Old
Speaker: Jennifer Chubb, GWU
1957 E Street, Room B16
Speaker: Andrew Kebo,
Old
Speaker: Joseph S. Miller,
http://www.math.uconn.edu/~josephmiller/
Title: Degrees
of unsolvability of continuous functions
1957
E Street, Room 211
Speaker: Simon Y. Berkovitch,
Computer Science, GWU
http://cs.seas.gwu.edu/people/faculty-detail.php?personID=3
Old
Speaker: Andrey Frolov,
Old
Speaker: Rehana
Patel,
OTHER LOGIC TALKS
Speaker: Lenore Blum, Distinguished
Career Professor of Computer Science,
Title: Computing over the Reals: Where Turing Meets
Abstract:
The classical (Turing) theory of computation has been
extraordinarily successful in providing the foundations and framework for
theoretical computer science. Yet its dependence on 0’s and 1’s is
fundamentally inadequate for providing such a foundation for modern scientific
computation where most algorithms—with origins in
Mike Shub,
Steve Smale and I introduced a theory of computation
and complexity over an arbitrary ring or field R. If R is Z2
= {0, 1}, the classical computer science theory is recovered. If R is
the field of real numbers,
Complexity classes P, NP and the fundamental question “Does P = NP?” can be formulated naturally over an arbitrary ring or field R. The answer to the fundamental question depends on the complexity of deciding feasibility of polynomial systems over R. When R is Z2, this becomes the classical satisfiability problem of Cook-Karp-Levin. When R is the field of complex numbers, the answer depends on the complexity of Hilbert’s Nullstellensatz.
The notion of reduction between discrete problems (e.g. between traveling salesman and satisfiability) has been a powerful tool in classical complexity theory. But now, in addition, the transfer of complexity results between domains (discrete and continuous) becomes a real possibility.
In this talk, I will discuss these results and indicate how basic notions from numerical analysis such as condition, round off and approximation are being introduced into complexity theory, bringing together ideas germinating from the real calculus of Newton and the discrete computation of computer science.
This talk will
be accessible to undergraduates.
Funger Hall, Room 223
Speaker: Natasha Dobrinen,
http://www.logic.univie.ac.at/~dobrinen/
Title: To infinity and beyond
Abstract: We give a brief history of set theory, starting with Cantor's investigations into the transfinite and building up to the three main lines of research in modern set theory, namely, inner models, large cardinals, and forcing. We show how these work together and include applications to other areas of mathematics. In conclusion, we present some areas of active research and future trends in modern set theory.
1957 E Street, Room 112
Speaker: Bjorn Kjos-Hanssen,
http://www.geocities.com/bjoernkjoshanssen/
Title:
Brownian motion and Kolmogorov complexity
Abstract: Brownian motion is a mathematical model of certain erratic behaviors, such as the fluctuation of stock prices. However, the model does feature slow times, during which the movement is more predictable. Probabilists have shown that the set of slow times has Lebesgue measure zero, but positive Hausdorff dimension.
Using methods of computability theory, we can say something about the individual real numbers t that occur as slow times. We show that they are all quite complicated, in the sense that their decimal expansions have high Kolmogorov complexity.
1957
Speaker: Timothy McNicholl,
Title: Computable aspects of inner functions
Abstract: The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also very useful in applied mathematics. Two foundational results in this theory are Frostman’s Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman’s Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u, the order of the zero at 0 (if there is any) and the Blaschke sum of u provide the exact amount of information necessary to compute the factorization of u. Along the way, we discuss some uniform computability results for Blaschke products. These results play a key role in the analysis of factorization. We use Type-Two Effectivity as our foundation.
Speakers: Sarah Pingrey and Jennifer Chubb, GWU
http://home.gwu.edu/~spingrey/
Sarah
and Jennifer will each give a half hour talk.
Titles: (first) Complexity of relations on computable structures, and (second) Strong reducibilities, scattered linear orderings, ranked sets, and Kolmogorov complexity
Abstract: Harizanov showed that the Turing degree spectrum of the w-part of a linear ordering of type w + w* is all of the limit computable Turing degrees. This is not the case for truth-table degrees. There is a computable enumerable set D such that D is not weak truth-table reducible to any initial segment of any computable scattered linear ordering. We will use algorithmic information theory to establish this result.