# Patrice Sawyer

### Full Professor

##### Department of Mathematics & Computer Science

##### Science, Engineering and Architecture

### Biography

Patrice Sawyer has been a faculty member of the Department of Mathematics and Computer Science at UPEI (1989-1990) and of the Department of Mathematics and Statistics at the University of Ottawa (1990-1994). Patrice then joined Laurentian University in 1994 in the Department of Mathematics and Computer Science.

Representative of Natural Science and Engineering Research Council (NSERC) at Laurentian University from 2000 to 2008, he also contributed as a member of several boards of directors of research organizations, including SNOLAB Institute, the Mining Innovation, Rehabilitation and Applied Research Corporation (MIRARCO) and the Shared Hierarchical Academic Research Computing Network (SHARCNET). Over the past 20 years, he has been active on many university committees, including as member of the Faculty Association Board.

A full professor in the Department of Mathematics and Computer Science, Patrice Sawyer has served as Chair of the department.

In 2006, he was appointed Dean of the Faculty of Science and Engineering. In 2008 he was appointed Acting Vice-President, first in the portfolio of Francophone Affairs and academic staff relations, and later as Vice-President of Research. Later, he added Francophone Affairs to his responsibilities.

After his administrative leave, he has returned as a faculty member in January 2016.

### Education

- BSc (Laval)
- PhD (McGill)

### Academic Appointments

Professor, Department of Mathematics and Computer Science, Laurentian University 1994-present

Professor, Department of Mathematics and Statistics, University of Ottawa 1990-1994

Professor, Department of Mathematics and Computer Science, University of Prince Edward Island 1989-1990

### Research

My main research topic is the harmonic analysis on symmetric spaces. The concept of a spherical function φ_{λ} is central to this area. In fact, the spherical functions φ_{λ}(e^{X}) play the same role as the exponential e^{iλ(x)} in the usual Fourier theory. I have several projects related to this theme:

**The heat equation:**

The starting point of this work was my doctoral thesis. The equation describing the heat diffusion in a body (in two or three dimensions for example) can be expressed in mathematical terms. This is true for spaces that have a different geometry than Euclidean spaces such as symmetric spaces of noncompact type which have a heat diffusion whose behavior is radically different than for Euclidean spaces.

**The product formula:**

This research is conducted in collaboration with Piotr Graczyk at the Université d’Angers. It addresses the following question: under which circumstances is the measure μ_{X,Y} defined by the equation by φ_{λ} (e^{X}) φ_{λ} (e^{Y}) = ∫_{a }φ_{λ} (e ^{H}) dμ_{X,Y}(H) absolutely continuous with respect to the Lebesgue measure? This is equivalent to deciding wehter the set a(e^{X}Ke^{Y}) has a non-empty interior where g = k_{1}^{a(g)} k_{2} (Cartan decomposition) with a(g) ε a^{+} , k _{1,} k _{2} ε K.**Generalized spherical functions (hypergeometric functions related to root systems):**

Lie algebras form a significant component of the study of symmetric spaces. Simple Lie algebras are distinguished by their root systems. Several researchers have investigated the spherical functions corresponding to symmetric spaces and generalized them by allowing root multiplicities to take values that do not correspond to symmetric spaces. By finding explicit formulas forthe spherical functions corresponding to the spaces **SL**(n,**F)** / **SU**(n,**F)** and **SO**(p, q) / **SO**(p) x **SO**(q), I was able to derive explicit formulas for the generalized spherical functions corresponding to the root systems of type A and type B.

__My publications are listed under the section "Publications" of this profile.__

I also published four articles not related to symmetric spaces (the order of authors is purely alphabetical):

- B. G. Ivanoff and P. Sawyer, Local
*time for processes indexed by a partially ordered set*,*Statist. Probab. Lett.*61 (2003), no. 1, 1–15. - H. Boudjellaba, B. MacGibbon and P. Sawyer, O
*n exact inference for change in a Poisson sequence, Comm. Statist. Theory Methods*30 (2001), no. 3, 407–434. - P. Sawyer,
*An old limit revisited*, Mathematical Gazette, March 2001, 41--44. : this paper is expository in nature. - M. Bunge and P. Sawyer,
*On connections, geodesics and sprays in synthetic differential geometry*, Cahiers de topologie et géométrie différentielle catégoriques, vol. 25, 1984, p. 221-258.

#### Awards

#### Publications

Except for the Actes (with M. Tremblay and S. Lafortune), the order of authors is strictly alphabetical.

P. Sawyer, *Laplace-type representation for some generalized spherical functions of type BC, *to appear in Colloqium Mathematicum, 1-19, 2018.

P. Sawyer, *A Laplace-type representation of the generalized spherical functions of type A*, Mediterranean Journal of Mathematics, August 2017, 14:147.

P. Sawyer, *An **analogue** to the Duistermaat–Kolk–Varadarajan estimate for the spherical functions associated with the root systems of type A*, Advances in Pure and Applied Mathematics, Vol. 7, No. 3, 165–175, 2016.

P. Sawyer, *Computing the Iwasawa decomposition of the classical Lie groups of noncompact type using the QR decomposition*, Linear Algebra and its Applications, Volume 493, 15 March 2016, 573–579, available online at http://authors.elsevier.com/sd/article/S00243795150072.

P. Graczyk and P. Sawyer, *The Convolution of orbital measures on symmetric spaces: a survey*,

Proceedings of the Conference Probability on Algebraic and Geometric Structures, Contemporary Mathematics, Vol. 668, 81-110, 2016.

P. Graczyk and P. Sawyer, *Convolution of orbital measures on symmetric spaces of type C _{p} and *

*D*, Aust. Math. Soc. 98 (2015), no. 2, 232–256.

_{p}P. Graczyk and P. Sawyer, *On the product formula on noncompact Grassmannian*, Colloq. Math.

133 (2013), 145-167.

P. Graczyk and P. Sawyer, *A Sharp Criterion for the Existence of the Density in the Product*

*Formula on Symmetric Spaces of Type An*, Journal of Lie Theory Volume 20 (2010), 751-76

Heldermann Verlag.

P. Graczyk and P. Sawyer*, Absolute continuity of convolutions of orbital measures on Riemannian*

*symmetric spaces*, Journal of Functional Analysis 259 (2010), 1759-1770.

P. Sawyer,* The heat kernel on the symmetric space SL(n;F)/SU(n;F),* in The Ubiquitous

Heat Kernel, Jay Jorgenson and Lynne Walling (eds), Contemporary AMS series, no. 398, 369-392, 2006.

P. Graczyk and P. Sawyer, *On the kernel of the product formula on symmetric spaces*, Journal

of Geometric Analysis, Vol. 14, 2004, 4, 653-672.

P. Sawyer, *La géométrie hyperbolique pour les non-initiés*, Actes de la 10ème Journée Sciences

et Savoirs, Acfas-Sudbury, Sudbury, April 2004.

A copy may be found at the Laurentian University institutional repository LU ZONE UL:

https://zone.biblio.laurentian.ca/dspace/handle/10219/62/

P. Sawyer, *The Abel transform on symmetric spaces of noncompact type *in Lie groups and symmetric spaces, American Mathematical Society Translation Series 2, 210, American Mathematical Society, Providence, RI, 2003, p. 331-355.

P. Graczyk et P. Sawyer, *Some convexity results for the Cartan decomposition*, Canadian

Journal of Mathematics, no. 5, 2003, p. 1000-1018.

P. Graczyk and P. Sawyer, *The product formula for the spherical functions on symmetric spaces*

*of noncompact type*, Journal of Lie Theory, vol. 13, no. 1, 2003, p. 247_261.

G. Ivanov and P. Sawyer, *L*o*cal time for processes indexed by partially ordered sets, *Statistics &

Probability Letters, vol. 61, 2003, p. 1-15.

P. Graczyk and P. Sawyer, *The product formula for the spherical functions on symmetric spaces*

*in the complex case*, Pacific journal of Mathematics, vol. 204, no. 2, 377-393, 2002.

P. Sawyer, *The spherical functions related to the root system B2*, Canadian Mathematical

Bulletin, vol. 45, no. 3, 2002, p. 436-447.

P. Sawyer, *The central limit theorem on SO0(p; q)/SO(p) x SO(q),* Journal of Theoretical

Probability, vol. 14, no. 3, 2001, p. 857-866.

H. Boudjellaba, B. McGibbon and P. Sawyer, *On inference for change in a Poisson sequence*,

Communications in Statistics - Theory and Methods, vol. 30, no. 3, 2001, p. 407-434.

P. Sawyer, *The asymptotic expansion of spherical functions on symmetric cones*, Pacific Journal

of Mathematics, vol. 200, no. 1, 2001, p. 251-256.

P. Sawyer, *An old visit revisited*, Mathematical Gazette, March 2001, p. 41-44.

P. Sawyer, *Spherical functions on* *SO**0(p; q)/ SO(p) x SO(q),* Canadian Mathematical Bulletin,

vol. 42, no. 4, 1999, p. 486-498.

P. Sawyer, *The eigenfunctions of a Schrödinger operator*, Quarterly Journal of Mathematics,

Oxford, vol. 50, no. 2, 1999, p. 71-86.

P. Sawyer, *Estimates for the heat kernel on SL(n;R)/SO(n)*, Canadian Journal of Mathematics,

vol. 49, no. 2, 1997, p. 359-372.

P. Sawyer, *Spherical functions on symmetric cones*, Transactions of the American Mathematical

Society, vol. 349, 1997, p. 3569-3584.

P. Sawyer, On an upper bound for the heat kernel on **SU**^{*}(2n)/**Sp**(n), Canadian Mathematical

Bulletin, vol. 37, no. 3, 1994, p. 408-418.

P. Sawyer, *The heat equation on spaces of positive definite matrices*, Canadian Journal of

Mathematics, vol. 44, no. 3, 1992, p. 624-651.

M. Bunge and P. Sawyer, *On connections, geodesics and sprays in synthetic differential geometry*, Cahiers de topologie et géométrie différentielle catégoriques, vol. 25, 1984, 221-258.

**Preprints:**

P. Sawyer, *A Laplace-type representation of the generalized spherical functions associated to the root systems oftype*A, 1-21, 2016.

P. Sawyer, *The strange consequences of using a blowtorch in hyperbolic space*, 1-14, 2015.

P. Sawyer,* The product formula for the classical symmetric spaces of noncompact type,* 1-29, 2015.

**Books**

M. Tremblay, S. Lafortune, P. Sawyer (editors), Les Actes de la 11ème Journée Sciences et Savoirs,

Acfas-Sudbury, Sudbury, April 2005, 197 p.

A copy may be found at the Laurentian University institutional repository LU ZONE UL:

https://zone.biblio.laurentian.ca/dspace/handle/10219/42/

S. Lafortune, P. Sawyer, M. Tremblay (editors), Les Actes de la 10ème Journée Sciences et Savoirs,

Acfas-Sudbury, Sudbury, April 2004, 201 p.

A copy may be found at the Laurentian University institutional repository LU ZONE UL:

https://zone.biblio.laurentian.ca/dspace/handle/10219/41/

P. Sawyer, Algèbre linéaire, Simon & Schuster Custom Publishing, 1998, 403 p.

**Book review**

CRM Proceedings & Lecture Notes, vol 28, Topics in Probability and Lie Groups: Boundary

Theory, edited by J. C. Taylor (2001). Revue par Patrice Sawyer pour les Notes de la Société

Canadienne des Mathématiques, Septembre 2002, Volume 34, No. 5.