Analysis Seminar: Thomas Hangelbroek @ Keller 313
Oct 19 @ 3:30 pm – 4:30 pm

Title: A curiosity of the trace operator II

Abstract: I’ll discuss the regularity of the trace operator on various smoothness spaces. In short, this is an operator which reduces (roughly) smoothness as measured in L_p by an order of 1/p. Its behavior on Sobolev spaces, especially on the Hilbert spaces W_2^s (i.e., where smoothness is measured in L_2), plays a critical role in approximation theory when boundaries are present, and in the stability, regularity and existence results for weak solutions for PDEs.

Previously I presented a somewhat negative results: that the trace operator from W_2^(1/2) to L_2 is not bounded (although it is bounded from W_2^{s+1/2} to W_2^s when s>0. In this talk, I’ll explain how a minor correction (a so-called “curiosity” according to Hans Triebel) works – namely, by reducing the domain to a certain Besov space and using results from the atomic decomposition of these spaces.

Logic Seminar: André Nies
Oct 20 @ 1:30 pm – Oct 20 @ 2:30 pm

Speaker: André Nies (University of Auckland), Fellow of the Royal Society of New Zealand

Title: Structure within the class of $K$-trivial sets

Abstract: The $K$-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity $K$ grows as slowly as possible. Since 2002, many alternative characterisations of this class have been found: properties such as low for $K$, low for Martin-Löf (ML) randomness, and basis for ML randomness, which state in one way or the other that the set is close to computable.

Initially the class looked quite amorphous. More recently, internal structure has been found. Bienvenu et al. (JEMS 2016) showed that there is a “smart” $K$-trivial set, in the sense that any ML-random computing it must compute all $K$-trivials. Greenberg, Miller and Nies (submitted) showed that there is a dense hierarchy of subclasses. Even more recent results with Turetsky combine the two approaches using cost functions.

The talk gives an overview and ends with open questions (of which there remain many).

Location: Keller Hall 303

Grad student seminar (all welcome) – Michelle Manes @ Keller 301
Oct 21 @ 12:30 pm – 1:30 pm

Hilbert’s third problem, scissors congruence, and the Dehn invariant

Logic Seminar: David Ross part II
Oct 27 @ 1:30 pm – Oct 27 @ 2:30 pm

TITLE: Asymptotic Fixed Points, Part II

ABSTRACT: Continuing the earlier seminar, I will give nonstandard proofs for
one or more (depending on time) results like the
one below (which consolidates and generalizes a number of recent results in
the area).


$(X,d)$ is a complete metric space,

$T:Xto X$ is continuous,

$phi, phi_n:[0,infty)to[0,infty)$, and
$phi_n$ converges to $phi$ uniformly on the range of $d$,

$phi$ is semicontinuous and satisfies $phi(s)0$,

$d(T^nx,T^ny)lephi_n(d(x,y))$ for all $x,yin X$ and $ninmathbb N$.

Moreover, suppose that for any $x,y$, $phi(t)lesssim t$ on infinite
elements of
$${^*d(T^nx,T^my) : m, n text{ hyperintegers}}.$$
Then $T$ has a unique fixed point $x_infty$, and for every $xin X, limlimits_{ntoinfty}T^n(x)=x_infty$. Moreover, this convergence is
uniform on bounded subsets of $X$.

Colloquium: Joe Gerver (Rutgers)
Oct 27 @ 3:30 pm – 4:30 pm

Speaker: Joe Gerver (Rutgers)

Title: Non-collision singularities in the n-body problem: history and recent progress

Abstract: In the 1890′s, Poincare asked whether, in the n-body problem with point masses and Newtonian gravitation, it is possible to have a singularity without a collision. This might happen, for example, if one or more bodies were to oscillate wildly, like the function sin (1/t). We will go over the history of this problem for the past 120 years, including some recent developments.

Colloquium: Michelle Manes (UH Mānoa) @ Keller 401
Oct 28 @ 3:30 pm – 4:30 pm

Speaker: Michelle Manes (UH Mānoa)

Title: Curve-based cryptography, a tour of recent developments

Abstract: Elliptic curve cryptography (ECC) was first proposed in the mid 1980s, but it took 20 years for ECC algorithms to become widely used. Researchers are currently laying the mathematical foundations for cryptosystems based on genus 2 and more recently on genus 3 curves. If we’re successful, these systems may be widely used in another 10 years or so. Many of the breakthroughs in this area have come from research collaborations forged at the Women in Numbers and Sage Days for Women conferences.

In this talk, I’ll give a brief introduction to the idea of cryptosystems based on the “discrete log problem,” including ECC and higher genus curves. I’ll trace the story of the recent results, and I’ll provide some mathematical details for the most recent work on genus 3 curves.

Colloquium: Pamela Harris (Williams)
Nov 4 @ 3:30 pm – 4:30 pm
Analysis Seminar: Wolfgang Erb
Nov 9 @ 3:30 pm – 4:30 pm

Title: Is polynomial interpolation really that bad?

Abstract: A myth in numerical analysis (according to a nice article of N. Trefethen) is the belief that polynomial interpolation has to be avoided in practice since it is not stable and converges in general badly to the interpolated function.

In this talk, we are going to shed light on this myth by considering different aspects of polynomial interpolation as numerical stability and convergence properties. We will discuss some ot the theories of Trefethen why polynomial interpolation has such a bad reputation. At the end of the talk, I will give some examples how Chebyshev polynomials can be used efficiently to interpolate data points on Lissajous curves.