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.

Master’s Defense – David DeVine @ Keller 301
Nov 14 @ 3:00 pm – 4:00 pm

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

Link to Master’s project

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.

Master’s Defense – Christina Mende @ Keller 301
Nov 15 @ 4:00 pm – 5:00 pm


Link to Master’s project

In 2004, H. Farkas found a series of identities which relate the convolution of a certain arithmetic function with an analogue of the classical σ-function. In 2009, P. Guerzhoy and W. Raji interpreted series of identities of this kind using generating functions and modular forms. Their results pertain to primes p ≡ 3 mod 4. In this paper, we address the primes p ≡ 5 mod 8 and obtain four new series of similar identities. Our methods are close to those employed by Guerzhoy and Raji.

Number theory seminar: Ken Ono (Emory) @ Keller 301
Dec 1 @ 3:00 pm – 4:00 pm

Title: Zeta polynomials for modular forms

Abstract: The speaker will discuss recent work on Manin’s theory of zeta polynomials for modular forms. He will describe recent results which confirm Manin’s speculation that there is such a theory which arises from periods of newforms. More precisely, for each even weight $k > 2$ and newform $f$, the speaker will describe a canonical polynomial $Z_f(s)$ which satisfies a functional equation of the form $Z_f(s) = Z_f(1-s)$, and also satisfies the Riemann Hypothesis: if $Z_f(\rho) = 0$, then $\Re(\rho) = 1/2$. This zeta function is arithmetic in nature in that it encodes the moments of the critical values of $L(f, s)$. This work builds on earlier results of many people on period polynomials of modular forms. This is joint work with Seokho Jin, Wenjun Ma, Larry Rolen, Kannan Soundararajan, and Florian Sprung.