UIUC Dept. of Mathematics Seminar Calendar

      March 2012             April 2012              May 2012      
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Thursday, April 19, 2012

Number Theory Seminar
11:00 am in 243 Altgeld Hall, Thursday, April 19, 2012

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Submitted by ford.

Andrew Schultz (Wellesley College, MA)
A generalization of the Gaussian Formula and q-analog of Fleck's congruence
Abstract: In the early 1900's, Fleck proved that alternating sums of binomial coefficients taken across particular residue classes modulo a prime number are highly divisible by that prime number. In this talk, I'll discuss some recent work for analogous sums of q-binomial coefficients, and we'll see that these give "half" of a generalization of Fleck's result. This represents a project that I worked on with (then UIUC undergrad) Robert Walker.

NetMath Lunch Seminar
12:05 pm in 102 Altgeld Hall, Thursday, April 19, 2012

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Submitted by gfrancis.

Andrew Schultz [email] (Wellesley College, MA)
NetMath and C&M as a Catalyst for Pedagogical Innovation
Abstract: I came to UIUC as a postdoc in 2007 and was assigned a Calculus & Mathematica course because of an image I had on my webpage. Only later did I find out that the perspectives of Uhl, Davis and Porta influenced the way I was taught linear algebra at Davidson College, and that my time as an instructor in the C&M series would in turn shape my own perspective on effective teaching practices. In this talk I'll discuss how teaching a NetMath or C&M course can impact one's perspective on teaching strategies in the "traditional" classroom.

Group Theory
1:00 pm in 347 Altgeld, Thursday, April 19, 2012

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Submitted by nmd.

Pere Menal Ferrer (Universitat Autonoma de Barcelona)
Reidemeister torsion for hyperbolic 3-manifolds
Abstract: Reidemeister torsion is an invariant defined for a CW-complex and a linear representation of its fundamental group. It was first defined in the 1930s by Reidemeister, de Rham and Franz to classify lens spaces in dimension 3, and since then it has proven to be a powerful invariant. In this talk, I will first give a brief review of Reidemeister torsion, and how to define it for a hyperbolic 3-manifold. Then I will introduce a certain class of invariants ${ T_n (M) }$ attached to a hyperbolic manifold $M$, which are defined as the Reidemeister torsion of $M$ with respect to the composition of the holonomy representation of $M$ and the $n$-dimensional fundamental representation of $\mathrm{SL}(n, C)$. I will show that the sequence $\{ \log |T_n(M)| / n^2 \}$ converges to $-\mathrm{Vol}(M)/ 4\pi$ (this is an extension of a result by W. Müller which deals with closed manifolds). Finally, I will discuss how the sequence $\{ T_n (M) \}$ determines and is determined by the complex length spectrum of $M$. This is joint work with Joan Porti.

Probability Seminar (special event)
2:00 pm in 347 Altgeld Hall, Thursday, April 19, 2012

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Submitted by jathreya.

David Stork (Distinguished Research Scientist and Research Director, Rambus Labs, Computational Sensing and Imaging Initiative)
Lake Wobegon Dice
Abstract: We present sets of n non-standard dice-Lake Wobegon Dice-having the following paradoxical property: On every (random) roll of a set, each die is more likely to roll greater than the set average than less than the set average; in a specic statistical sense, then, each die is "better than the set average." We define the Lake Wobegon Dominance of a die in a set as the probability the die rolls greater than the set average minus the probability the die rolls less than the set average. We further define the Lake Wobegon Dominance of the set to be the dominance of the set's least dominant die and prove that such paradoxical dominance is bounded above by $(n-2)/n$ regardless of the number of sides s on each die and the maximum number of pips p on each side. A set achieving this bound is called Lake Wobegon Optimal. We give a constructive proof that Lake Wobegon Optimal sets exist for all $n \geq 3$ if one is free to choose $s$ and $p$. We also show how to construct minimal optimal sets, that is, that set that requires the smallest range in the number of pips on the faces. We determine the frequency of such Lake Wobegon sets in the $n = 3$ case through exhaustive computer search and nd the unique optimal $n = 3$ set having minimal $s$ and $p$. We investigate symmetry properties of such sets, and present equivalence classes having identical paradoxical dominance. We construct inverse sets, in which on any roll each die is more likely to roll less than the set average than greater than the set average, and thus each die is "worse than the set average." We show the unique extreme "worst" case, the Lake Wobegon Pessimal set. We speculate on the application of such paradoxical Lake Wobegon Dominance for collusion among agents, such as economic agents, each seeking to be "above average" as often as possible. [Joint work with Jorge Moraleda]

Mathematics Colloquium
4:00 pm in 245 Altgeld Hall, Thursday, April 19, 2012

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Submitted by kapovich.

Kevin Ford (Department of Mathematics, University of Illinois at Urbana-Champaign)
Prime chains and applications
Abstract: A sequence of primes $p_1,...,p_k$ is a "prime chain" if $p_j|(p_{j+1}-1)$ for each $j$. For example: 3, 7, 29, 59, 709. We describe new estimates for counts of prime chains satisfying various properties, e.g. the number of chains with $p_k < x$ ($k$ variable) and the number of chains with $p_1=p$ and $p_k \le x$. We discuss some applications of these estimates, in particular the settling of a 50-year old conjecture of Erdos that $\phi(a)=\sigma(b)$ has infinitely many solutions ($\phi$ is Euler's function, $\sigma$ is the sum of divisors function). We also focus on the distribution of $H(p)$, the length of the longest chain ending at a given prime $p$. $H(p)$ is also the height of the "Pratt tree" for $p$, the tree structure of all chains ending at $p$. We give new, nontrivial bounds for $H(p)$, valid for almost all $p$, and settle a conjecture of Erdos, Granville, Pomerance and Spiro from 1990. We introduce and analyze a random model of the Pratt tree, based on branching random walks, which leads to some surprising conjectures about the distribution of $H(p)$. Finally, we give an application to groups with "perfect order subsets" and discuss various open problems in the area.

CAS/MillerComm 2012
5:30 pm in Room 62, Krannert Art Museum, 500 E. Peabody Drive, Champaign, Thursday, April 19, 2012

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Submitted by seminar.

David Stork (Distinguished Research Scientist and Research Director, Rambus Labs, Computational Sensing and Imaging Initiative)
When Computers Look at Art: Image Analysis in Humanistic Studies of the Visual Arts
Abstract: What can computers reveal about images that even the best-trained connoisseurs, art historians and artists cannot? How do these computer methods work? How much more powerful and revealing will these methods become? In short, how is computer image analysis changing our understanding of art? This profusely illustrated lecture for non-scientists will include works by Jackson Pollock, Vincent van Gogh, Jan van Eyck and others. You may never see paintings in the same way again. This CAS/MillerComm 2012 Lecture is hosted in part by the Department of Mathematics and the Krannert Art Museum.