Welcome to my professional Web page |
Currently, I am an Associate Professor in the Department of Physics and Astronomy and an Adjunct Professor in the Department of Mathematics at the University of Manitoba. I am also the Past Director of the Winnipeg Institute for Theoretical Physics (Director: January 1, 2015- December 31, 2016) and an associate at the Center for Advanced Mathematical Sciences at the American University of Beirut.
I.
Books
II.
Papers
1.
Calculus
on a non-Archimedean field extension of the real numbers: The intermediate
value theorem, mean value theorem, inverse function theorem and implicit
function theorem, 14^{th}
International Conference on p-Adic Functional Analysis, Aurillac, France,
June 30-July 5, 2016.
2.
On the Levi-Civita fields: introduction and
survey of recent research, 14^{th}
International Conference on p-Adic Functional Analysis, Aurillac, France,
June 30-July 5, 2016.
3.
One-variable and Multi-variable Integral
Calculus over the Levi-Civita Field and Applications, NUMTA2016
(Numerical Computations: Theory and Applications), The 2^{nd}
International Conference and Summer School, Calabria, Italy, June 19-25,
2016.
4.
Characterization
of compact and self-adjoint operators, and study of positive operators on a
Banach space over a non-Archimedean field, International Conference on p-Adic
Mathematical Physics and its Aplications, Belgrade, Serbia, September 7-12,
2015.
5.
New
results on the Lebesgue-like measure and integration theory on the Levi-Civita
field and applications, 13^{th}
International Conference on p-Adic Functional Analysis, Paderborn, Germany,
August 12-16, 2014.
6.
On
positive operators on a Banach space over the complex Levi-Civita field, The Seventh Conference on Function Spaces,
Southern Illinois University- Edwardsville, Illinois, USA, May 20-24, 2014.
7.
Preliminaries
in non-Archimedean Functional Analysis, The Seventh Conference on Function Spaces,
Southern Illinois University- Edwardsville, Illinois, USA, May 20-24, 2014.
8.
Analysis on non-Archimedean ordered field extensions of
the real numbers and applications, NUMTA2013
(Numerical Computations: Theory and Applications) International Conference and
Summer School, Falerna, Italy, June 16-23, 2013.
9. B*-algebras of operators and study of positive operators on a free Banach space of countable type over the complex Levi-Civita field, 12th International Conference on p-Adic Functional Analysis, University of Manitoba, Winnipeg, Canada, July 2-6, 2012.
10. (Co-author, talk given by Jose Aguayo) Characterization of Compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field, 12th International Conference on p-Adic Functional Analysis, University of Manitoba, Winnipeg, Canada, July 2-6, 2012.
11. (Co-author, talk given by Todd Sierens) On locally uniformly differentiable functions: the Inverse Function Theorem and the Implicit Function Theorem in a non-Archimedean setting, 12th International Conference on p-Adic Functional Analysis, University of Manitoba, Winnipeg, Canada, July 2-6, 2012.
12. Absolute and relative extrema, the mean value theorem and the inverse function theorem for analytic functions on a Levi-Civita field, 11th International Conference on p-Adic Functional Analysis, Université Blaise Pascal, Clermont-Ferrand, France, July 5-9, 2010.
13. Analysis on a non-Archimedean field extension of the real numbers and applications, The Sixth Conference on Function Spaces, Southern Illinois University- Edwardsville, Illinois, USA, May 18-22, 2010.
14. Analysis on the Levi-Civita field, a brief overview, Tenth International Conference on p-Adic and Non-Archimedean Analysis, Michigan State University, East Lansing, Michigan, USA, June 30- July 3, 2008.
15. Intermediate value theorem for analytic functions on a Levi-Civita field, Ninth International Conference on P-adic Functional Analysis, University of Concepcion, Concepcion, Chile, July 10-14, 2006.
16. Analytical properties of power series on Levi-Civita fields, Eighth International Conference on P-adic Functional Analysis, Clermont-Ferrand, France, July 5-9, 2004.
17. Measure theory and integration on the Levi-Civita field, Seventh International Conference on P-adic Functional Analysis, Nijmegen, The Netherlands, June 17-21, 2002.
18. The differential algebraic structure of the Levi-Civita field and applications, Ninth International Colloquium on Numerical Analysis and Computer Science with Applications, Plovdiv, Bulgaria, August 12-17, 2000.
19. Convergence on the Levi-Civita field and study of power series, Sixth International Conference on P-adic Functional Analysis, Ioannina, Greece, July 2-9, 2000.
20. Power series on the Levi-Civita field, Eighth International Colloquium on Numerical Analysis and Computer Science with Applications, Plovdiv, Bulgaria, August 13-18, 1999.
Contributed Talks at Conferences
Invited Talks, Seminars and Colloquia at
Universities
My research interests and activities include various areas of non-Archimedean Analysis: power series and analytic functions, measure theory and integration, optimization, existence and uniqueness of solutions of differential equations, complex analysis, multivariable analysis and functional analysis. The focus of my research has been on the Levi-Civita fields which were first introduced by the Italian mathematician Tullio Levi-Civita at the end of the nineteenth century. Of those Levi-Civita fields, one (which we denote by R) is of particular interest; it is shown to be the smallest non-Archimedean field extension of the real numbers that is complete in the order topology and real closed. In fact, R is small enough so that the numbers of the field can be implemented on a computer; and this allows for many useful applications, one of which is the fast and accurate computation of the derivatives of real-valued functions up to high orders.
We studied two topologies on R: the valuation topology induced by the order on the field, and another weaker topology induced by a family of semi-norms, which we call weak topology. We showed that each of the two topologies results from a metric on R, that the valuation topology is not a vector topology while the weak topology is, and that R is complete in the valuation topology while it is not in the weak topology. Then we studied the properties of both topologies in detail; in particular, we gave simple characterizations of open, closed, and compact sets in both topologies. Finally, we showed that the metric which induces the weak topology is translation invariant.
We studied convergence of sequences and series in both topologies mentioned above, which led to an exhaustive study of power series. A handful of people had investigated power series on the Levi-Civita fields before, but all the previous studies had been restricted to the special case of power series with real coefficients. We have dropped that restriction and showed that power series on Levi-Civita fields have all the nice smoothness properties that real power series have. In particular, they satisfy the intermediate value theorem, the extreme value theorem, the mean value theorem and the inverse function theorem; they are infinitely often differentiable; and they are re-expandable around any point within their domain of convergence.
While it is a known fact that conventional continuity or differentiability are not sufficient to guarantee that a function on a closed interval of a non-Archimedean field be bounded or satisfy any of the common theorems of real calculus, we have shown that under mild conditions, differentiability is sufficient for the function to assume all intermediate values and a differentiable inverse function. We also showed that conventional differentiability is not the right one to study optimization questions on non-Archimedean fields in general; and based on a stronger concept of differentiability, we studied finite-dimensional optimization both with and without constraints. In both cases, we derived necessary and sufficient conditions of first and second order for a function to have a local minimum at a point of its domain.
We developed a measure theory and integration on the Levi-Civita field R. We introduced a measure that proved to be a natural generalization of the Lebesgue measure on the field of the real numbers and have similar properties. Then we introduced a family of simple functions from which we obtained a larger family of measurable functions and derived a simple characterization of such functions. We studied the properties of measurable functions, we showed how to integrate them over measurable sets, and we showed that the resulting integral satisfies similar properties to those of the Lebesgue integral of Real Analysis.
Most recently, together with my collaborators Jose Aguayo and Miguel Nova from Concepcion (Chile), we developed an operator theory on a Banach space over C:=R+iR. Let c_0 denote the space of all null sequences of elements of C. The natural inner product on c_0 induces the sup-norm of c_0. We showed that c_0 is not orthomodular; then we characterized those closed subspaces of c_0 with an orthonormal complement with respect to the inner product. Such a subspace, together with its orthonormal complement, defines a special kind of projection, the so-called normal projection. We presented a characterization of such normal projections as well as a characterization of other kinds of operators, the self-adjoint and compact operators on c_0. Then we worked on some B*-algebras of operators, including those mentioned above; we studied normal and Hilbert-Schmidt operators; and finally, we studied the properties of positive operators, which we then used to introduce a partial order on the B*-algebra of compact and self-adjoint operators on c_0 and studied the properties of that partial order.
While the Levi-Civita field R is interesting to study in detail for the reasons stated above, I have recently started expanding my research focus to include any non-Archimedean field extension
of the real numbers that is real closed and complete in the topology induced by the order and whose Hahn group is Archimedean; such a field is denoted by F. For example, with two NSERC USRA students, I proved the inverse function theorem and implicit function theorem for locally uniformly differentiable functions from F^{n} to F^{n} and from F^{n} to F^{m }(m < n), respectively. Then, in a followup paper, we studied the properties of locally uniformly differentiable functions on F or F^{n}; and most recently, we proved a local version of the mean value theorem and Taylor's theorem for locally uniformly differentiable functions on F. Enlarging the scope of my research will make it more interesting to a wider audience of mathematicians and will thus open the door to new collaborations in non-Archimedean Analysis.
Training
of Highly Qualified Personnel (at U of M)
I.
Graduate
Students
1. Darren Flynn (September 2014- ); PhD candidate (Physics).
2. William Grafton (September 2013- August 2016); M.Sc. in Mathematics.
3. Angel Barria (January 2013- ); PhD candidate (Mathematics).
4. Darren Flynn (July 2012- August 2014); M.Sc. in Physics.
5. Andrew Senchuk (January 2010- ; I am co-advisor, with advisor: Prof. Gerald Gwinner); PhD candidate (Physics).
II.
Undergraduate
Students
o Summer Students
1. Gidon Bookatz (2016); NSERC-USRA student in the joint Math-Physics Honours program.
2. Ryan Sherbo (2015); NSERC-USRA student in the joint Math-Physics Honours program.
3. Gidon Bookatz (2015); NSERC-USRA student in the joint Math-Physics Honours program.
4. Gidon Bookatz (2014); NSERC-USRA student in the joint Math-Physics Honours program.
5. Gidon Bookatz (2013); NSERC-USRA student in the joint Math-Physics Honours program.
6. William Grafton (2012); undergraduate summer student in the joint Math-Physics Honours program.
7. James Roberts (2011); undergraduate summer Honours student, Department of Physics at the University of Winnipeg.
8. William Grafton (2011).
9. Todd Sierens (2010); undergraduate summer student in the joint Math-Physics Honours program; currently a PhD student at the University of Waterloo and the Perimeter Institute.
10. Todd Sierens (2009); NSERC-USRA student in the joint Math-Physics Honours program.
11. Trevor Rempel (2009); NSERC-USRA student in the joint Math-Physics Honours program; currently a PhD student at the University of Waterloo and the Perimeter Institute.
o Undergraduate Honour Thesis:
12. Cheng Tang (2012-2013).
I.
At the
University of Manitoba (May 2008- present)
o
Winter
2017: PHYS 2490, Theoretical Physics II
o Fall 2016:
Ø PHYS 2390, Theoretical Physics I
Ø PHYS 3650, Classical Mechanics II
o Winter 2016: PHYS 2490, Theoretical Physics II
o Fall 2015:
Ø PHYS 1020, General Physics I
Ø PHYS 2390, Theoretical Physics I
o Winter 2015: PHYS 2490, Theoretical Physics II
o Fall 2014:
Ø PHYS 1020, General Physics I
Ø PHYS 2390, Theoretical Physics I
o Winter 2014: PHYS 2490, Theoretical Physics II
o Fall 2013:
Ø PHYS 1020, General Physics I
Ø PHYS 2390, Theoretical Physics I
o Summer
2013: MATH 8430, Non-Archimedean
Operator Theory (a reading course)
o Winter
2013:
Ø MATH 8430, Non-Archimedean Analysis (a reading course)
Ø PHYS 2490, Theoretical Physics II
Ø PHYS 3640, Electro- and Magnetodynamics and Special Relativity
o Fall 2012: PHYS 2390, Theoretical Physics I
o Winter 2012:
Ø PHYS 2490, Theoretical Physics II
Ø PHYS 3640, Electro- and Magnetodynamics and Special Relativity
o Fall 2011: on sabbatical leave
o Winter 2011:
Ø PHYS 2490, Theoretical Physics II
Ø PHYS 3640, Electro- and Magnetodynamics and Special Relativity
o Fall 2010: PHYS 7590, Electromagnetic Theory
o Winter 2010:
Ø PHYS 2490, Theoretical Physics II
Ø PHYS 3640, Electro- and Magnetodynamics and Special Relativity
o Fall 2009: PHYS 7590, Electromagnetic Theory
o Winter 2009:
Ø PHYS 2490, Theoretical Physics II
Ø PHYS 3640, Electro- and Magnetodynamics and Special Relativity
o Spring 2008
Ø MATH 133, Calculus and Analytic Geometry
Ø MATH 391, Writing in the Mathematical Sciences
o Fall 2007
Ø MATH 137, Applied Calculus I
Ø MATH 651, Elements of Modern Analysis
o Spring 2007
Ø MATH 133, Calculus and Analytic Geometry
Ø MATH 391, Writing in the Mathematical Sciences
o Fall 2006
Ø MATH 137, Applied Calculus I
Ø MATH 551, Methods of Classical Analysis
o Spring 2006
Ø MATH 133, Calculus and Analytic Geometry
Ø MATH 391, Writing in the Mathematical Sciences
o Fall 2005
Ø MATH 137, Applied Calculus I
Ø MATH 599, Special Topics: Methods of Classical Analysis
o Spring 2005
Ø MATH 137, Applied Calculus I
o Fall 2004
Ø MATH 137, Applied Calculus I
Ø MATH 430, Multivariable Calculus
o Spring 2004
Ø MATH 137, Applied Calculus I
Ø MATH 435, Introduction to Real Variables
o Fall 2003
Ø MATH 137, Applied Calculus I
Ø MATH 311, Linear Algebra
o 2002-2003: MATH 234, Multivariable Calculus
o 2011-2002: MATH 234, Multivariable Calculus
o Spring 2000, Fall 2000 and Spring 2001: MATH 235, Differential Equations.
I.
At the
University of Manitoba (May 2008- present)
a. PhD Students Advisory Committees
i. L.J. Zhou, PhD in Physics: May 2016- present
ii. Brad Cownden, PhD in Physics: November 2015- present
iii. Erica Franzmann, PhD in Astrophysics: August 2015- present
iv. Ievgen Bilokopytov, PhD in Mathematics: November 2014- present
v. Mark
McCrea, PhD in Physics: May 2011- January 2017 (successfully defended his
thesis).
b. PhD and M.Sc. Thesis Internal Examiner
i. Mahnaz
Alavinejad, M.Sc. in Mathematics: 2016
ii. Jesse Bellec, M.Sc. in Physics: 2015
iii. Lindsay Simpson, M.Sc. in Mathematics: 2015
iv. Mahdi Kamaee, M.Sc. in Chemistry: 2014
v. Fereshteh Nazari, M.Sc. in Mathematics: 2014
vi. Bryan Penfound, M.Sc. in Mathematics: 2010
vii. Chandra Podder; PhD in Mathematics: 2010
viii. Mohammad Safi; PhD in Mathematics: 2010.
c. Departmental Committees
i. Nominating Committee: 2014-2015
ii. Term Work Appeals (Chair): 2012-2017
iii. Honours/Majors Committee (Chair): 2009-2015
iv. Curriculum Committee: 2008-2011 (Member); 2011-2017 (Chair)
d. Faculty of Science Committees
i. Hiring Committee (Spousal Assistant Professor Position, Dept. of Mathematics): March 2017
ii. Faculty of Science Executive Council, July 2016- June 2018
iii. Hiring Committee (Assistant Professor Position, Dept. of Statistics): August 2015- March 2016
iv. Hiring Committee (Assistant Professor Position, Dept. of Mathematics): October 2014- April 2015
v. Past Director of the Winnipeg Institute for Theoretical Physics: January 2017- December 2018
vi. Director of the Winnipeg Institute for Theoretical Physics: January 2015- December 2016
vii. Director Elect of the Winnipeg Institute for Theoretical Physics: January 2013- December 2014
viii. Committee on Courses and Programs (COCAP): 2011-2017
ix. The Local Discipline Committee: 2008-2017
e. International Committees and Professional
Service
i. Member of the Scientific Committee for the 15^{th} International Conference on p-Adic Functional Analysis (Poznan, Poland, July 2-6, 2018).
ii. Member of the Scientific Committee for the 14^{th} International Conference on p-Adic Functional Analysis (Aurillac, France, July 4-9, 2016).
iii. Member of the Scientific Committee for NUMTA2016: Numerical Computations: Theory and Algorithms (Pizzo Calabro, Italy, June 20-24, 2016)
iv. Member of the Scientific Committee for the 13^{th} International Conference on p-Adic Functional Analysis (University of Paderborn, Germany, August 12-16, 2014).
v. Organizer of a special session on non-Archimedean Functional Analysis at the Seventh Conference on Function Spaces, Southern Illinois University at Edwardsville, May 20-24, 2014.
vi. Member of the Scientific Committee for NUMTA2013: Numerical Computations: Theory and Algorithms (Falerna, Italy, June 17-23, 2013)
vii. Organizer
and chair of the Scientific Committee for the 12^{th} International
Conference on p-Adic Functional Analysis (University of Manitoba, July 2-6,
2012)
viii. Member of the Scientific Committee for the 11^{th} International Conference on p-Adic Functional Analysis (Universite Blaise Pascal, France, July 5-9, 2010)
ix. Co-organizer and member of the Scientific Committee for the 10^{th} International Conference on p-Adic and Non-Archimedean Analysis (Michigan State University, USA, June 30-July 3, 2008)
x. Reviewer for Zantrablatt Math and Mathematical Reviews (more than 25 papers since 2009)
xi. Referee of several papers for Math journals (15 papers since 2008.)
xii. Referee of 4 research proposals (3 for Qatar Foundation and 1 for the Austrian Science Fund.)
f. Community Service: Member of the Board
of Directors for Campus Daycare May 2011- August 2014.
g. Outreach: Volunteered at info days
every year (2009-2014).
II.
At Western
Illinois University (August 2003- May 2008)
a. Graduate Students Advisory Committees
i. 2003-2004:
Javid Siddique; M.Sc. in Mathematics.
ii. 2003-2004:
Duygu Inceoz; M.Sc. in
Mathematics.
b. Thesis External Examiner
i. 2008:
Dodzi Attimu; PhD in Mathematics; Howard University (US).
c. Departmental Committees
i. 2007-2008
1. Graduate Committee (Chair)
2. Personnel Committee
3. Curriculum Committee
ii. 2006-2007
1. Graduate Committee (Chair)
2. Colloquium Committee
3. Computer Affairs Committee
4. Curriculum Committee
iii. 2005-2006
1. Graduate Committee (Chair)
2. Colloquium Committee
3. Curriculum Committee
iv. 2004-2005
1. Graduate Committee (Chair)
2. Colloquium Committee
3. Curriculum Committee
4. Upper Division Curriculum Committee
v. 2003-2004
1. High School Relations Committee
2. Library Committee (Chair)
3. Upper Division Curriculum Committee
As the Chair of the Graduate
Committee (Fall 2004-Spring 2008), I led the efforts of that committee in
recruiting and advising graduate students as well as assisting the Department Chair
with various matters pertaining to the graduate program. I have led the efforts
of the Graduate Committee in preparing the proposals for revisions in the
graduate program including new structure, courses and tracks to better serve
the needs of the incoming graduate students. I gathered ideas, thoughts and
suggestions from the various groups in the Department about the structure and
the contents of the courses in the revised program and designed a web page for
that purpose to facilitate the discussions and the exchange of ideas in the
Department. As a result, proposals for a revised program were submitted and
approved by the Dean and the Graduate Council; and the revised program started
in Fall 2006.
d. College (Arts and Science) and University
Committees
i. 2007-2008
1. University Research Council
2. Evidence of English Proficiency and
Minimum TOEFL Requirements Committee
3. College of Arts & Sciences
Graduate Studies Committee
ii. 2006-2007
1. University Research Council
2. Evidence of English Proficiency and Minimum
TOEFL Requirements Committee
3. College of Arts & Sciences
Graduate Studies Committee
iii. 2005-2006
1.
College
of Arts & Sciences Graduate Studies Committee
III.
At Michigan
State University: Coordinator of the
Math Learning Center (2002-2003), supervising all the tutors and teaching assistants who worked at the
center.
o Nominated for the J. S. Frame Teaching Excellence Award for Faculty, Spring 2003, Department of Mathematics, Michigan State University.
o Nominated for the J. S. Frame Teaching Excellence Award for Faculty, Spring 2002, Department of Mathematics, Michigan State University.
o Hariri Foundation Fellowship to study in the USA (M.Sc.: 1988-1990 and PhD: 1993-1997); see this article, p.4.
o Member of The Honor Society of Phi Kappa Phi.
o Member of Phi Beta Delta, the Honor Society for International Scholars.
o BS in Physics with High Distinction, American University of Beirut, June 1988.
o Philip Hitti Award, American University of Beirut, June 1988, for graduating with the highest average in the Faculty of Arts and Sciences.
o Dean's Honor List, faculty of Arts and Sciences, American University of Beirut, 1986-1988.
o Malcolm
Kerr Award, American University of Beirut, 1986-1988.
o American Mathematical Society
o Canadian Association of Physicists
o Canadian Mathematical Society
o Arabic: fluent (reading, speaking and writing).
o English: fluent (reading, speaking and writing).
o French:
good (reading, speaking and writing).