UIUC Dept. of Mathematics Seminar Calendar

       May 2012              June 2012              July 2012      
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
        1  2  3  4  5                   1  2    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
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 20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
 27 28 29 30 31         24 25 26 27 28 29 30   29 30 31

Tuesday, January 17, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, January 17, 2012

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Michael Tychonievich (Ohio State University)
Metric Properties of Sets Definable in the Expansion of the Real Field by a Logarithmic Spiral
Abstract: We discuss some metric properties of sets definable in certain expansions of the real field, including expansions of the real field by logarithmic spirals. For example, in these structures each bounded definable curve has finite length if and only if the dimension of its frontier is 0.

Tuesday, January 31, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, January 31, 2012

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Dan Grayson (UIUC)
Voevodsky's new foundations for mathematics
Abstract: Voevodsky's "Homotopy Type Theory" uses topology to provide new models for checking the consistency of a "type theory" close to what is currently implemented in the mathematical proof-checking computer program "coq". The new models allow the introduction of new axioms into the theory. These ideas promise to dramatically simplify the computerized checking of the proofs of modern mathematics, perhaps inaugurating the era where mathematicians commonly check their work as they proceed. In this talk a relative newcomer to the field will give an elementary account of the new theory and explain what it has to do with topology.

Tuesday, February 7, 2012

Friday, February 17, 2012

Tuesday, February 21, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, February 21, 2012

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Ward Henson (UIUC)
Uncountably categorical Banach space structures
Abstract: Model theory is applied to (unit balls of) Banach spaces (and structures based on them) using the $[0,1]$-valued continuous version of first order logic. A theory $T$ of such structures is said to be $\kappa$-categorical if $T$ has a unique model of density $\kappa$. Work of Ben Yaacov and Shelah-Usvyatsov shows that Morley's Theorem holds in this context: if $T$ has a countable signature and is $\kappa$-categorical for some uncountable $\kappa$, then $T$ is $\kappa$-categorical for all uncountable $\kappa$. Known examples of uncountably categorical such structures are closely related to Hilbert space. After the speaker called attention to this phenomenon, Shelah and Usvyatsov investigated it and proved a remarkable result: if $M$ is a nonseparable Banach space structure (with countable signature) whose theory is uncountably categorical, then $M$ is prime over a Morley sequence that is an orthonormal Hilbert basis of length equal to the density of $M$. There is a wide gap between this result and verified examples of uncountably categorical Banach spaces, which leads to the question: can a stronger such result be proved, in which the connection to Hilbert space structure is clearly expressed in the geometric language of functional analysis? The main part of this talk will focus on some new examples of uncountably categorical Banach spaces that the speaker has studied. This is based on joint work with Yves Raynaud.

Tuesday, February 28, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, February 28, 2012

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Slawomir Solecki (Department of Mathematics, University of Illinois at Urbana-Champaign)
Point realizations of Boolean actions
Abstract: I will show that if $M$ is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from $M$ to $U(1)$ on a separable probability algebra which preserves the measure and yet does not admit a point realization in the sense of Mackey. This is in contrast with Mackey's point realization theorem for locally compact, second countable groups. The proof of the above theorem goes through showing certain results concerning the infinite dimensional Gaussian measure space $({\mathbb C}^{\mathbb N},\gamma_\infty)$ which contrasts the Cameron--Martin Theorem. I will place the main result in the background of recent work on point realization and lack thereof for various classes of Polish groups. These results are due to Becker, Glasner, Tsirelson, Weiss, Kwiatkowska and myself. This is a joint work with Justin Moore.

Tuesday, March 6, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, March 6, 2012

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Lou van den Dries (Department of Mathematics, University of Illinois at Urbana-Champaign)
The structure of approximate groups according to Breuillard, Green, Tao
Abstract: Roughly speaking, an approximate group is a finite symmetric subset A of a group such that AA can be covered by a small number of left-translates of A. Last year the authors mentioned in the title established a conjecture of H. Helfgott and E. Lindenstrauss to the effect that approximate groups are ``finite-by-nilpotent''. This may be viewed as a sweeping generalisation of both the Freiman-Ruzsa theorem on sets of small doubling in the additive group of integers, and of Gromov's characterization of groups of polynomial growth. Among the applications of the main result are a finitary refinement of Gromov's theorem and a generalized Margulis lemma conjectured by Gromov. Prior work by Hrushovski on approximate groups is fundamental in the approach taken by the authors. They were able to reduce the role of logic to elementary arguments with ultra products. The point is that an ultraproduct of approximate groups can be modeled in a useful way by a neighborhood of the identity in a Lie group. This allows arguments by induction on the dimension of the Lie group. I will give two talks: the one on Tuesday will describe the main results, and the sequel on Friday will try to give a rough idea of the proofs.

Tuesday, March 13, 2012

Tuesday, April 3, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, April 3, 2012

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Robert E. Jamison (Clemson/UIUC)
A Dependency Calculus for Finitary Closures
Abstract: A closure system consists of a ground set $X$ together with a family $\mathscr{C}$ of closed subsets of $X$. The only requirements are that $\mathscr{C}$ is closed under arbitrary intersections and contains $X$. Thus each subset $S$ of $X$ lies in a smallest closed set $\mathscr{C}(S)$. The map $S \to \mathscr{C}(S)$ is the closure operator. The closure operator is finitary provided whenever $p \in \mathscr{C}(S)$, there is a finite subset $E$ of $S$ with $p \in \mathscr{C}(E)$. In this talk a first order logic for finitary closure operators will be presented. This first order logic can be used to describe and systematize the study of most important properties of finitary systems. In particular, I will describe a classification scheme for many of the important classes of finitary closures (matroids, antimatroids, partial order convexity, etc). Moreover, I will describe several metatheorems concerning classical convexity invariants such as the Helly and Radon numbers.

Tuesday, April 10, 2012

Friday, April 20, 2012

Tuesday, April 24, 2012

Logic Seminar
1:00 pm in Altgeld Hall, Tuesday, April 24, 2012

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Anush Tserunyan (UCLA)
Finite generators for countable group actions
Abstract: Consider a Borel action of a countable group $G$ on a standard Borel space $X$. A countable Borel partition $P$ of $X$ is called a generator if $GP=\{ gA: g \in G, A \in P\}$ generates the Borel $\sigma$-algebra of $X$. Existence of such $P$ of cardinality $n$ is equivalent to the existence of a $G$-embedding of $X$ into the shift $n^G$. For $G=Z$, the Kolmogorov-Sinai theorem implies that finite generators don't exist in the presence of an invariant probability measure with infinite entropy. It was asked by Weiss in the late 80s, whether the nonexistence of any invariant probability measure would guarantee the existence of a finite generator. We show that the answer is positive in case $X$ admits a $\sigma$-compact topological realization (e.g. if $X$ is a $\sigma$-compact Polish $G$-space). We also show that finite generators always exist in the context of Baire category thus answering a question of Kechris. In fact, we show that if $X$ is a Polish $G$-space having infinite orbits, then there is a 4-generator on an invariant comeager set.

Friday, April 27, 2012

Logic Seminar
4:00 pm in 347 Altgeld Hall, Friday, April 27, 2012

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Eva Leenknegt (Purdue)
In search of p-adic minimality: An exploration of weak p-adic structures
Abstract: Consider a structure (F,L), where F is a field and L is a language that is 'related' to the language of rings, in the sense that the Lring-definable subsets of F coincide with the L-definable subsets of F (that is, we require (F,L) to be 'Lring-minimal'). When F is a real closed field, a structure satisfying this property will be o-minimal, so all tools of o-minimality, such as the cell decomposition theorem, are at our disposal when studying such structures. The situation is less clear when F is a p-adic(ally closed) field. If L is an expansion of the ring languages, then (F,L) will be P-minimal, but very little is known in general for weaker structures (reducts of the ring language) When will a reduct L of the ring language give rise to an Lring-minimal structure? In the o-minimal case, the answer is easy: the only requirement is that the order should be definable in L. Our first challenge will be to find a p-adic equivalent of this 'minimal language' (<). Once such a language has been found, one can start constructing examples of weak p-adic Lring- minimal structures. While a general theory is still far away, individual examples show that there are some fundamental differences when comparing p-adic and o-minimal reducts of the ring language. I will give some examples of this. One of the questions that comes up is the existence of cell decomposition: could it possibly be true that every p-adic Lring-minimal language has cell decomposition? I will discuss some (partial) answers to this question, and show how we can use this to study examples of weak structures.

Tuesday, May 1, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, May 1, 2012

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Laurentiu Leustean (Institute of Mathematics of the Romanian Academy)
Proof mining in nonlinear analysis
Abstract: The program of proof mining is concerned with the extraction of hidden finitary and combinatorial content from proofs that make use of highly infinitary principles. This new information can be both of quantitative nature, such as algorithms and effective bounds, as well as of qualitative nature, such as uniformities in the bounds or weakening the premises. Thus, even if one is not particularly interested in the numerical details of the bounds themselves, in many cases such explicit bounds immediately show the independence of the quantity in question from certain input data. This line of research, developed by Ulrich Kohlenbach in the 90's, has its roots in Georg Kreisel's program on unwinding of proofs, put forward in the 50's. In this talk I will present applications of proof mining to the asymptotic behavior of nonexpansive iterations and nonlinear generalizations of ergodic averages.

Tuesday, September 4, 2012

Tuesday, September 11, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, September 11, 2012

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Andrew Arana (UIUC Philosophy and Math)
Transfer in algebraic geometry - Part II
Abstract: The focal question of this talk is to investigate the value of transfer between algebra and geometry, of the sort exemplified by the Nullstellensatz. Algebraic geometers frequently talk of such transfer principles as a "dictionary" between algebra and geometry, & claim that these dictionaries are fundamental to their practice. We'll first need to get clear on what such transfer consists in. We'll then investigate what how such transfer might improve how knowledge is gathered in algebraic geometric practice. --- This talk is a continuation of the talk given in the Logic seminar last week. Enough of a survey of what has come before will be given so that people who missed the first talk, can still attend this talk with profit.

Tuesday, September 18, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, September 18, 2012

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Mia Minnes (UC San Diego)
Algorithmic Randomness via Random Algorithms
Abstract: Algorithmic randomness defines what it means for a single mathematical object to be random. This active area of computability theory has been particularly fruitful in the past several decades, both in terms of expanding theory and increasing interaction with other areas of math and computer science. Randomness can be equivalently understood in terms of measure theory, descriptive complexity, and martingales. In this context, we present a novel definition of betting strategies that uses probabilistic algorithms also studied in complexity theory. This definition leads to new characterizations of several central notions in algorithmic randomness and addresses Schnorr's critique, a longstanding philosophical question in algorithmic randomness. Moreover, these techniques suggest new approaches for tackling one of the biggest open questions in the field (KL = ML?). This is joint work with Sam Buss.

Thursday, October 11, 2012

Friday, October 12, 2012

Tuesday, October 16, 2012

Tuesday, October 23, 2012

Tuesday, October 30, 2012

Tuesday, November 6, 2012

Tuesday, November 13, 2012

Logic Seminar
1:00 pm in 345 Altgeld Hall, Tuesday, November 13, 2012

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Tobias Kaiser (Universit�t Passau)
A very first step towards an algebraic understanding of the ring of analytic functions that are globally subanalytic
Abstract: We are interested in the ring of functions that are analytic and globally subanalytic on a given globally subanalytic domain. The motivation comes from the following two results: The ring of functions that are analytic on a neighbourhood of a compact subanalytic set is Noetherian (see [J. Frisch: Points de platitude d'un morphisme d'espaces analytiques complexes, Inventiones math. 4, 118-138 (1967)]) and the ring of Nash functions (i.e. functions that are analytic and semialgebraic) on a semialgebraic domain is Noetherian (see [J.-J. Risler: Sur l'anneau des fonctions de Nash globales, Ann. Sci. Ecole Norm. Sup. 8, 365-378 (1975)]). To deal with the rings in the subanalytic case it is natural to try rst to understand the local rings given by germs at boundary points of the domain. The most promising tool for that is the rectilinearization theorem proven by Bierstone& Milman and Parusinski. There rings of multivariate Puiseux series comes into the game. We describe these rings as twisted group rings.