Khodr Shamseddine's Professional Web Page

Welcome to my professional Web page

Currently, I am a professor and Associate Head for Undergraduate Studies (and Chair of the Undergraduate Committee, formerly the Curriculum Committee) in the Department of Physics and Astronomy, an Adjunct Professor in the Department of Mathematics and an Ian C. P. Smith Integrated Science Faculty Scholar at the University of Manitoba. I am also a former Director of the Winnipeg Institute for Theoretical Physics (January 1, 2015- December 31, 2016) and an associate at the Center for Advanced Mathematical Sciences at the American University of Beirut. Prior to arriving at the University of Manitoba, I was a tenured associate professor in the Department of Mathematics at Western Illinois University.

Contact Information	Education	Refereed Publications
Invited Talks	Contributed Talks	Seminars and Colloquia
Research Interest	Training of HQP	Conference Organization
Teaching	Service	Honor Societies/Awards
Professional Societies	Languages	Research Group Photo
2018 Merit Award in Teaching, Service and Research

Contact Information

Address: Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
Phone: (204)474-6207
Fax: (204)474-7622
Email: khodr.shamseddine@umanitoba.ca
Web: www2.physics.umanitoba.ca/u/khodr

Education

Dual Ph.D. in Physics and Mathematics, Michigan State University. Dissertation title: New Elements of Analysis on the Levi-Civita Field.
M.Sc. in Physics, Michigan State University (GPA: 4.0/4.0).
B.Sc. in Physics (minor in Mathematics), American University of Beirut (with High Distinction).

Refereed Publications

I. Books

Advances in Ultrametric Analysis, Proceedings of the 14^th International Conference on p-Adic Functional Analysis, Alain Escassut, Cristina Perez-Garcia, and Khodr Shamseddine, editors, Contemporary Mathematics, American Mathematical Society, Volume 704, 2018, ISBN: 978-1-4704-3491-5.
Advances in Non-Archimedean Analysis, Proceedings of the 13^th International Conference on p-Adic Functional Analysis, Helge Glockner, Alain Escassut and Khodr Shamseddine, editors, Contemporary Mathematics, American Mathematical Society, Volume 665, 2016, ISBN: 978-1-4704-1988-2.
Advances in Ultrametric Analysis, Proceedings of the 12^th International Conference on p-Adic Functional Analysis, Khodr Shamseddine, editor, Contemporary Mathematics, American Mathematical Society, Volume 596, 2013, ISBN: 978-0-8218-9142-1.
Advances in p-Adic and Non-Archimedean Analysis, Proceedings of the 10th International Conference on p-Adic and Non-Archimedean Analysis, Martin Berz and Khodr Shamseddine, editors, Contemporary Mathematics, American Mathematical Society, Volume 508, 2010, ISBN: 978-0-8218-4740-4.

II. Papers

Taylor's theorem, the inverse function theorem and the implicit function theorem for weakly locally uniformly differentiable functions on non-Archimedean spaces, Khodr Shamseddine, submitted.
On computational applications of the Levi-Civita field, Darren Flynn and Khodr Shamseddine, Journal of Computational and Applied Mathematics, in print.
On the topological structure of the Hahn field and convergence of power series, Darren Flynn and Khodr Shamseddine, Indagationes Mathematicae, Volume 30 # 5, 2019, pp. 773-795.
On an operator theory on a Banach space of countable type over a Hahn field, Khodr Shamseddine and Changying Ding, Proceedings of the 11th ISAAC Congress (Växjö, Sweden, August 2017), Analysis, Probability, Applications, and Computation, April 2019, pp. 267-282.
Summary on non-Archimedean valued fields, Angel Barria Comicheo and Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 704 (Advances in Ultrametric Analysis), 2018, pp. 1-36.
Calculus on a non-Archimedean field extension of the real numbers: inverse function theorem, intermediate value theorem and mean value theorem, Gidon Bookatz and Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 704 (Advances in Ultrametric Analysis), 2018, pp. 49-67.
On Integrable Delta Functions on the Levi-Civita Field, Darren Flynn and Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 10 # 1, 2018, pp. 32-56.
Positive operators on a free Banach space over the Levi-Civita field, Jose Aguayo, Miguel Nova and Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 9 # 2, 2017, pp. 122-137.
A local mean value theorem for functions on non-Archimedean field extensions of the real numbers, Khodr Shamseddine and Gidon Bookatz, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 8 # 2, 2016, pp. 160-175.
Measure theory and Lebesgue-like integration in two and three dimensions over the Levi-Civita field, Khodr Shamseddine and Darren Flynn, Proceedings of the 13^th International Conference on p-Adic Functional Analysis, Contemporary Mathematics, American Mathematical Society, Volume 665 (Advances in Non-Archimedean Analysis), 2016, pp. 289- 325.
On the solutions of linear ordinary differential equations and Bessel-type special functions on the Levi-Civita field, Alpar Meszaros and Khodr Shamseddine, Journal of Contemporary Mathematical Analysis, Volume 50 # 2, 2015, pp. 53-62.
Analysis on the Levi-Civita field and computational applications, Khodr Shamseddine, Applied Mathematics and Computation, Volume # 255, 2015, pp. 44-57.
Inner product on B*-algebras of operators on a free Banach space over the Levi-Civita field, Jose Aguayo, Miguel Nova and Khodr Shamseddine, Indagationes Mathematicae, Volume 26 # 1, 2015, pp. 191-205.
A brief survey of the study of power series and analytic functions on the Levi-Civita fields, Khodr Shamseddine, Proceedings of the 12th International Conference on p-Adic Functional Analysis, Contemporary Mathematics, American Mathematical Society, Volume 596 (Advances in Ultrametric Analysis), 2013, pp. 269-280.
One-variable and multi-variable calculus on a non-Archimedean field extension of the real numbers, Khodr Shamseddine, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 5 # 2, 2013, pp. 160-175.
Characterization of compact and self-adjoint operators on Free Banach spaces of countable type over the complex Levi-Civita field, Jose Aguayo, Miguel Nova and Khodr Shamseddine, Journal of Mathematical Physics, Volume 54 # 2, 2013.
New results on integration on the Levi-Civita field, Khodr Shamseddine, Indagationes Mathematicae, Volume 24 # 1, 2013, pp. 199-211.
Preliminary notes on Fourier series for functions on the Levi-Civita field, Khodr Shamseddine and William Grafton, International Journal of Mathematical Analysis, Volume 6 # 19, 2012, pp. 941-950.
On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers, Khodr Shamseddine and Todd Sierens, ISRN Mathematical Analysis, Volume 2012, Article ID 387053, 20 pages.
Absolute and relative extrema, the mean value theorem and the inverse function theorem for analytic functions on a Levi-Civita field, Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 551 (Advances in Non-Archimedean Analysis), 2011, pp. 257-268.
Nontrivial order preserving automorphisms of non-Archimedean fields, Khodr Shamseddine, Contemporary Mathematics, American Mathematical Society, Volume 547 (Function Spaces in Modern Analysis), 2011, pp. 217-225.
On the topological structure of the Levi-Civita field, Khodr Shamseddine, Journal of Mathematical Analysis and Applications, Volume 368, 2010, pp. 281-292.
Analysis on the Levi-Civita field, a brief overview, Khodr Shamseddine and Martin Berz, Contemporary Mathematics, American Mathematical Society, Volume 508 (Advances in p-Adic and Non-Archimedean Analysis), 2010, ISBN 978-0-8218-4740-4, pp. 215-237.
The implicit function theorem in a non-Archimedean setting, Khodr Shamseddine, Trevor Rempel and Todd Sierens, Indagationes Mathematicae, Volume 20 # 4, 2009, pp. 603-617. [This paper is the result of my work with two NSERC-USRA students at the University of Manitoba in Summer 2009]
Intermediate value theorem for analytic functions on a Levi-Civita field, Khodr Shamseddine and Martin Berz, Bulletin of the Belgian Mathematical Society-Simon Stevin, Volume 14, 2007, pp. 1001-1015.
Generalized power series on a non-Archimedean field, Khodr Shamseddine and Martin Berz, Indagationes Mathematicae, Volume 17 # 3, 2006, pp. 457-477.
Analytical properties of power series on Levi-Civita fields, Khodr Shamseddine and Martin Berz, Annales Mathématiques Blaise Pascal, Volume 12 # 2, 2005, pp. 309-329.
Constrained second order optimization on non-Archimedean fields, Khodr Shamseddine and Vera Zeidan, Indagationes Mathematicae, Volume 14 # 1, 2003, pp. 81-101.
Measure theory and integration on the Levi-Civita field, Khodr Shamseddine and Martin Berz, Contemporary Mathematics, American Mathematical Society, Volume 319 (Ultrametric Functional Analysis), 2003, ISBN: 0-8218-3320-0, pp. 369-387.
Intermediate values and inverse functions on non-Archimedean fields, Khodr Shamseddine and Martin Berz, International Journal of Mathematics and Mathematical Sciences, Volume 30 # 3, 2002, pp. 165-176.
On the existence and uniqueness of solutions of differential equations on the Levi-Civita field, Khodr Shamseddine, International Journal of Differential Equations and Applications, Volume 4 # 4, 2002, pp. 375-386.
Infinite dimensionality of the solution space of y’=0 on non-Archimedean fields, Khodr Shamseddine, International Journal of Differential Equations and Applications, Volume 4 # 1, 2002, pp. 25-30.
One-dimensional optimization on non-Archimedean fields, Khodr Shamseddine and Vera Zeidan, Journal of Nonlinear and Convex Analysis, Volume 2 # 3, 2001, pp. 351-361.
Convergence on the Levi-Civita field and study of power series, Khodr Shamseddine and Martin Berz, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Proceedings of the Sixth International Conference on P-adic Analysis, July 2-9, 2000, ISBN 0-8247-0611-0, pp. 283-299.
The differential algebraic structure of the Levi-Civita field and applications, Khodr Shamseddine and Martin Berz, International Journal of Applied Mathematics, Volume 3 # 4, 2000, pp. 449-464.
Power series on the Levi-Civita field, Khodr Shamseddine and Martin Berz, International Journal of Applied Mathematics, Volume 2 # 8, 2000, pp. 931-952.
Exception handling in derivative computation with non-Archimedean calculus, Khodr Shamseddine and Martin Berz, Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, A. Griewank, eds., SIAM, Philadelphia, Penn, 1996, pp. 37-51.
COSY INFINITY and its applications to nonlinear dynamics, Martin Berz, Kyoko Makino, Khodr Shamseddine, Georg Hoffstatter and Weishi Wan, Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, A. Griewank , eds., SIAM, Philadelphia, Penn, 1996, pp. 363-367.

Invited Talks at Conferences

1. Calculus theorems for locally uniformly differentiable functions on a non-Archimedean ordered field extension of the real numbers, Seventh International Conference on p-adic Mathematical Physics and its Applications, Covilha, Portugal, September 30-October 4, 2019.

2. One-variable and Multi-variable Integral Calculus over the Levi-Civita Field and Applications, Sixth International Conference on p-adic Mathematical Physics and its Applications, Mexico City, Mexico, October 23-27, 2017.

3. On the Levi-Civita Fields: Introduction and Summary of Selected Recent Research, 11^th Congress of the International Society for Analysis, its Applications and Computations (ISAAC), Växjö, Sweden, August 14-18, 2017.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. Preliminaries in non-Archimedean Functional Analysis, The Seventh Conference on Function Spaces, Southern Illinois University- Edwardsville, Illinois, USA, May 20-24, 2014.

11. 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.

12. 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.

13. (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.

14. (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.

15. 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.

16. 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.

17. 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.

18. 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.

19. Analytical properties of power series on Levi-Civita fields, Eighth International Conference on P-adic Functional Analysis, Clermont-Ferrand, France, July 5-9, 2004.

20. Measure theory and integration on the Levi-Civita field, Seventh International Conference on P-adic Functional Analysis, Nijmegen, The Netherlands, June 17-21, 2002.

21. 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.

22. 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.

23. 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

Elements of an operator theory on the space c₀ over a non-Archimedean valued field, 46^th Canadian Operator Symposium, University of Manitoba, June 4-8, 2018.
The Non-Archimedean Field R, Overview and Applications, International Conference on Scientific Computations, LAU, Beirut, Lebanon, March 1999.
Non-Archimedean Structures as Differentiation Tools, Second LAAS International Conference on Computer Simulation, AUB, Beirut, Lebanon, September 1997.
Exception Handling in Derivative Computation with Non-Archimedean Calculus, SIAM 96: Second International Workshop on Industrial and Applied Mathematics, Santa Fe, New Mexico, February 1996.

Invited Talks, Seminars and Colloquia at Universities

Department of Physics and Astronomy and Winnipeg Institute for Theoretical Physics (joint colloquium), University of Manitoba, March 8, 2019.
Department of Mathematics, Universidad de Concepcion, Concepcion, Chile, March 23, 2018.
Department of Mathematics, American University of Beirut, Beirut, Lebanon, August 31, 2017.
Numerical Calculus Laboratory, University of Calabria, Rende, Italy, June 28, 2016.
Department of Mathematics, American University of Beirut, Beirut, Lebanon, July 22, 2015.
Department of Mathematics (Functional Analysis seminar, part II), University of Manitoba, March 17, 2015.
Department of Mathematics (Functional Analysis seminar, part I), University of Manitoba, March 10, 2015.
Department of Mathematics, University of Manitoba, March 21, 2014.
Department of Physics, University of Regina, March 7, 2014.
Department of Physics and Engineering Physics, University of Saskatchewan, March 6, 2014.
Department of Mathematics & Statistics (Algebra seminar), University of Saskatchewan, March 6, 2014.
Department of Physics and Astronomy and Winnipeg Institute for Theoretical Physics (joint colloquium), University of Manitoba, April 10, 2013.
Science Department, Texas A & M University in Qatar, October 23, 2012.
Department of Mathematics, Western Illinois University, August 31, 2012.
Departments of Mathematics (Joint Mathematics Colloquium), Universidad del Bio-Bio and Universidad de Concepcion, Concepcion, Chile, December 5, 2011.
Department of Mathematics, American University of Beirut, Beirut, Lebanon, July 21, 2011.
Department of Mathematics, University of Wisconsin- Eau Claire, December 17, 2008.
Winnipeg Institute of Theoretical Physics, November 5, 2008.
Winnipeg Institute of Theoretical Physics, October 22, 2008.
Department of Mathematics, University of Wisconsin- Eau Claire, July 7, 2007.
Department of Mathematics, University of Wisconsin- Eau Claire, June 21, 2007.
Department of Physics & Astronomy, University of Manitoba, November 24, 2006.
Department of Mathematics, University of Manitoba, November 24, 2005.
Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, August 12, 2005.
Center for Research in Applied Mathematics & Statistics (CRAMS), Business and Computer University College, Beirut, Lebanon, January 7, 2005.
Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, January 5, 2005.
Department of Mathematics, Western Illinois University, November 11, 2004.
Department of Mathematics, University of Wisconsin- Eau Claire, July 13, 2004.
Department of Physics and Astronomy, Michigan State University, June 18, 2004.
Department of Mathematics, University of Wisconsin- Eau Claire, April 30, 2004.
Department of Mathematics, Western Illinois University, February 5, 2004.
Department of Mathematics, Western Illinois University, February 14, 2003.
Department of Mathematics and Computer Science, Lawrence Technological University, Michigan, May 14, 2002.
Department of Mathematics, Michigan State University, October 8, 2001.
Center for Advanced Mathematical Sciences (CAMS), American University of Beirut, Beirut, Lebanon, July 19, 2000.
National Superconducting Cyclotron Laboratory, Michigan State University, November 24, 1999.
Department of Mathematics, American University of Beirut, Beirut, Lebanon, March 24, 1999.

Research Interests

My research interests and activities include various areas of non-Archimedean Analysis: one-variable and multi-variable calculus, power series and analytic functions, measure theory and integration, optimization, existence and uniqueness of solutions of differential equations, complex analysis, and functional analysis over non-Archimedean valued fields. 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 ordered field extension of the real numbers that is complete in the topology induced by the order 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 have studied in my research group 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 dropped that restriction and showed that power series on the 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 ordered 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. We generalize the results of to two and three dimensions. In particular, we defined a Lebesgue-like measure on R² (resp. R³). Then we defined measurable functions on measurable sets using analytic functions in two (resp. three) variables and showed how to integrate those measurable functions using iterated integration. The resulting double (resp. triple) integral satisfies similar properties to those of the single integral as well as those properties satisfied by the double and triple integrals of real calculus.

Together with my collaborators Jose Aguayo and Miguel Nova from Concepcion (Chile), we developed an operator theory on the Banach space c₀ over C:=R+iR, where c₀ denotes the space of all null sequences of elements of C. The natural inner product on c₀ induces the sup-norm of c₀. We showed that c₀ is not orthomodular; then we characterized those closed subspaces of c₀ 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 c₀. 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 c₀ 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 also expanded 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, we studied the properties of weakly locally uniformly differentiable functions at a point or on an open subset of F or Fⁿ and we proved local versions of the intermediate value theorem, the mean value theorem and Taylor's theorem for weakly locally uniformly differentiable functions on F. We also proved the inverse function theorem and implicit function theorem for weakly locally uniformly differentiable functions from Fⁿ to Fⁿ and from Fⁿ to F^m(m < n), respectively. Moreover, the work on the topological structure as well as on the integration theory and its applications on the Levi-Civita field R has recently been extended to the field F.

Training of Highly Qualified Personnel (at U of M)

I. Postdoctoral Fellows:

1. Angel Barria Comicheo (March 1-August 31, 2020); Department of Mathematics and Department of Physics and Astronomy (joint appointment).

2. Angel Barria Comicheo (March 1-August 31, 2019); Department of Mathematics.

II. Graduate Students

1. Darren Flynn (September 2014- August 2019); PhD in Physics.

2. Angel Barria Comicheo (January 2013- October 2018); PhD in Mathematics.

3. Andrew Senchuk (January 2011- May 2018; I was co-advisor, with advisor: Prof. Gerald Gwinner); PhD in Physics.

4. William Grafton (September 2013- August 2016); M.Sc. in Mathematics.

5. Darren Flynn (July 2012- August 2014); M.Sc. in Physics.

III. Undergraduate Students

o Summer Students

1. Jeremy Croitor; NSERC indigenous -USRA student in the joint Math-Physics Honours program.

2. Yudong Chen (2019); MITACS Globalink intern (from China).

3. Zhenghang Du (2019); MITACS Globalink intern (from China).

4. Changying Ding (2017); MITACS Globalink intern (from China).

5. Jeremy Friesen (2017); NSERC-USRA student in the joint Math-Physics Honours program.

6. Gidon Bookatz (2016); undergraduate summer student in the joint Math-Physics Honours program.

7. Ryan Sherbo (2015); NSERC-USRA student in the joint Math-Physics Honours program.

8. Gidon Bookatz (2015); NSERC-USRA student in the joint Math-Physics Honours program.

9. Gidon Bookatz (2014); NSERC-USRA student in the joint Math-Physics Honours program.

10. Gidon Bookatz (2013); NSERC-USRA student in the joint Math-Physics Honours program.

11. William Grafton (2012); undergraduate summer student in the joint Math-Physics Honours program.

12. James Roberts (2011); undergraduate summer Honours student, Department of Physics at the University of Winnipeg.

13. William Grafton (2011).

14. 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.

15. Todd Sierens (2009); NSERC-USRA student in the joint Math-Physics Honours program.

16. 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:

17. Cheng Tang (2012-2013).

Teaching

I. At the University of Manitoba (May 2008- present)

o Fall 2020: PHYS 3496, Mathematical Physics II

o Fall 2019: PHYS 3496, Mathematical Physics II

o Winter 2019: PHYS 2496, Mathematical Physics I

o Fall 2018: PHYS 3496, Mathematical Physics II

o Winter 2018: on sabbatical leave

o Fall 2017: Teaching credit as an Ian C. P. Smith Integrated Science (SIS) Faculty Scholar.

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

At Western Illinois University (August 2003- May 2008)

o Spring 2008

Ø MATH 133, Calculus and Analytic Geometry

Ø MATH 391, Writing in the Mathematical Sciences

o Fall 2007