We discuss asymptotic-in-time behavior of time-like constant meancurvature hypersurfaces in Minkowski space. These objects model extended relativistic test objects subject to constant normal forces, and appear in the classical field theory foundations of the theory of vibrating strings and membranes. From the point of view of their Cauchy problem, these hypersurfaces evolve according to a geometric system of quasilinear hyperbolic partial differential equations. Inthis talk we will focus on three explicit solutions to the equations:the Minkowski hyperplane, the static catenoid, and the expanding de Sitter space. Their stability properties in the context of the Cauchy problem will be discussed, with emphasis on the geometric origins of the various mechanisms and obstacles that come into play.
Tuesday, February 10, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan McNew – Dartmouth College
We look at two combinatorial problems which can be solvedusing careful
estimates for the distribution of smooth numbers. Thefirst is the
Ramsey-theoretic problem to determine the maximal size ofa subset of of
integers containing no 3-term geometric progressions.This problem was
first considered by Rankin, who constructed such asubset with density
about 0.719. By considering progressions among thesmooth numbers, we
demonstrate a method to effectively compute thegreatest possible upper
density of a geometric-progression-free set.Second, we consider the
problem of determining which prime numberoccurs most frequently as the
largest prime divisor on the interval[2,x], as well as the set prime
numbers which ever have this propertyfor some value of x, a problem
closely related to the analysis offactoring algorithms.
Given a holomorphic map of C^m to itself that fixes a point, what happens to points near that fixed point under iteration? Are there points attracted to (or repelled from) that fixed point and, if so, how? We are interested in understanding how a neighborhood of a fixed point behaves under iteration. In this talk, we will focus on maps tangent to the identity. In dimension one, the Leau-Fatou Flower Theorem provides a beautiful description of the behavior of points in a full neighborhood of a fixed point. This theorem from the early 1900s continues to serve as inspiration for this study in higher dimensions. In dimension 2 our picture of a full neighborhood of a fixed point is still being constructed, but we will discuss some results on what is known, focusing on the existence of a domain of attraction whose points converge to that fixed point.
Monday, February 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Timo Eirola – Aalto University, Helsinki, Finland
We consider three different approaches to solve the equations for electron density around nuclei particles.
First we study a nonlinear eigenvalue problem and apply Quasi-Newton methods to this.
In many cases they turn to behave better than the Pulay mixer, which widely used in physics community.
Second we reformulate the problem as a minimization problem on a Stiefel manifold.
One that formed from mxn matrices with orthonormal columns.
Then for Quasi-Newton techniques one needs to transfer the secant conditions to the new tangent space, when moving on the manifold. We also consider nonlinear conjugate gradients in this setting.
This minimization approach seems to work well especially for metals, which are known to be hard.
Third (if time permits) we add temperature (the first two are for ground state). This means that we need to include entropy in the energy and optimize also with respect to occupation numbers.
Joint work with Kurt Baarman and Ville Havu.
Monday, February 9, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Will Kazez – University of Georgia
I will discuss Eliashberg and Thurston's theorem that C^2 taut foliations can be approximated by tight contact structures. I will try to explain the importance of their work and why it is useful to weaken their smoothness assumption. This work is joint with Rachel Roberts.
Monday, February 9, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xiaodong Li – University of Pennsylvania
Low-rank structures are common in modern data analysis and signal processing, and they usually
play essential roles in various estimation and detection problems. It is challenging to recover the underlying
low-rank structures reliably from corrupted or undersampled measurements. In this talk, we will introduce
convex and nonconvex optimization methods for low-rank recovery by two examples.
The first example is community detection in network data analysis. In the literature, it has been formulated
as a low-rank recovery problem, and then SDP relaxation methods can be naturally applied. However,
the statistical advantages of convex optimization approaches over other competitive methods, such as spectral
clustering, were not clear. We show in this talk that the methodology of SDP is robust against arbitrary
outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small
fraction of outliers. We also demonstrate that a degree-corrected version of SDP works well for a real-world
network dataset with a heterogeneous distribution of degrees.
Although SDP methods are provably effective and robust, the computational complexity is usually high
and there is an issue of storage. For the problem of phase retrieval, which has various applications and
can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by
nonconvex optimization. We prove that our method converges reliably to the original signal. It requires far
less storage and has much higher rate of convergence compared to convex methods.
Thursday, February 5, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Olivia Prosper – Dartmouth College
Sir Ronald Ross’ discovery of the transmission mechanism of malaria in 1897 inspired a suite of mathematical models for the transmission of vector-borne disease, known as Ross-Macdonald models. I introduce a common formulation of the Ross-Macdonald model and discuss its extension to address a current topic in malaria control: the introduction of malaria vaccines. Following over two decades of research, vaccine trials for the malaria vaccine RTS,S have been completed, demonstrating an efficacy of roughly 50% in young children. Regions with high malaria prevalence tend to have high levels of naturally acquired immunity (NAI) to severe malaria, leading to large asymptomatic populations. I introduce a malaria model developed to address concerns about how these vaccines will perform in regions with existing NAI, discuss some analytic results and their public health implications, and reframe our question as an optimal control problem.