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

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

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).

Suppose

$(X,d)$ 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 $X$.

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.

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.

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.

**Title: ***Locomotion and Rotation with three stiff
legs at Low Reynolds Number*

**Abstract. **

For biological organisms the ability to turn and reorient in space is of vital importance to their evolutionary fitness. Motivated by the kinematics of swimming crustaceans, this paper analyzes the hydrodynamics of a theoretical tripodal organism whose legs extend radially from a spherical body with small radius. Each leg moves sinusoidally about a specified time-averaged angle relative to the swimmer’s orientation. Arguments of symmetry are presented to establish expectations about the swimmer’s kinematic dynamics; then, applying classical results from slender-body theory to the model we specify a resistance matrix and present numerical results to the equations of motion depending on the amplitude, phase, and average angle for each leg. As the prescribed phase shift of each leg is varied the model predicts that maximal turning effciency occurs when the phase

difference between adjacent legs is 2π/3 with maximal net translation occurring coincidentally.