UIUC Dept. of Mathematics Seminar Calendar

    September 2012          October 2012          November 2012    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                    1       1  2  3  4  5  6                1  2  3
  2  3  4  5  6  7  8    7  8  9 10 11 12 13    4  5  6  7  8  9 10
  9 10 11 12 13 14 15   14 15 16 17 18 19 20   11 12 13 14 15 16 17
 16 17 18 19 20 21 22   21 22 23 24 25 26 27   18 19 20 21 22 23 24
 23 24 25 26 27 28 29   28 29 30 31            25 26 27 28 29 30   
 30

Thursday, October 18, 2012

Number Theory Seminar
11:00 am in 241 Altgeld Hall, Thursday, October 18, 2012

Del

Edit

Copy

Submitted by ford.

Michael Filaseta (Univ. South Carolina Math.)
49598666989151226098104244512918
Abstract: If $p$ is a prime with decimal representation $d_{n} d_{n-1} \dots d_{1} d_{0}$, then a theorem of A. Cohn implies that the polynomial $f(x) = d_{n} x^{n} + d_{n-1}x^{n-1} + \cdots + d_{1}x + d_{0}$ is irreducible. One can view this result as following from the fact that if $g(x) \in \mathbb Z[x]$ with $g(0) = 1$, then $g(x)$ has a root in the disk $D = \{ z \in \mathbb C: |z| \le 1 \}$. On the other hand, that such a $g(x)$ has a root in $D$ has little to do with $g(x)$ having integer coefficients. In this talk, we discuss a perhaps surprising result about the location of a zero of such a $g(x)$ that makes use of its coefficients being in $\mathbb Z$ and discuss the implications this has on generalizations of Cohn's theorem. A variety of open problems will be presented. This research is joint work with a former student, Sam Gross.

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, October 18, 2012

Del

Edit

Copy

Submitted by kapovich.

Alex Furman (University of Illinois at Chicago)
Classifying lattice envelopes for (many) countable groups
Abstract: Let $\Gamma$ be a given countable group. What locally compact groups $G$ contain a lattice (not necessarily uniform) isomorphic to $\Gamma$ ? In a joint work with Uri Bader and Roman Sauer we answer this question for a large class of groups including Gromov hyperbolic groups and many linear groups. The proofs use a range of facts including: recent work of Breuillard-Gelander on Tits alternative, works of Margulis on arithmeticity of lattices in semi-simple Lie groups, and a number of quasi-isometric rigidity results.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, October 18, 2012

Del

Edit

Copy

Submitted by aimo.

Raanan Schul (SUNY Stony Brook)
Lipschitz Functions vs. Projections
Abstract: We discuss joint work with Jonas Azzam. ``All Lipschitz maps from $R^7$ to $R^3$ are orthogonal projections''. This is of course quite false as stated. It turns out however, that there is a surprising grain of truth in this statement. We show that all Lipschitz maps from $R^7$ to $R^3$ (with 3-dimensional image) can be precomposed with a map $g:R^7\to R^7$ such that $f\circ g$ will satisfy, when we write the domain as $R^4\times R^3$ and restrict to $E$, a large portion of the domain, that $f\circ g$ will be constant in the first coordinate and biLipschitz in the second coordinate. Geometrically speaking, the map $g$ distorts $R^7$ in a controlled manner, so that the fibers of $f$ are straightened out. Our results are quantitative. The target space can be replaced by any metric space! The size of the set $E$ on which behavior is good is an important part of the discussion and examples such as Kaufman's 1979 construction of a singular map $[0,1]^3$ onto $[0,1]^2$ are an important enemy. On route we will discuss a new extension theorem which is used to construct the bilipschitz map $g$, improving results of Jones (88) and David (88). In particular, if $g:R^7\to R^7$ is a Lipschitz map, then it agrees with a globally defined biLipschitz map $\hat{g}:R^7\to R^7$ on a large piece of the domain. This was previously known only by increasing the dimension of the target space of $\hat{g}$ (David and Semmes, 91).

Mathematics Colloquium
4:00 pm in Altgeld Hall 245, Thursday, October 18, 2012

Del

Edit

Copy

Submitted by kapovich.

Alex Furman (University of Illinois at Chicago)
Groups with good pedigrees, or superrigidity revisited
Abstract: In the 1970s G.A. Margulis proved that certain discrete subgroups (namely lattices) of such Lie groups as SL(3,R) have no linear representations except from the given imbedding. This phenomenon, known as superrigidity, has far reaching applications and has inspired a lot of research in such areas as geometry, dynamics, descriptive set theory, operator algebras etc. We shall try to explain the superrigidity of lattices and related groups by looking at some hidden symmetries (Weyl group) that they inherit from the ambient Lie group. The talk is based on a joint work with Uri Bader.