharmacology, a project was started with a view to building models which incorporate transient regimes in their representations of absorption mechanisms.
Machine learning algorithms allow a computer to learn from examples. This field of research is at the intersection of artificial intelligence, statistical inference, and numerical optimization. Learning algorithms are particularly useful when we don't have enough explicit knowledge about a problem in order to directly write a program that solves it, but where enough examples illustrating the task to perform are available. In this context, learning means choosing a function from a set of functions according to the expectation of a criterion (the quality of the solution found by the computer on a particular example). However, since the true probability distribution of the examples is unknown, this expectation cannot be computed, only approximated by its empirical value on the observed data. The real difficulty of learning is therefore to generalize, i.e., to transfer information from the observed data to new cases. The research of Yoshua Bengio is focused on certain types of learning algorithms (in particular, artificial neural networks, and hidden Markov models) and their applications (to pattern recognition, speech recognition, computer vision, monitoring industrial processes, and prediction and decision taking from financial time-series).
Nantel Bergeron works in three main areas:
Abraham Broer is primarily interested in connections between algebraic geometry and representation theory. He currently studies decomposition classes: these are subvarieties of a semisimple Lie algebra with elements having similar Jordan normal forms. In this study, interesting variations of determinantal varieties and cotangent bundles of flag varieties come up.
Important tools in this study are the various vanishing theorems he proved for the higher sheaf cohomology and Dolbeault cohomology of vector bundles on spaces like the cotangent bundle of a flag variety.
Applications are in normality problems of nilpotent varieties; in the structure of rings of differential operators on homogeneous spaces, in particular their multiplicities and PRV-determinants; in Springer's representations of the Weyl group on cohomology groups; and in the combinatorial theory of the hyperplane arrangements coming up in Lie theory.
A different interest lies in invariant theory. He recently found some basic results on the invariant theory of finite groups acting over a field of positive characteristic.
A recent paper [IEEE Transactions on Automatic Control 42 (1997) 1394-1407] written in collaboration with Yu. S. Ledyaev, E. Sontag and A. Subbotin solves a well-known and long-standing question in control theory: we give a constructive proof of the fact that any asymptotically controllable system admits a retour d'état which stabilises it. In general, it is necessary that this retour d'état be discontinuous. One can then prove its robustness by some new and apparently very promising techniques, and establish interesting relationships with the regularity of eventual Liapunov functions.
Project 1: Shape optimization, intrinsic differential geometry, and asymptotic theory of thin shells
The central theme of this research program is the optimization with respect to the shape or the geometry of a domain on which is defined a or a system of partial differential equations. This type of problem is generic in shape and structural optimization (aeronautics, thermal problems, image processing, etc.). At the theoretical level it is necessary to give a meaning to derivatives with respect to the shape and construct appropriate topologies on families of subsets. Among these, topologies induced by distance functions or families of functions parametrised by sets and embedded in a function space are of special interest. For instance, the algebraic distance function provides a very nice tool for the differential calculus on differential submanifolds. It makes it possible to deal with the theory of shells in a completely intrinsic fashion and extend the shape calculus to differential equations on submanifolds.
In a first step we have been able to give a completely intrinsic theory of thin shells for polynomial models. This led to the development of an intrinsic theory of Sobolev spaces on C1,1 submanifolds of RN as well as to the associated functional analysis and the analogues of the classical inequalities (Korn, Poincaré, etc.).
In a second step, we have developed intrinsic tools to deal with the asymptotic problem (singular perturbation with respect to the thickness). Each polynomial model converges to an asymptotic model which is a coupled system of two equations: a membrane equation and a bending equation. For the P(2,1) model we recover the asymptotic equations generally accepted in mechanics. However we get the general case in which a new coupling term appears in the bending equation. This generalizes the two known cases: the plate and the bending dominated case. The new coupling term involves the mean curvature and the membrane energy, both zero in the so-far-known cases.
Within this program, A. Raoult (Grenoble, France), M. Bernadou (Pole Léonard de Vinci and INRIA, France) and M. Fortin (Laval) visited the CRM. The doctoral student J. Zhao (McGill) has completed his Doctoral thesis on the intrinsic theory of nonlinear shells and the Master student S. Roy (Montréal) works on the construction and approximation of submanifolds from distance functions. This constitutes an essential step in the optimal design of shells.
Project 2: Numerical methods in impulsive differential equations
In order to preserve some accuracy, the solution of impulsive differential equations is usually approximated by low order variable step methods where the discretization nodes are chosen at the occurrence of impulses. In impulsive control problems this typically leads to a slow progression of the method or even a standstill when a train of impulses has an accumulation point. Joint work with F. Dubeau (Sherbrooke) has led to the use of high-order one step methods on a fixed node discretization. Despite the fact that the solution is at most of bounded variation (not even continuous), we obtain a nodal convergence rate which linearly increases with the order of the approximation scheme. We theoretically and numerically show that the predicted orders are optimal. This goes against the established perception that when the solution is nonsmooth, the order of the error cannot be improved by going to higher order schemes. So these results provide an interesting approach to impulsive control problems which typically occur in aeronautics.
Project 3: Numerical methods for thin shells
The numerical approximation of the solution of thin shells represents an important international research activity and an even more important part of the software development business for structures. There is also an urgent need for models and methods to handle multilayer shells made of composite materials or shells controlled by piezoelectric sensors and actuators. A specific GIREF based project for the design of the body of all-purpose vehicles is in the planning phase. (GIREF is the Groupe Interdisciplinaire de Recherche en Élements Finis.)
Cooperation has been initiated with M. Fortin (Laval) in an effort to combine the numerical and software development expertise of the GIREF and the new intrinsic tools developed in Montréal. Thanks to a CRM-CERCA postdoctoral student, Gen Yang, the program is now progressing well.
One critical problem in the approximation of thin shells is the numerical locking phenomenon which partly arises from the fact that we are dealing with a singular perturbation problem. F. Brezzi (Pavia, Italy), D. Arnold (Penn. State, USA) and their collaborators have found a clever and constructive way to get around this difficulty for plates. We have been able to show that the constructions extend to thin shells and that a whole family of mixed locking-free methods can now be used. Yet these techniques are also strongly dependent on the knowledge of the asymptotic system. As we have now identified the general asymptotic model, it becomes possible to adapt this work to still problematic cases. Other types of approximations are also currently being considered.
Using supersymmetry it is possible to generalize in a non-trivial way the Korteweg-de Vries equation (KdV) to an integrable system of two coupled differential equations (Mathieu). Knowing that the supersymmetry can itself be extended (parasupersymmetry and fractional supersymmetry [Durand, Vinet]), it is natural to look for generalizations to integrable systems of several coupled differential equations. The formalism of fractional superspace introduced by Durand allows such a generalization in a natural way. This result is reached using the fractional extension of supersymmetry, the Hamiltonian structure of the fractional pseudo-classical mechanics and the fractional generalization of superextension of Virasoro algebra (and/or its q-deformations).
Richard Fournier and his collaborator (St. Ruscheweyh) are working at describing explicitly the values omitted by various normalized classes of univalent functions on the unit disk in the complex plane. It seems that these values might be described in simple terms by certain combinations of Taylor coefficients of the functions. Moreover it appears that the omitted values characterize, in a certain sense, various classes of univalent functions, for example the convex ones. This work had led to new inequalities on Taylor coefficients and the modulus of convex conformal transformations. It is hoped that these results can be used to solve some problems on homographic transformations of convex univalent functions.
Anyons are two-dimensional objects exhibiting fractional statistics, whose exchange properties are governed by the braid group rather than the permutation group. These anyonic structures seem to be a natural framework in which to realise the usual quantum groups and their supersymmetric analogues. This was done during the year for the quantum unitary affine groups. Another project, joint with V. Hussin, was the classification of the quantum deformations of the supergroup Gl(1|1), using the technique of R-matrices.
In the study of integrable systems as well as in the classification of conformal field theories, W-algebras play a central role. Their q-deformations appear in the quantisation of systems such as the Calogero-Moser and the Ruijsenaars-Schneider models. Particular emphasis was placed on the clarification of the relation between elliptic algebras and q -deformations of W-algebras. This gave rise to several publications. In the same area, a joint project with L. Vinet concerned the application of finite W-algebras to the study of Lie algebras.
The critical point theory of continuously differentiable functionals and the set-valued analysis are two important and active domains in mathematics. Marlène Frigon's work is concerned with the development of a critical point theory for set-valued functionals. This theory will then be applied to partial differential inclusions.
Langis Gagnon is principal specialist at Lockheed Martin Canada and is affiliated to the CRM. His research bears on the analysis of radar and infrared images and data fusion for airborne command and control systems. In accordance with LMC's mandate with the CRM, he helps to promote industrial research by co-supervising students, with, so far, four students of J. Patera and one ncm2 postdoctoral fellow having benefited from the opportunity. These students are working on enhancement of radar images by multiresolution filtering, extraction of characteristics in the infrared images of ships, the use of the Karhunen-Loeve transform as a tool for automatic recognition and multiresolution detection of military vehicles in radar images.
Bernard Goulard is in the third year of a NSERC-Industry partnership project. The purpose of this project is to extend the capacity of Atlantic Nuclear Services' monitoring and diagnosis systems by incorporating into them some Artificial Neural Networks (ANN). In collaboration with Y. Bengio, J-M. Lina and P. Turcotte, this has led to the implanting of a modular ANN based on a "mixture of experts," to classify the different regimes of a reactor. The distribution of the data densities was modelled by a parametrised mixture of Gaussians; the iterative EM (Expectation -Maximisation) algorithm was used, after having been modified to take into account the ambiguity attached to the classification of data. In collaboration with J-M. Lina, Bernard Goulard is also working on the application of wavelets to image processing; in fact, a one -year extension of the project is being requested from NSERC to allow for the adaptation of the methods developed for teledetection. In another area, he is continuing his study, jointly with R. Roy and a student (A. Qaddouri), of parallel iterative processes in the solution of Boltzmann's equations of transport, and their applications to the characterisation of the distribution of neutrons in a reactor. They are now applying their methods to large scale problems (large spatial domains and numerous energy groups).
Michel Grundland's research in the last few years has dealt with symmetry-reduction methods and Riemann-invariant methods and their application to equations of nonlinear field theory, condensed matter physics, as well as fluid dynamics. The development of these methods has provided several new tools to study nonlinear phenomena in physics, especially those described by multidimensional systems of partial differential equations (pde) that were not solved by other methods (like inverse scattering). Grundland's research program can be divided into 4 projects:
The research interests of Gena Hahn lie in discrete mathematics. There are two main directions, infinite graphs and graph homomorphisms. The former includes mostly studies of ends in countable graphs (work with J. ·iráÀ, Bratislava and F. Laviolette, Université de Montréal) and of infinite tournaments (in progress, with R. E. Woodrow, Calgary, and P. Ille, C.N.R.S. Marseille). There is also progress, with N. Sauer, Calgary, on the question of when a property of an infinite graph is also true for some finite subgraph including a given finite set. Graph homomorphisms are studied in the guise of the various chromatic numbers and in connection with the ultimate independence ratio of a graph. They are also treated in surveys (with C. Tardif, Bielefeld, published, and with G. MacGillivray, Victoria, in progress). Part of the interest is the potential application to design and communication algorithms in networks, a subject much studied in computer science. Related to this is also the revived interest in Cayley graphs as models of networks.
During the past year, John Harnad's main research interest were all related to the modern theory of integrable systems. The topics studied were:
A recent work, in collaboration with A.R.Its, carries on the study of dual isomonodromic deformations, but also initiates a new program relating the latter to computation of correlation functions in integrable quantum and statistical models and the spectral distributions of random matrices, in which a special class of Fredholm integral operators arise, whose Fredholm determinants are the correlation functions in question. These are computed through the Riemann -Hilbert problem "dressing method," adapted to the case of isomonodromic deformations, leading to integral representations of importance in the calculation of asymptotics of such correlation functions. A key result derived in this work is the fact that the "dual" isomonodromic representations, deduced generally from the R-matrix structure, follow in this context from the invariance of the Fredholm determinant under Fourier transform of the integral kernel.
Jacques Hurtubise' research work deals with geometrical and topological aspects of objects originating from mathematical physics. His projects are divided into two rather disjoint topics.
The first one studies the relationship between the solution spaces of several field equations of mathematical physics like those of the sigma model or Yang -Mills equations, and the functional spaces in which they lie. The questions are mostly topological in nature, like the proof of topological stability theorems. These theorems have been extended this year to the most general case known today. The solution spaces are here characterized as minima or critical sets of an action functional, and the techniques used in the proofs involve also analytic subtleties from the calculus of variations.
The second one addresses the algebra-geometric properties of completely integrable mechanical systems. An invariant has been recently introduced that allows for a measurement of the complexity of a large number of these mechanical systems; whenever this complexity is minimal, the system possesses very natural coordinate systems that seem to be related to its quantisation.
During the last twenty years, the theory of Lie groups and algebras has been extendedin many directions. One of them deals with the supersymmetric theories and the notions of Lie supergroups and superalgebras. Since it is concerned with a unified description of fermionic and bosonic objects, one has to work with commuting and anticommuting variables. An interesting question deals with the resolution of nonlinear differential equations with such variables. V. Hussin, in collaboration with her student A. Ayari, has given an extension of the concept of symmetries for such equations and has obtained new invariant solutions.
Another theory developed along the last decade is the one of quantum groups, which arestructures that appear in statistical mechanics, for example. Now, one deals with deformations of Lie groups by changing the type of the objects one is working with. A systematic method for investigating such deformations has been used for the non- semisimple Lie algebras. V. Hussin, in collaboration with L. Frappat, A. Lauzon and G. Rideau, has given a classification of the possible deformations of some 3 and 4 dimensional algebras.
In joint work with Ian Anderson, Niky Kamran completed a detailed analysis of the cohomology of the variational bicomplex and the geometric integrability property for nonlinear hyperbolic differential systems in the plane. He also initiated with Thierry Robart the study of analytic Lie pseudogroups of infinite type as infinite-dimensional local Lie groups modelled on suitable locally convex topological vector spaces. Finally, he pursued his ongoing research projects with Peter Olver on quasi-exactly solvable spectral problems for differential operators, and with Keti Tenenblat on the Laplace transformation for submanifolds of projective space.
Systems of equations over a group have been widely studied and are currently one of the main directions in combinatorial group theory. To approach Tarski's problem about the elementary equivalence of free groups of different ranks, it is necessary to study sets of solutions of equations over a free group. It is possible to transcribe into the theory of free groups, and more general torsion-free hyperbolic groups (in Gromov's sense), some basic elementary algebraic geometry and give analogies of notions of the Zariski topology, irreducibility and Hilbert's Nullstellensatz.
O. Kharlampovich indicates, in collaboration with A. Myasnikov, an algorithm to represent the set of solutions to an arbitrary system of equations over a free group as the union of a finite number of irreducible components in the Zariski topology (quadratic and multiquadratic equations play an important role here). They give a description of the irreducible components. They describe all finitely generated fully residually free groups as subgroups of groups obtained from a free group by a finite number of extensions of centralisers. As a result, they describe all groups that are universally equivalent to a free group.
During 1996-1997 Anatol Kirillov's primary interests have revolved around the interplay of representation theory of quantum groups and classical Lie algebras, algebraic geometry, combinatorics and exactly solvable models of statistical physics. In particular, the following results were obtained:
Algebraic geometry. Quantum cohomology of flag varieties and Schubert calculus. Construction and study of quantum Schubert and quantum Grothendieck polynomials, quantum Schur and quantum Macdonald functions. Algebraic construction of quantum cohomology ring of flag variety, and quantum Orlik-Solomon algebra. New approach to classical and quantum Schubert calculus based on the study of certain non--commutative algebras. Proof of quantum Cauchy identity and quantum Pieri rule.
Representation theory and integrable models. Proof of the Macdonald conjecture on integrality of double Kostka coefficients. Description of spectral decomposition of integrable lattice models with applications to the representation theory of affine Lie algebras.
Gilbert Labelle works in the area of algebraic combinatorics, in particular towards the development of a theory of asymptotic structures and a classification of combinatorial structures by their stabilisers. A particular structure he has considered is that of quadtrees and hyperquadtrees, studying their asymptotic properties and their statistical properties. Using symbolic computation, he is also developing the analysis of combinatorial species.
We are interested in the connection between groups and Lie algebras which was established by Wilhem Magnus. Each central series of a group gives a graded Lie algebra. This Lie algebra is very difficult to calculate in general. In the case of the lower central series of a free group we get a free Lie algebra a result of Magnus and Witt. In the case of a group defined by a single relation we have determined the Lie algebra associated to the lower central series, in the case where the group is torsion free and the Lie algebra associated to the lower p-central series for p sufficiently large. The Lie algebras that we obtain are defined by a single relator and we determine the Poincaré series of these algebras.
These results are true also for pro-p-groups. Recently, with Helmut Koch and Suzanne Kukkuk, we have determined the Lie algebra associated to the lower central series of the Galois group of the maximal p-extension of a local field in the case p not equal to 2.
In the above results the groups in question were of cohomological dimension 1 or 2. It would be interesting to extend our results to the case of a group of cohomological dimension 3.
Partition functions for the Ising model on a two-dimensional lattice with boundaries depend on the state of the spin field at the boundary. For example, on a cylinder, the partition functions for both constant and free spin fields are known. But what are the partition functions for other boundary configurations? What are the relevant physical quantities at the boundary, i.e. those that have a limit as the size of the lattice goes to infinity? What are their distribution? Several physical arguments indicate that the critical behaviour of the Ising model is conformally invariant in the bulk. Does conformal invariance hold for physical properties at the boundary? In what sense? Robert Langlands and Yvan Saint-Aubin are exploring these questions using the theory of conformal fields and computer simulations.
Christian Léger's research is on the use of resampling methods in statistics. These methods use the power of the computer to approximate the distribution of an estimator to construct, for instance, a confidence interval for an unknown parameter. To validate these methods, asymptotic theory as well as computer simulations are used. In the last few years, Léger has studied the use of resampling methods, such as the bootstrap and cross-validation, to choose a tuning parameter for nonparametric estimators. In a recent paper, it was shown that the rate of convergence of the estimator played an important role in the success of these methods in choosing the tuning parameter. More precisely, this paper explains why cross-validation works in choosing a tuning parameter when the problem is "hard," but does not when the problem is "easy."
Polyominoes are important combinatorial structures for mathematical physics. They appear naturally in polymer models and the study of percolation. Recent work of the Bordeaux and Australian schools have given an enumeration with respect to area, perimeter and other finer parameters, for many classes of polyominoes having minimal convexity properties. In a geometrical or combinatorial context, it is natural to consider convex polyominoes up to a reflection or a rotation, i.e. as objects free to move in space. Pierre Leroux is currently working at enumerating them, as orbits under the action of the dihedral group on convex polyominoes. Due to Burnside's lemma, this involves the enumeration of the various symmetry classes of convex polyominoes. Many of these classes are intimately related to certain classical families of discrete models in statistical mechanics. For example, the class of convex polyominoes with a diagonal symmetry is related to that of directed and convex polyominoes (or animals) with a compact diagonal source.
Sabin Lessard's research interests include a wide variety of population genetic models and the concomitant evolutionary dynamics. His ultimate goals are: a) to explain the maintenance of variability in biological populations, b) to develop mathematical and statistical techniques to analyse population genetic structures, c) to deduce general evolutionary principles, and d) to study populations with complex interactions between individuals.
Most special functions of mathematical physics admit q-analogs, namely deformations involving a parameter q. Just as Lie algebras provide a unifying framework for discussing special functions, q-deformations of these algebras provide a unifying framework for discussing q-special functions. In collaboration with Luc Vinet (CRM) and Roberto Floreanini (Trieste), Jean LeTourneux carries out a systematic investigation of the quantum algebraic interpretation of the q-special polynomials encompassed in the scheme of Askey-Wilson polynomials.
According to the Efimov effect, a three-body system has an infinite number of bound states when it involves two-body interactions that marginally bind the two-body system. Formal proofs of this effect are too complex to provide any physical intuition. Simpler proofs, given for special cases within the framework of the Born-Oppenheimer approximation, break down as soon as one goes beyond the lowest order approximation. With Bertrand Giraud (Saclay) and Yukap Hahn (Univ. of Connecticut), Jean LeTourneux investigates a certain number of questions raised by this situation.
In the course of 1997, this group's research on wavelets was oriented towards statistical analysis and the modelling of stochastic processes. With image processing as its main application, this research follows on the lines of work done two years ago on orthogonal bases of dyadic complex wavelets, also known as Daubechies wavelets. The mathematical properties of these bases were the subject of intense study by the PHYSNUM group at the CRM; in particular, M. Ben Slimane, a postdoctoral fellow studied the question of the regularity of this type of wavelet. This allowed a better interpretation of the roles of amplitude and phase in complex wavelets. One of the theory's main achievements rests on the wavelets being an unconditional basis for Besov spaces. The norms associated to these spaces can be expressed in a simple fashion in terms of the amplitudes of the coefficients in a wavelet basis, and this property underlies the famous work of Lucier, de Vore, Donoho and Johnstone on robust estimators for signals with a background noise. We have used their analysis in the context of complex wavelets, putting the emphasis on the information contained in the phase of the coefficients. With a Bayesian formalism, this allowed a computation of maximum likelihood estimators for the amplitude of modes in complex wavelets, parametrised by the phases of the coefficients. This work was then applied to problems in image processing, and allowed the research program in this area to become more tightly focused, around questions in medical imaging and in satellite imaging. In the former, our collaboration with radiologists at the Notre Dame hospital now concentrates on the statistical analysis of stereotactic images of biopsies. In the second case, our expertise in the statistics of complex signals led to the study of satellite interferometry images. These projects were the source of two summer studentships in 1997 (P. Scott, of McMaster University, and S. Demers, from Laval University) The computer analysis was taken in hand by P. Turcotte, using his J. Wave package, dedicated to the multi-scale analysis of complex images.
Michel Mayrand works on the Command and Control System (CCS) for the Canadian Patrol Frigates (CPF). The approach used by the Research team of which he is a member consists in decomposing the different subsystems of the CCS, namely Multi-Sensor Data Fusion, Situation and Threat Assessment, and Resource Management, into a set of agents which perform small specific tasks and interact in a common fashion via a controller and a Blackboard architecture. The Blackboard itself is a global database which is available to all Knowledge Sources (the agents) in the system, and which contains all the active problem-specific data objects needed by, or produced by, the agents, and which are part of the solution space to the problem. Compared with traditional programming, the natural modularity of this solution makes it easier to maintain and to parallelise using multithreading. Special efforts have been made to improve the speed of this expert system for manipulating large amounts of abstract data and for solving large and complex problems. This is achieved by implementing data-driven algorithms such as the Rete Algorithm for activating rules efficiently. This expert system is also designed to support forward and backward chaining for problem solving.
The research of John McKay for the year continued to focus on two questions: computational Galois theory and moonshine.
Computational Galois groups is a topic of research which has progressed to the stage that we have named descriptions of all the transitive permutation groups of degree up to 15. A coauthor has extended the list of groups to degree 31. Although computing monodromy groups of polynomials is relatively easy as it is defined in terms of homotopy, computing Galois groups over Q is far harder and is ultimately limited by the size of polynomials/Q that can be factored. Good upper bounds for the Galois group (or easily computed properties that bound it) are much needed.
The name "moonshine" refers to a series of questions surrounding modular functions and the representations of finite groups, in particular the Monster. Norton defined a class of functions known as replicable functions which generalizes the class of Hauptmodules, which in turn generalizes the elliptic modular function, j(z). By generalizing Dedekind's construction of j(z), and working with differential equations, we are able to determine many useful invariants of Hauptmodules. These invariants, of a geometric and number theoretic nature, are of much interest. An equivalent formulation leads to a dynamical system for each Hauptmodule. These dynamical systems are each associated to a non-associative commutative algebra. A possible explanation of moonshine is the existence of a 24-dimension manifold such that the Monster group can act on its loop space but such has not been found yet.
There are several linked aspects to this research program, with the first being the dominant one:
A work with Paul Cohn, which gives a construction of the free field, an algorithm for its word problem and a primary decomposition of noncommutative fractions, is in course of publication. The same is true for an article with Christian Kassel, in algebraic K-theory, where the authors give a Coxeter-like presentation for the semi-direct product of the symmetric group and the Steinberg group (type A), together with investigations of the analogue in their case of the K2. With the same coauthor and Alain Lascoux, Christophe Reutenauer is presently working on a parametrisation of Schubert cells, resting on the special matrices used in the previous work.
One of the long-term goals of Christiane Rousseau's research program is the completion of the proof for the existence part of Hilbert's 16th problem for quadratic systems, i.e. to show that there exists a uniform bound for the number of limit cycles in a quadratic system. This project, initiated in 1991 with F. Dumortier and R. Roussarie, is progressing steadily. An important step made recently by Rousseau and H. Zoladek by exploiting simultaneously Khovanskii and Bautin's techniques for the centres and Roussarie's techniques for blowing up of families, allows one to hope for a complete solution in the coming three to ten years.
All the techniques introduced here have an intrinsic interest going far beyond their application to the above problem. With Roussarie, Rousseau has applied some of them to the study of certain homoclinic loops in 3-dimensional space, and their Ph.D. student, L.S. Guimond, is making further progress in that direction.
Another aspect of Rousseau's research project will be devoted to algebra-geometric methods applied to the study of polynomial vector fields. She is working on the problem of the centre (in collaboration with D. Schlomiuk) and on the geometric characterization of isochrone vector fields (with P. Mardeiç and L. Moser-Jauslin).
This study of polynomial vector fields has a direct impact on still another aspect: the study of singularities of vector fields of higher codimension (typically larger than or equal to 3). The bifurcations of these singularities are organizing centres of bifurcation diagrams occurring in many applied models.
Roch Roy's research deals with the modelling of time series. Although the analysis and modelling of time series is a classical field in statistics, its remains of current interest since it is applied in many scientific disciplines. His recent research work was concentrated on the following projects:
In biomathematics, David Sankoff works on algorithms for the analysis of DNA sequences and he has, within the context of the human genome project, extended this discipline to the development of methods for studying genome evolution resulting from the process of chromosomal rearrangement. This has resulted in the development of algorithms (in collaboration with John Kececioglu and Gopalakrishnan Sundaram) for sorting permutations using a small set of operations: reversals, transpositions, translocations. Sankoff and Vincent Ferretti study syntenic sets of genes in collaboration with Joseph Nadeau, a geneticist at Case Western Reserve, and several mathematics and statistics students. In phylogeny, Sankoff and Ferretti have developed a method of nonlinear phylogenetic invariants.
In sociolinguistics, David Sankoff directs a program whose goal is a rigorous statistical methodology for the analysis of syntactic variation and phonology in spoken language, based on computerized transcriptions of corpora of free speech. With David Rand, he developed and distributed a software package (GoldVarb) for linguistic data analysis. His empirical interests include bilingual syntax, specifically methods for distinguishing alternating borrowing codes, and the study of particles of speech.
Dana Schlomiuk works on local and global problems on families of planar analytic vector fields. In this project, interdisciplinary methods intervene (algebro-geometric or of commutative algebra, bifurcation theory, holomorphic foliations, etc.).
The project concerns the global analysis of families of quadratic vector fields (with J.Pal, Y.Dupuis, J.Surprenant) and the problem of the centre (with L. Farell). Results were obtained on the understanding of the dynamics and global geometry of the classes of systems which were studied.
Presently, work continues on the study of ideals intervening in the problem of the centre, on the algebraic invariant curves of polynomial systems, in particular on the multiplicity of such curves and on its role in the integrability of the systems.
Elisa Shahbazian is the director of research and development at Lockheed Martin Canada. In particular she is coordinating a team of ten researchers working in the areas of data fusion and image processing. Current research projects include: image processing and pattern recognition, with the collaboration of Prof. Jirí Patera and three students; real-time analysis of data for the Canadian Patrol Frigate (CPF); validation of data fusion for aerial reconnaissance; data fusion and image processing for aerial reconnaissance; a feasibility study for the CP-140 patrol aircraft; pursuit and targeting using MSDF.
The field of nonsmooth analysis, pioneered by F.H. Clarke in the 1970's, provides a "calculus" for functions which are nondifferentiable and possibly not even continuous, and which are therefore not amenable to treatment by standard (i.e. smooth) methods. On the geometric side, there have been many important applications of this theory in recent years, notably in optimization, control, and general dynamical systems (invariance theory and existence of equilibria). Ron Stern, in collaboration with F.H. Clarke, Yu. S. Ledyaev, P.R. Wolenski, and J.J. Ye, has been contributing in these areas in recent years. At present, a general problem Stern is working on is the construction of control feedback laws in certain control problems, using the tools of nonsmooth analysis.
The combinatorial problem of optimally arranging a set of large antennae used in radio astronomy has led to the notions of perfect systems of difference sets and of additive sequences of permutations. The question of existence for the systems can be considered as solved by Kotzig and Laufer, but enumeration and construction remain to be studied. For this construction, the additive sequences are a precious tool, but they themselves pose difficult problems of existence and enumeration. For sequence bases with no repetitions, the list of all sequences of length 2 has been compiled by Abraham, Kotzig and Turgeon up to cardinality eight. Among the bases of cardinality eight, Abraham and Turgeon discovered an equivalence relation by means of linear transformations. The sequences of length 2 with repetitive bases are now known up to cardinality six. Much progress has also been accomplished concerning sequences of length greater than 2.
Pierre Valin directs research on data fusion at Lockheed Martin Canada for the maritime surveillance aircraft CP-140. Part of this work is in collaboration with Professors D. Grenier and M. Lecours of the Department of Electrical Engineering and Computer Science at Université Laval. He is also scientific advisor for the production of a synthetic aperture radar for the same plane, for which Lockheed Martin Canada is the main contractor. In physics, his research now centres on hadronic multiplicities.
The main objectives of Luc Vinet's research projects are:
Last year, in collaboration with his Ph.D. student Luc Lapointe, Luc Vinet made a major step towards obtaining an algebraic solution of the Calogero-Sutherland model, and in so doing proved long-standing conjectures on some of the most important symmetric polynomials in algebraic combinatorics. With Roberto Floreanini (Trieste) and Jean LeTourneux, Luc Vinet has pursued his systematic investigation of the quantum algebraic interpretation of q-special functions. He has also undertaken a study of difference equations from the symmetry point of view.
Lie groups as symmetry groups of differential equations provide powerful tools for solving such equations, especially when combined with singularity theory and other attributes of modern integrability theory. Pavel Winternitz, together with Decio Levi (University of Rome III) and Luc Vinet, is developing a formalism that should be equally useful for treating difference equations. Two different approaches are being considered simultaneously. One applies to differential difference equations, involving both continuous and discrete variables. Transformations involving the continuous variables are treated via Lie algebras, the discrete ones are treated globally. In the second approach all variables are continuous, but their increments are discrete, i.e. differences figure instead of derivatives. The symmetry group is then constructed via "discrete prolongation" techniques, adapted from the usual Lie techniques used for differential equations. In order to recover all Lie point symmetries of a differential equation in the continuous limit, it turns out to be necessary to consider a much larger class of symmetries in the discrete case. They act simultaneously on the entire lattice, not just at one point.
Necessary and sufficient conditions will be provided for the existence of consistent estimates in nonparametric regression when the independent variable is a random vector. The approach will be based on the fact that estimation of a regression function may be seen as estimation of several parameters. The conditions obtained in this case by Kakutani and Shepp will be used to derive the results also for the case of a regression type function.
The research of Jean-Paul Zolésio centres on questions of modelling in continuum mechanics, more particularly on problems involving free surfaces and free boundaries. The particular techniques developed in this research program are ones which deal with shape optimisation, and involve questions of both a theoretical and practical nature. Problems considered include:
Parts F, G and H of this program are the result of collaboration with M. Delfour.
27 March 1998, webmaster@CRM.UMontreal.CA