I am an Associate Professor of Mathematics and the Chair of the Mathematics Program at Centre College. I teach courses in the Mathematics, the Computer Science, and the Philosophy Program.
I can be reached via e-mail at alexmcal@centre.edu or in my office in 117 Olin Hall during my office hours. You can also reach me via any of the following:

Math Department Centre College 600 West Walnut Street Danville, KY 40422-1394

(859) 238 - 5408 (office) (859) 238 - 5314 (dept) (859) 236 - 9610 (fax) (859) 936 - 9651 (home) preferably for time = t where 9 AM < t < 9 PM

I love teaching - that's one of the main reasons came to Centre College. You can reach course web pages via the following links:

I am also a big proponent of reading and, in particular, of reading mathematics textbooks. And not just the exercises in the text, but all the ideas and examples and definitions and theorems that come before the exercises. Some of my thoughts and reflections on this topics can be found at: Reading Your Mathematics Textbook.

KYMAA Talk: slides

My greatest love in mathematics is mathematical logic and my specialty in logic is Computability Theory. The fundamental notion in Computability Theory is the formalization of the intuitive notion of an algorithm. Numerous formal definitions were introduced by Church, Gödel, Kleene, and Turing in the 1930's and 1940's and were all eventually shown to be equivalent. Perhaps the most commonly used formalization is that of a Turing machine, which is essentially a computer program idealized to ignore time and space limitations.

Among other things, we seek to classify and understand the relative information content of various countable sets based on their ability to compute one another. In particular, X is computable in Y if there is a Turing machine (or computer program) which can identify the elements of X, given access to complete information about the elements of Y. This notion induces an equivalence relation on the sets of natural numbers whose equivalence classes are partially ordered and possess a rich algebraic structure which has been intensely studied.

My personal research interests have grown out of studying the relative computability of various models and enumerations of families of sets. A model is a set with some relations which satisfy certain properties; e.g., groups are models of group theory and fields are models of field theory. An enumeration of a family of sets is a list of precisely the sets in the family. If you look at my publications, you will see that I have demonstrated various conditions under which certain structures and enumerations can and can not compute other structures, enumerations, and sets.

I have also studied the degree-theoretic structure of the families of sets that arise in these contexts. These families of sets have nice definitions, but these definitions also limit our ability to perform certain constructions. Although much is known, there are open questions about some of these families of sets; e.g., does there exist a set in the family which is Turing incomparable with a given set in the family?

There are many interesting, open questions which are similar in nature to those we have mentioned above and on which we are currently working. A brief technical list of my research interests would include the following:

• enumerations
• models and completions of Peano arithmetic, set theory and other theories with a certain "richness"
• Scott sets and weak Scott sets
• families of sets representable and weakly representable in theories,
• models possessing certain weak saturation properties
• Turing minimality of models and enumerations

The following is a list of my publications and manuscripts; if you're interested in a copy of a paper, or if you have any questions or comments, please contact me at alexmcal@centre.edu.

Research Publications

• Turing upper bounds for countable jump ideals,
  in Nonstandard Models of Arithmetic and Set Theory, Contemporary Mathematics,
  ISSN: 0271-4132, Volume 361, October 2004, 129 - 143.
  submitted April 2003, accepted June 2003, revised October 2003.
• Bounded Scott Set Saturation,
  Mathematical Logic Quarterly, Volume 48, Number 2, 2002, 245 - 259.
  submitted December 2000, revised April 2001, accepted May 2001.
• Computability in structures representing a Scott set,
  Archive for Mathematical Logic, Volume 40, 2001, 147 - 165.
  submitted October 1997, accepted May 1999.
• Completions of PA: models and enumerations of representable sets,
  Journal of Symbolic Logic, Volume 63, Number 3, September 1998, 1063 - 1082.
  submitted January 1996, revised January 1997, accepted March 1997.
• Evaluation of high-level bound-bound and bound-continuum hydrogenic oscillator
  strengths by asymptotic expansion, (K. Omidvar and A.M. McAllister),
  Physical Review A, Volume 51, Number 2, Febuary 1995, 1063 - 1066.

• The Turing partial order of w-models defined via completions of arithmetic,
  prepared Summer 2001, 15 pages.
• Scott set saturation and the relative computability of models and enumerations,
  prepared Summer 1998, 35 pages.

• The Kentucky Section goes high-tech: The tale of an e-newsletter,
  Focus of the MAA, Volume 23, Number 2, February 2003, 8 - 9.
  submitted, revised, accepted December 2002.
• An Experiment that Worked: Revising the Calculus Curriculum,
  (with William W. Johnston and John H. Wilson)
  Focus of the MAA, Volume 21, Number 8, November 2001, 8 - 9.
  submitted, revised, accepted September 2001.

• An Introduction to LaTeX and VTeX, prepared August 2003, 12 pages.
• Introduction to Maple, prepared August 2002, 4 pages.
  
• A Dreamweaver Primer, prepared May 2002, 28 pages.
  
• My Understanding of the ACS Information Fluency Project, prepared February 2002, 2 pages.
  
• Reading Your Mathematics Textbook, prepared September 1999, 2 pages.
  

• A Survey of Advanced Mathematical Thought,
  with William W. Johnston
  A textbook for a transition course surveying mathematics.
  Anticipated publication by Oxford University Press in September 2008.
• Schaum's Easy Outlines: Logic,
  McGraw-Hill, New York, 2005. ISBN: 0-07-145535-3.
  An abridgement based on Schaum’s Outline of Theory and Problems of Logic, Second Edition
  by John Nolt, Dennis Rohatyn, and Achille Varzi, McGraw-Hill, New York, 1998.
  Abridgment Editor: Alex M. McAllister, 135 pages.
  submitted January 2005, accepted February 2005, published September 2005.
  

• #74 Danny, Vantage Point, Volume 36, Number 2, January 2003.
• #87 Questions, Vantage Point, Volume 36, Number 2, January 2003.
  

I am married to Julie Eberhardt McAllister. We met each other at Virginia Tech where Julie earned her B.S. in Industrial and Systems Engineering. Julie completed two years towards a Masters of Divinity at the University of Notre Dame and completed her Master's degree in theology at St. Michael's College in Burlington, Vermont. Julie is currently on a hiatus from serving as an adult and youth minister to care for our children and parents (and me!).

We have a son named Benjamin Martin who turned eleven last May, a son named Daniel John who turned nine last July, and a daughter named Ella Margaret turned three in November. We also have two puppies - BJ and Bridget. Our family enjoys soccer, scouts, walking, festive holidays, and good friends. We're also in the middle of gutting and rebuilding a 100 year old house that we hope to live in soon. The rest of our family resides throughout the Southeast in Alabama, Georgia, Maryland, and West Virginia.

Some of my hobbies include biking, running, playing guitar, woodworking, doing yardwork, watching movies, and playing with my wife and my kids. I also love to play pick-up and intramural soccer games with Centre students -- come join us on Sunday afternoons on the intramural fields behind the pool.

Some more biographical information is available on the mathematics department webage at: Mathematics Biographies.

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