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Carlos E. Ortiz
Professor, Computer Science and Mathematics
Boyer 112b (215) 572-4057 By Appointment
About Me
Although I have a degree in Civil Engineering, I decided to pursue graduate studies in pure mathematics. I got my Ph.D. in mathematics from the University of Wisconsin-Madison in 1997.
I have been very lucky to be at Arcadia University. I got to know, mentor and interact with a diverse body of students during my years here. Students have challenged and encouraged me to critically look at my teaching and mentoring, but more importantly, they have profoundly influenced my life outlook.
I have been interested in teaching and pedagogy since my undergraduate years and I participated in different pedagogical initiatives over my 20 years at Arcadia, including Calculus Reform, The Justice Collective, and the development of First Year Seminars at Arcadia. Presently I am interested in incorporating into the general curriculum of Arcadia the key tools of Big Data and Data Mining. I feel that proficiency in these areas will be essential to every college graduate in the very near future.
Over my years at Arcadia, I was also given the freedom and support to explore many areas of human knowledge where mathematics, and in particular model theory, can be applied. This has led me to work and publish in diverse areas of mathematics and different scientific fields such as Functional Analysis, Theoretical Computer Science, Genomics and Finance. Presently I do research in Finite Model Theory and in Finance. My project for the coming years is to learn and study neural networks and deep learning in A.I. from the viewpoint of finite model theory.
Areas Of Focus
Theoretical Computer Science, Finite model Theory, Applications of Logic to Functional Analysis, Applications of logic diverse fields such as to Finance Theory, International Math Education, Genomics.
Languages
English, French, Spanish
Education History
University of Wsconsin-Madison 1997
Ph.D., Major in Mathematics
Publications
First Order Extensions of Residue Classes and Uniform Circuit Complexity
An Innovative Form of Credit Enhancement For Securitized Reverse Mortgages:
When do Securitized Reverse Mortgages Become Liabilities
Securitization of Financial Asset/Liability Products with Longevity Risk
Approximate Formulae for a Logic that Captures Classes of Computational Complexity
Delta Hedging a Portfolio of Servicing Rights under Gamma and Vega Constraints with Optimal Fixed Income Securitie
Can ARMs Mortgage Servicing Portfolios be Delta Hedged under Gamma Constraints?
Approximate Formulae for a Logic that Captures Classes of Computational Complexity
Methods of class field theory to separate logics over finite residue classes and circuit complexity
Beer Annuities: Hold the interest and Principal
Reverse Engineering of Prepayment Functions and the Potential Applications for MSRs and IOs Valuation
Drosophila Muller F Elements Maintain a Distinct Set of Genomic Properties Over 40 million Years of Evolution
Research Summary
I am interested in theoretical computer science and in applications of model theory to different areas of mathematics as well as in other fields.
Currently I am working in descriptive complexity theory, which is an area of theoretical computer science that studies computational complexity classes from the point of view of finite model theory. My work focuses on developing tools to achieve separation of complexity classes by studying the model theory of finite-model logics associated with these classes. Some of my papers on this topic are:
1. Expressive Power and Complexity of a Logic with Quantifiers that Count Proportions of Sets, with A. Arratia, Journal of Logic and Computation, 16 (2006), 817-840,
2. First Order Extensions of Residue Classes and Uniform Circuit Complexity, with A. Arratia, Logic, Language, Information and Computation, WoLLIC 2013, Lecture Notes in Computer Sciences (L. Libkin et al editors), 8071 (2013), 49-63.
I am also looking into applications of logic to classical mathematical fields. For my dissertation I explored the model theory of analytic structures by studying the possibility of obtaining classical model theoretic results, such as the Omitting Types Theorem, in a very expressive infinitary logic that includes negation, infinite conjunction and existential quantification over infinitely many variables. The main tool that I developed was the notion of approximate formulas for the formulas in this logic. Some of the results obtained can be found in:
1. An Omitting Types Theorem for Normed Spaces”, a paper that studies the existence of a powerful Omitting Types Theorem for the logic of approximate formulas in normed spaces. This paper appeared in the Annals of Pure and Applied Logic, 108 (2001),
2. "Uniform Versions of Infinitary Properties in Banach Spaces", a paper that studies the following situation: Suppose that there is a statement P true in all the metric structures of a class, under which conditions do we know that the "uniform version" of this statement holds? And what is the "uniform version"? The answer is that this situation holds for a very large set of properties (namely, the ones that are expressible in an infinitary language L). In this paper I also obtain a general recipe to obtain the “uniform” version of the formulas in the language L.
I have also done work in applications of mathematical logic to diverse fields such as Finance Theory and Genomics. With my colleagues at Arcadia we have also reflected on the internationalization of the mathematics curriculum at college level.