Seminar Calendar
for Logic Seminar events the year of Wednesday, November 4, 2009.

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Questions regarding events or the calendar should be directed to Tori Corkery. 
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Tuesday, January 27, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, January 27, 2009
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Submitted by ssolecki.
Lionel Nguyen Van The (University of Calgary)
Weak partition properties of the Urysohn space
Abstract: In 1994, Odell et Schlumprecht built a uniformly continuous map from the unit sphere of the Hilbert space into the unit interval and which does not stabilize on any isometric copy of the sphere. The purpose of this talk is to consider similar problems when the Hilbert space is replaced by the Urysohn metric space. This is a joint work with N. Sauer.

Wednesday, January 28, 2009

Logic Seminar
4:00 pm   in 306 Altgeld Hall,  Wednesday, January 28, 2009
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Submitted by ssolecki.
Lionel Nguyen Van The (University of Calgary)
Partition properties of the Urysohn sphere

Tuesday, February 3, 2009

Logic Seminar
1:00 pm   Tuesday, February 3, 2009
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Submitted by w-henson.
No meeting today; see Friday at 4:00

Friday, February 6, 2009

Logic Seminar
4:00 pm   in 345 Altgeld Hall,  Friday, February 6, 2009
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Submitted by ssolecki.
Alice Medvedev (UIC)
Some tractable trivial minimal sets in ACFA
Abstract: Let (K, sigma) be a model of ACFA of characteristic 0, and let f(x) be a polynomial over K. We completely describe the definable structure on the Lascar-rank 1 set f^sharp defined by sigma(x)= f(x). Chatzidakis and Hrushovski proved the Zilber Trichotomy conjecture in ACFA, and characterized the functions that give fieldlike f^sharp. Since all of these are isomorphic to the fixed field of sigma, it is easy to see that the definable structure on them is terribly complicated. My thesis characterized the functions that give grouplike f^sharp, and Chatzidakis, Hrushovski, and others showed that the corresponding groups are one-based. We now complete the picture by describing the quantifier-free definable relations on f^sharp, which pins down the algebraic closure operator; the (lack of) quantifierfully definable relations on f^sharp, which shows that most of them are strongly minimal; and the non-orthogonality relation between different such sets, which turns out to be occasionally definable and otherwise describable.

Tuesday, February 17, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, February 17, 2009
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Submitted by ssolecki.
Stevo Todorcevic (University of Toronto/University of Paris 7)
Generic Banach spaces

Tuesday, February 24, 2009

Logic Seminar
1:00 pm   in Altgeld Hall,  Tuesday, February 24, 2009
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Submitted by ssolecki.
No meeting today; see Friday at 4:00

Friday, February 27, 2009

Logic Seminar
4:00 pm   in 345 Altgeld Hall,  Friday, February 27, 2009
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Submitted by ssolecki.
C. Ward Henson (UIUC)
Highly homogeneous metric structures
Abstract: In general, this talk is about separable metric structures whose theories in continuous logic are omega-categorical and admit quantifier-elimination. There are several interesting examples that arose "in nature." There are good characterizations of omega-categoricity (when the signature is countable) and of QE. The Fraisse construction has a natural extension to the metric setting and it has produced a few more examples. Moreover, there are lots of interesting open questions. In particular, the classification program that has generated so much interesting mathematics in the classical setting has not yet been taken up in a serious way.

Tuesday, March 3, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, March 3, 2009
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Submitted by ssolecki.
Isaac Goldbring (UIUC)
Structure theorems for locally compact local groups admitting a contractive local automorphism
Abstract: An automorphism F of a locally compact group G is said to be contractive if, for all x in G, F^n(x) approaches 1 as n approaches infinity; here F^n denotes the nth iterate of F. G is said to be contractible if it admits a contractive automorphism. It is known that a connected contractible locally compact group is a Lie group. It is also known that an arbitrary contractible locally compact group is isomorphic to the direct product of its connected component G_1 and a totally disconnected, closed, normal subgroup D of G; both G_1 and D are contractible as well (so G_1 is a Lie group). S. Selivanova asked whether similar theorems hold for locally compact local groups which admit contractible local automorphisms. In this talk, I will present local analogues of the aforementioned results. The key idea is to use a theorem of Malcev to show a locally compact local group which admits a contractible local automorphism is actually contained in a locally compact group and then extend the local automorphism to an automorphism of the global group. In general, the extension isn't an autormorphism, but rather a pseudo-automorphism, that is, a continuous, open, injective group homomorphism. I will discuss conditions which ensure that a psuedo-automorphism is actually an automorphism. If the induced pseudo-automorphism is actually an automorphism, then one can deduce structure theorems for locally compact local groups admitting contractive local automorphisms from their global counterparts. This is joint work with Lou van den Dries.

Tuesday, March 10, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, March 10, 2009
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Submitted by ssolecki.
No meeting today; see Friday at 4:00

Friday, March 13, 2009

Logic Seminar
4:00 pm   in 345 Altgeld Hall,  Friday, March 13, 2009
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Submitted by ssolecki.
Slawomir Solecki (UIUC)
The Loomis--Whitney inequality, concentration of measure, and dynamics of L_0
Abstract: I will present a generalization of the Loomis--Whitney inequality, which is deduced from Lieb's inequality, and which leads to a concentration of measure result in product spaces. This concentration result is then applied to prove extreme amenability of L_0(phi,H), where phi is a pathological submeasure and H is an amenable, locally compact group. This result extends, with a different proof, a classical theorem of Herer and Christensen on extreme amenability of L_0(phi,the reals) for pathological phi. I will also explain connections of this work with the earlier work of Farah and myself.

Tuesday, March 17, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, March 17, 2009
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Submitted by ssolecki.
Justin T. Moore (Cornell University)
Fast growth in Folner sets for Thompson's group F
Abstract: While it is not known whether Thompson's group F is amenable, I will establish a lower bound on the diameter of Folner sets. In particular, I will demonstrate the following: there is a constant C > 1 such that if A is a C^(-4^n)-Folner set in F, then A has diameter at least H(n), where H(0) = 0 and H(n+1) = 2^(H(n)).

Thursday, March 19, 2009

Logic Seminar
4:00 pm   in 345 Altgeld Hall,  Thursday, March 19, 2009
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Submitted by ssolecki.
Justin T. Moore (Cornell University)
Fast growth in Folner sets for Thompson's group F: the proofs

Tuesday, March 31, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, March 31, 2009
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Submitted by ssolecki.
Aleksandra Kwiatkowska (UIUC)
Point realizations of Boolean actions and groups of isometries I
Abstract: I will present the following result: given a Polish group G of isometries of a locally compact separable metric space, each measure preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. The proof of our result relies on a new characterization of Polish groups of isometries of locally compact separable metric spaces, which I will state. I will also present background of the above results. This is a joint work with Slawek Solecki

Tuesday, April 7, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, April 7, 2009
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Submitted by ssolecki.
Slawomir Solecki (UIUC)
Point realizations of Boolean actions and groups of isometries II
Abstract: This is a continuation of the seminar talk of April 1. I will recall the main results stated in that talk, but I will spend most of the hour on outlining their proofs. These involve the solution to Hilbert's fifth problem, the Ryll Nardzewski fixed point theorem, and some connections with pro-Lie groups. This is a joint work with Ola Kwiatkowska.

Tuesday, April 14, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, April 14, 2009
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Submitted by ssolecki.
Aleksandra Kwiatkowska (UIUC)
Near actions and almost cocycles
Abstract: By a classical theorem due to Mackey, every near-action of a locally compact Polish group has a point realization, and by a recent theorem due to Glasner and Weiss, the same is true for the group of permutations of natural numbers. We can ask similar questions for other types of objects, for example, for cocycles. I will present a recent theorem due to Becker which says that there is an almost cocycle of the group of permutations of natural numbers that is not equivalent to a strict cocycle.

Tuesday, April 21, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, April 21, 2009
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Submitted by ssolecki.
Robert Kaufman (UIUC)
Closest points in metric spaces
Abstract: Extremal problems are traditional in analysis and appear in classical existence theorems. Critical attention to existence of extremals is more recent. Proximal sets are a model for minimum problems. Let F be a closed set in a Polish space and 'Prox' the set of points not in F, but admitting a nearest point in F. Then every analytic set can be realized--isometrically--as 'Prox' in a suitable extension. Only rudiments on metric spaces are used.

Tuesday, April 28, 2009

Logic Seminar
1:00 pm   in 345 Altgeld Hall,  Tuesday, April 28, 2009
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Submitted by ssolecki.
No meeting today; see Thursday at 4

Thursday, April 30, 2009

Logic Seminar
4:00 pm   in 345 Altgeld Hall,  Thursday, April 30, 2009
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Submitted by ssolecki.
Christian Rosendal (UIC)
Complete metric groups acting on trees
Abstract: We study to which extent complete metric groups decompose as free products with amalgamation and will show that under fairly mild conditions on the group, any such decomposition will be compatible with the topological structure of the group.

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.