Math Faculty |
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| Steven Buechler Peter Cholak Julia F. Knight Sergei Starchenko |
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| Philosophy Faculty | ||||||||||||||||||||||||
| Timothy Bays Patricia Blanchette Michael Detlefsen Curtis Franks |
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| Graduate Students | ||||||||||||||||||||||||
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| Jacob Heidenreich was enrolled in a joint Mathematics and Philosophy Ph.D. program. Andrew Arana was the first Ph.D. graduate of this program. For more details contact Julia Knight and/or Michael Detlefsen. For the Philosophy graduate students see the Department of Philosophy home page. | ||||||||||||||||||||||||
| Some recent mathematical logic Ph.D's (With latest known job information) |
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| Activities |
| Throughout the academic year the group maintains a weekly Logic Seminar. When we have an out-of-town speaker we normally go out as group. An up-to-date listing can be found off the Mathematics Department home page. In addition to the basic graduate logic course and two undergraduate courses in mathematical logic, an advanced graduate topics class is offered almost every semester. The logic group holds a logic picnic every year in the late summer at Warren dunes. |
| Our Research From the graduate bulletin: |
| The research in mathematical logic at Notre Dame is mainly in two broad areas: computability theory and model theory. Computability theory concerns computability and complexity, often measured by Turing degree. A set is computable if there is a program for computing its characteristic function on an ideal computer that never crashes. Set A is Turing reducible to set B if there is a program for computing the characteristic function of A on a computer equipped with a CD-ROM giving the characteristic function of B. Turing reducibility is a partial ordering on the set of subsets of the natural numbers, and the Turing degrees are the equivalence classes of the corresponding equivalence relation. A set is computably enumerable if it is the range of a computable function, or, equivalently, the domain of a partial computable function. The set e of all computably enumerable subsets of the natural numbers forms a lattice under the operations of union and intersection. Soare showed that the collection of "maximal" sets is a definable orbit in e. There is ongoing work on automorphisms, and the relation between complexity and structural properties, definable in the lattice. Well-known theorems may pose interesting problems in computability. This is true, in particular, for Ramsey's theorem, on which there is recent work. There has been quite a lot of work on computability and complexity in familiar kinds of mathematical structures-groups, linear orderings, Boolean algebras, etc. Much of this work has involved connections between definability and complexity. There has also been work on complexity of models of arithmetic. The standard model, consisting of the natural numbers with addition and multiplication, is computable; i.e., the operations are computable. Tennenbaum showed that no non-standard model can be computable. A recent result says that for any non-standard model, there is an isomorphic copy of strictly lower Turing degree. The other broad area of active work is model theory, particularly classification theory and o-minimality. In recent years, method developed in the context of stability theory have been used to analyze structures such as pseudofinite fields, pseudoalgebraically closed fields, difference fields, and quadratic forms over finite fields. This research has yielded applications to arithmetic number theory. Model-theorists now have a good understanding of how these dependence relations fit in a general framework. Ongoing work generalizes techniques from the geometrical stability theory of superstable theories to this broader class. This research is likely to give insight into the model-theoretic properties of bilinear forms and groups definable in structures such as those mentioned above. The standard example of an o-minimal structure is the field of real numbers. In the early 1980's, it was noticed that many properties of semialgebraic sets (sets definable in the field of reals) can be derived from a very few axioms, essentially the axioms defining o-minimal structures. After Wilkie proved that the exponential field of real numbers is o-minimal, the subject has grown rapidly. From a model-theoretic point of view, these structures resemble strongly-minimal structures, and many tools and methods of classification theory can be adapted to o-minimal structures. This remarkable combination of tools from stability theory and methods of semi-algebraic and subanalytic geometry provides elegant and surprisingly efficient applications not only in real algebraic and real analytic geometry, but also in analytic-geometric categories (e.g., groups of Lie type) over arbitrary real closed fields. |
| Other local sites of interest to logicians |
| The Notre Dame Journal of Formal Logic Computability Theory Home Page |
Last updated:
Tuesday, October 30, 2007 9:45 AM
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