In this talk, I will give an overview of my work on non-Archimedean ordered field extensions of the real numbers that are real closed and complete in the order topology. The smallest such field, the Levi-Civita field *R*, is small enough to allow for the calculus on the field to be implemented on a computer and used in computational applications such as the fast and accurate computation of the derivatives of real functions up to very high orders.Most recently, we developed an operator theory on the space *C:=R+iR* of null sequences of elements of the complex Levi-Civita field C := R + iR. A natural inner product can be defined on c_{0}, which induces the sup-norm of c_{0}; but unlike classical Hilbert spaces, c_{0} is not orthomodular with respect to this inner product, so I will characterize those closed subspaces that have orthonormal complements. I will also present characterizations of normal projections, adjoint and self-adjoint operators, and compact operators as well as the properties of positive operators on c_{0}.

**Speaker**: Dr. Khodr Shamseddine, Prairie Physics Seminar, Department of Physics, University of Manitoba

**Bio**:

B.Sc. in Physics (with a minor in Mathematics): American University of Beirut (1988)

M.Sc. in Physics: Michigan State University (1990)

Dual PhD in Physics and Mathematics: Michigan State University (1999)

Postdoctoral position: Michigan State University, January 2000- August 2003

Tenure-track assistant professor: Department of Mathematics, Western Illinois University, August 2003- August 2007

Tenured associate professor: Department of Mathematics, Western Illinois University, August 2007- May 2008

Tenure-track associate professor: Department of Physics and Astronomy and adjunct professor: Department of Mathematics at the University of Manitoba, May 2008- present; tenured in 2014; moved from Illinois to Manitoba to join my wife- also a professor in the Department of Physics and Astronomy.

My research is on non-Archimedean field extensions of the real numbers (i.e. fields that contain the real numbers but also numbers that are infinitely small and infinitely large) and their applications in Physics.