|
4:00pm to 5:30pm - RH 340 N - Logic Set Theory Michael Hehmann - (UC Irvine) Algorithmic Randomness We give an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.
|
|
2:00pm to 2:50pm - RH 440 R - Dynamical Systems Marlies Gerber - (University of Indiana) Non-classifiability of Kolmogorov Diffeomorphisms We consider the problem of classifying Kolmogorov automorphisms (or K-automorphisms for brevity) up to isomorphism or up to Kakutani equivalence. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and K-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. Therefore one might hope to extend Ornstein’s classification of Bernoulli shifts up to isomorphism by a numerical Borel invariant to a classification of K-automorphisms by some type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to K-automorphisms, considered as a subset of the Cartesian product of the set of K-automorphisms with itself, is a complete analytic set, and hence not Borel. Moreover, we prove this remains true if we restrict consideration to K-automorphisms that are also C∞ diffeomorphisms. In addition, all of our results still hold if “isomorphism” is replaced by “Kakutani equivalence”. This shows in a concrete way that the problem of classifying K-automorphisms up to isomorphism or up to Kakutani equivalence is intractable. These results are joint work with Philipp Kunde. |
|
3:00pm to 4:00pm - RH 306 - Number Theory John Yin - (University of Wisconsin) A Chebotarev Density Theorem over Local Fields I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei. |
|
9:00am to 9:50am - Zoom - Inverse Problems Gunther Uhlmann - (University of Washington) Nonlocality Helps |
|
1:00pm - 306 Rowland Hall - Harmonic Analysis Lars Becker - (Bonn) A degree one Carleson operator along the paraboloid Carleson proved in 1966 that the Fourier series of any square integrable |
|
1:00pm - RH 510R - Algebra Vladimir Baranovsky - (UCI) Bar and cobar constructions and their relationship with Koszul algebras |
|
4:00pm to 4:50pm - 306 RH - Colloquium Marlies Gerber - (Indiana University) Anti-classification Results in Smooth Ergodic Theory There are two well-known equivalence relations on the family of measure-preserving ergodic automorphisms (of a given Lebesgue space): isomorphism and Kakutani equivalence. Two such automorphisms are said to be isomorphic if there is a measure-preserving relabeling of the points in the space that changes one automorphism into the other. Kakutani equivalence is a weaker equivalence relation in which we are also allowed to replace the automorphisms by their first return maps to measurable subsets of the space before relabeling the points. We consider the complexity of the problem of classifying ergodic automorphisms up to isomorphism or up to Kakutani equivalence. For example, are these equivalence relations, considered as subsets of the Cartesian product of the space of ergodic automorphisms with itself, Borel sets? The first breakthroughs were the anti-classification results of Foreman, Rudolph, and Weiss for isomorphism of ergodic automorphisms, and the subsequent anti-classification results of Foreman and Weiss for isomorphism of ergodic automorphisms that are also smooth diffeomorphisms preserving a given smooth measure on a manifold. I will describe further anti-classification results for the Kakutani equivalence relation, and for both isomorphism and Kakutani equivalence, when we restrict to smooth diffeomorphisms with additional properties, such as the mixing property or the K property. |
|
1:00pm - DBH 1200 - Graduate Seminar Manny Reyes - (UC Irvine) A brief tour of noncommutative algebra |