Faculty & Contact

My research activities are in the areas of computational topology and imaging science. In the domain of computational topology, my research objectives are to develop efficient algorithms for

• the computation of topological invariants and structures such as the homologies and cohomologies of spaces and maps, and
• the computation, the characterization, and the representation of critical points and critical regions in scalar and vector fields by means of Morse theory and Conely index theory.

Topological invariants such as homology have been used recently in a wide variety of applications in domains such as dynamical systems, and image processing and recognition. In dynamics, typical problems are translated into problems in topology where invariants such as the Conley index are computed using homology algorithms. In digital image analysis, topological invariants are useful in shape description, indexation, and classification. Scalar and vector fields are used to represent data in different applications areas like geographic information systems and the charaterization of the critical points of the data constitutes a fundamental technique for the study of the important features of the data and its visualization. Other areas of applications include computer graphics, computer aided-design (CAD) and electrical engineering to name but a few. The necessity of improved algorithms for the computation of the structures mentioned above appears evident as new applications of computational topology arise in research for very large data sets. Although several algorithms and software packages have been developed for this purpose, there is still a lot of room for improvement as processing very large data sets is often very time and memory-consuming.

I currently supervise (and co-supervise with colleagues from the Université de Sherbrooke) a group of MSc and Ph.D. students working on these problems and on other problems directly related to computer vision such as shape description and recognition and segmentation using deformable models techniques. I am interested in expanding my research team and I welcome new students interested in working on these issues. I provide for my students an exciting research environment with lab facilities at Bishop’s University and at Université de Sherbrooke where I am a member of two research groups

• The Imaging and Vision Group – MOIVRE (MOdélisation en Imagerie, Vision et RÉseaux de neurones), and
• The Computational Topology Group – GRTC (Groupe de recherche en topologie computationnelle).

See Dr. Allili’s website for Publications list.

Papers on arXiv

Representation Theory of Finite Dimensional Algebras

Dr. Huard’s current research focuses on the finitistic dimension conjecture.

The finitistic dimension conjecture was first publicized by Bass in 1960. For an artin algebra A we define the finitistic dimension dimension of A to be fin.dim. A =sup {pd M such that M is a finitely generated right A-module and pd M is finite} where pd M denotes the projective dimension of M. Bass conjectured that for A as above, fin.dim. A is always finite. Since then, this conjecture has been verified for few classes of algebras. Our long term objectives are thus to better understand the finitistic dimension of an algebra, and to provide the community with new tools to study this invariant. We also want to determine new classes of algebras with finite finitistic dimension. This is of significant importance since the more diverse algebras that can be found for which the conjecture holds, the greater the chance that a general proof of the conjecture will occur. In order to achieve these goals, we study the finitistic dimension from three different perspectives. The first is through the relative homology theory developed by Auslander and Solberg. The second approach is through modules of infinite projective dimension. Finally, we study it from the point of view of stratifying systems.

Partners

Dr. Huard is a member of the Bishop’s-Sherbrooke research group in representation theory of algebras which consists of five professors and around 15 graduate students.
He is also a member of the Groud d’algèbre et théorie de nombre de l’ISM
(Institut des Sciences Mathématiques).

Student involvement

Dr. Huard is working with undergraduate students from Bishop’s University and graduate students from Université de Sherbrooke. Each year the research group gives summer research grants to undergraduate students interested in pursuing research in algebra. Since 2006, Dr. Huard organizes a mathematics camp held every two year at Bishop’s University. The 25 best CEGEP mathematics students from Québec, selected following a provincial examination, are invited to attend this unique 10 day event where the focus is put on mathematics.

Huard, F. “Tilted gentle algebras.” /Comm. Algebra/ 26, 63-72.

Huard, F. and Liu, S. 1999. “Tilted special biserial algebras.” /J.Algebra/ 217, 679-700.

Huard, F. and Liu, S. 1999. “Tilted string algebras.” /J. Pure and Applied Algebra,/153 (2000) 151-164

Huard, F. 2000. “One-point extensions of quasi-tilted algebras by projectives.” /Comm. Algebra, /(29) (7) (2001) 3055-3060

Huard, F. 2001. “Tilted algebras having underlying graph Dn”, Ann. Sc. Math. Québec

Brewster, R., Dedic R. and Huard, F. 2005. “Edge coloured homomorphisms and the recognition of bound quivers” /Discrete Math/(297) (2005) 13-25

Huard, F., Lanzilotta M. and Mendoza O. 2007. “A new approach to the finitistic dimension” J. Algebra

Huard, F., Lanzilotta M. and Mendoza O. 2007. “Finitistic dimension through infinite projective dimension” Bulletin of the London Math. Soc.

G. M. L. Gladwell, T. H. Jones, and N. B. Willms, A Test Matrix for an Inverse Eigenvalue Problem, Journal of Applied Mathematics, vol. 2014, Article ID 515082, 6 pages, 2014. doi:10.1155/2014/515082

Trevor H. Jones, Heat Kernel for Open Manifolds, Differential Geometry and its Applications, 28 (2010), pp. 518-522

His main interest research interest is Algebra, and more precisely Representation Theory of Algebras.

Funding:
Since 2010, he holds an NSERC (Natural Sciences and Engineering Research Council of Canada) research grant: $150,000 over 8 years.

Publications

Since 2003, he has been the author or co-author of 12 published research papers.

Published Papers

I. Assem, G. Dupont, R. Schiffler and D. Smith, Friezes, strings and cluster variables, Glasg. Math. J. 54 (2012), no. 1, 27-60.

A. Buan, O. Iyama, I. Reiten and D. Smith, Mutation of cluster-tilting objects and potentials, Amer. J. Math., 133 (2011), no. 4, 835-887.

I. Assem, C. Reutenauer and D. Smith, Friezes, Adv. Math. 225 (2010), no. 6, 3134-3165.

J.C. Bustamante, J. Dionne and D. Smith, (Co)homology theories for oriented algebras, Comm. Algebra, 27 (2009), no.5, 1516-1544.

J. Dionne, M. Lanzilotta and D. Smith, Skew group algebras of piecewise hereditary algebras are piecewise hereditary, J. Pure Appl. Algebra, 213 (2009), 241-249.

D. Smith, On tilting modules over cluster-tilted algebras, Illinois J. Math. 52 (2008), no. 4, 1223-1247.

D. Smith, Almost laura algebras, J. Algebra, 319 (2008), no. 1, 432-456.

M. Lanzilotta and D. Smith, Laura algebras and quasi-directed components, Colloq. Math., 105 (2006), no. 2, 179-196.

J. Dionne and D. Smith, Articulations of algebras and their homological properties, J. Algebra Appl., 5 (2006), no. 3 1-15.

I. Assem, F. U. Coelho, M. Lanzilotta, D. Smith and S. Trepode, Algebras determined by their left and right parts. Algebraic structures and their representations, 13-47, Contemp. Math., 376, Amer. Math. Soc., Providence, RI, 2005.

D. Smith, On generalized standard Auslander-Reiten components having only finitely many non-directing modules, J. Algebra, 279 (2004), no. 2, 493-513.

J-C. Bustamante, J. Dionne and D. Smith, Ordonnés de chaînes et algèbres d’incidence, Ann. Sci. Math. Québec 27 (2003), no. 1, 1-11.

Other publications

D. Smith, Algèbres de type laura, algèbres de groupes gauches et groups de (co)homologie, Ph.D. Thesis (2006), Université de Sherbrooke.

D. Smith, Articulation d’algèbres et propriétés homologiques, Master Thesis (2003), Université de Sherbrooke.