Seminar Calendar
for Logic Seminar events the next 12 months of Saturday, August 1, 2009.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery. 
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Friday, August 28, 2009

Model Theory and Descriptive Set Theory
4:00 pm   in 345 Altgeld Hall,  Friday, August 28, 2009
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Submitted by w-henson.
van den Dries / Henson / Solecki (UIUC Math)
Sums and Products -- organizational meeting
Abstract: This semester we will focus our Friday logic seminars on a new development of general mathematical interest, involving descriptive set theory, combinatorics, number theory, and model theory. At this organizational meeting, a plan for the semester will be worked out. The first paper we'll read is "Borel subrings of the reals" by G.A. Edgar and Chris Miller. A main result of this paper is that if E is a proper subring of the field of real numbers that is also a Borel set, then the Hausdorff dimension of E is 0. Indeed, they show that for all n, the set E^n has Hausdorff dimension 0 as a subset of R^n. Further work to which our seminar will aim involves papers by Jean Bourgain, Terry Tao, and others, and especially a preprint by Udi Hrushovski. All interested people are warmly invited to participate.

Tuesday, September 1, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, September 1, 2009
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Submitted by w-henson.
Lou van den Dries (UIUC Math)
Low real numbers
Abstract: This is an account of the recent paper "Computable functions of reals" by K. Tent and M. Ziegler. They define a very restrictive notion of computable real number, called "low", and they use general results about this notion to show that periods in the sense of Zagier (certain kinds of integrals) are low. I"ll show that exponentially-algebraic numbers are low.

Tuesday, September 8, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, September 8, 2009
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Submitted by w-henson.
Slawek Solecki (UIUC Math)
Avoiding families and Tukey functions
Abstract: We study Tukey functions from the ideal of all closed nowhere dense subsets of the Cantor set. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of the natural numbers. In connection with these results, we study combinatorial properties of families G of clopen subsets of the Cantor set with the property that for each nowhere dense set there is a set in G not intersecting it. We call such families avoiding. This a joint work with Stevo Todorcevic.

Tuesday, September 15, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, September 15, 2009
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Submitted by w-henson.
Michael Tychonievich (Ohio State)
Defining Additive Subgroups of the Reals from Convex Subsets
Abstract: Let G be a subgroup of the additive group of real numbers and let C be an infinite, convex subset of G. We show that G is definable in in the real field expanded by a predicate for C and that the set of integers is definable if G has finite rank. This has a number of consequences for expansions of certain o-minimal structures on the real field by multiplicative groups of complex numbers.

Tuesday, September 22, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, September 22, 2009
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Submitted by w-henson.
Aaron Hill (UIUC Math)
In a Polish group, is the set of squares Borel?
Abstract: For a Polish group G we consider whether or not the set of squares is Borel. We will briefly discuss the general situation and then focus on two specific Polish groups: The infinite symmetric group (in which the collection of squares is Borel) and the group of isometries of Baire space (in which the collection of squares is non-Borel).

Tuesday, September 29, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, September 29, 2009
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Submitted by w-henson.
Alex Berenstein (Univ. de los Andes, Bogota)
Reflexive representability and stable metrics
Abstract: Let G be a topological group. We say that G is reflexively representable if there is a topological isomorphism of G into the isometry group of a reflexive Banach space with the weak (or the strong) operator topology. We show that a metrisable group is reflexively representable if and only if its metric is uniformly equivalent to a stable metric. This is joint work with Itai Ben Yaacov and Stefano Ferri.

Tuesday, October 6, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, October 6, 2009
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Submitted by w-henson.
no meeting this week

Tuesday, October 13, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, October 13, 2009
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Submitted by w-henson.
Yevgeniy Gordon (EIU)
What does G. Birkhoff's Ergodic Theorem mean for very big finite spaces? (part I)
Abstract: The trivial proof of the ergodic theorem for a finite set X and a permutation s of X shows that for an arbitrary real-valued function f on X, the sequence of ergodic means A_n(f,s) stabilizes for large n. We show that if X is a very big finite set and f(x) is much less than the size of X for almost all x, then A_n(f,s) stabilizes for a significantly long segment of big numbers n that are, however, much less than the size of X. This statement has a natural rigorous formulation in terms of nonstandard analysis, which is, in fact, equivalent to the ergodic theorem for infinite probability spaces. Its standard formulation in terms of sequences of finite probability spaces is complicated. We discuss also a new notion of approximation of dynamical systems by finite dynamical systems, whose definition in terms of nonstandard analysis is much easier and much more natural than in classical terms. This is joint work with Lev Glebsky and Ward Henson.

Tuesday, October 20, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, October 20, 2009
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Submitted by w-henson.
Yevgeniy Gordon (EIU)
What does G. Birkhoff's Ergodic Theorem mean for very big finite spaces? (part II)
Abstract: The trivial proof of the ergodic theorem for a finite set X and a permutation s of X shows that for an arbitrary real-valued function f on X, the sequence of ergodic means A_n(f,s) stabilizes for large n. We show that if X is a very big finite set and f(x) is much less than the size of X for almost all x, then A_n(f,s) stabilizes for a significantly long segment of big numbers n that are, however, much less than the size of X. This statement has a natural rigorous formulation in terms of nonstandard analysis, which is, in fact, equivalent to the ergodic theorem for infinite probability spaces. Its standard formulation in terms of sequences of finite probability spaces is complicated. We discuss also a new notion of approximation of dynamical systems by finite dynamical systems, whose definition in terms of nonstandard analysis is much easier and much more natural than in classical terms. This is joint work with Lev Glebsky and Ward Henson.

Tuesday, October 27, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, October 27, 2009
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Submitted by w-henson.
Ahuva Shkop (UIC Math)
Schanuel's conjecture, Shapiro's conjecture, and an actual theorem
Abstract: In the 50's, Shapiro conjectured that if two exponential polynomials in one variable which are each sums of terms of the form exp(a+bz) have no common factors, then they have only finitely many common zeros. The goal of this talk is to prove that a special case of this conjecture holds in Pseudoexponentiation as well as in any other algebraically closed exponential field of characteristic zero satisfying Schanuel's conjecture.

Tuesday, November 3, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, November 3, 2009
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Submitted by w-henson.
Kostyantyn Slutskyy (Department of Mathematics, University of Illinois)
From Ramsey Theorem to Ramsey Groups
Abstract: We introduce a notion of a Ramsey action, inspired by the classical Ramsey theorem. In short, given an action of the group one now colours k-element linear orderings by open subsets in the phase space. The set M is then called monochromatic if the intersection of all colours of its k-element subsets is non-empty. We show how this notion generalizes classical Ramsey theorem, discuss some of its basic properties and illuminate its connection to ergodic theory.

Tuesday, November 10, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, November 10, 2009
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Submitted by w-henson.
Joseph Flenner (Notre Dame)
Relative decidability of henselian valued fields
Abstract: While logic has produced many results about the p-adics, among them a decision procedure due to Paul Cohen, the general theory of henselian valued fields presents an inherent difficulty: they are built on structures of arbitrary complexity in the residue field and value group. Ax-Kochen and Ersov, however, proved their completeness result for some henselian valued fields relative to the theories of the residue field and value group, and more recently, there have been some relative quantifier elimination theorems of Kuhlmann and others. In this spirit, we describe a structure of leading terms associated to a valued field, and outline a proof of decidability for henselian valued fields of characteristic 0 relative to the leading term structures.

Thursday, November 12, 2009

Joint Group Theory - Logic Seminar talk
1:00 pm   in Altgeld Hall 347,  Thursday, November 12, 2009
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Submitted by kapovitc.
Paul Schupp (Department of Mathematics, University of Illinois)
Some Questions at the Frontiers of Computer Science
Abstract: In this talk I want to survey some quite different questions which involve all three of the areas mention in the title. First of all, I will discuss the recent remarkable characterization of amenability for finitely generated groups in terms of the classic properties of cellular automata defined over the Cayley graph of the group. Then I will discuss the "remarkable rigidity of randomness" in group theory: In particular, the strong Mostow-type isomorphism rigidity of random groups and the "unreasonably low generic-case complexity" of group-theoretic decision problems. If time permits I will discuss the formal language-theoretic characterizations of virtually free and basic groups and the monadic and temporal logic of such groups.

Tuesday, November 17, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, November 17, 2009
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Submitted by w-henson.
Justin Moore (Cornell)
Forcing Axioms and the Continuum Hypothesis.
Abstract: Woodin asked whether there are two $\Pi_2$-sentences $\psi_i$ $i = 0,1$ such that for each $i$, it is forcible that $H(\omega_2)$ satisfies $\psi_i$ and $2^{\aleph_0} = \aleph_1$ but such that $\psi_0 \land \psi_1$ proves $2^{\aleph_0} > \aleph_1$. This is a precise formulation of the vague question ``Is there an optimal forcing axiom which is consistent with the Continuum Hypothesis?'' I will discuss recent joint work with D. Aspero and P. Larson in which we demonstrate a positive solution to Woodin's problem (and hence that there is no optimal forcing axiom consistent with CH).

Tuesday, December 1, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, December 1, 2009
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Submitted by w-henson.
Jana Marikova (WIU)
O-minimal residue fields of o-minimal fields, I.
Abstract: Let R be an o-minimal field with a proper convex subring V, and let k be the corresponding residue field with residue map st: V \to k. We show that a certain first order axiom scheme in the language of (R,V) singles out exactly the structures (R,V) such that k_{ind}, the residue field with structure induced from R via st, is o-minimal. It has been shown in previous work that if (R,V) satisfies the above mentioned axiom scheme, then k_{ind} is o-minimal. The other direction is new and the proof uses a recent result by Shiota.

Tuesday, December 8, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, December 8, 2009
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Submitted by w-henson.
Jana Marikova (WIU)
O-minimal residue fields of o-minimal fields, II.
Abstract: Let R be an o-minimal field with a proper convex subring V, and let k be the corresponding residue field with residue map st: V \to k. We show that a certain first order axiom scheme in the language of (R,V) singles out exactly the structures (R,V) such that k_{ind}, the residue field with structure induced from R via st, is o-minimal. It has been shown in previous work that if (R,V) satisfies the above mentioned axiom scheme, then k_{ind} is o-minimal. The other direction is new and the proof uses a recent result by Shiota.