Matrices of Nonnegative Rank 3
- Series
- Convex Algebraic Geometry Reading Seminar
- Time
- Friday, February 13, 2015 - 23:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Robert Krone – Georgia Tech
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Introduction to regularity theory of second order Hamilton-Jacobi-Bellman equations.
- Series
- PDE Working Seminar
- Time
- Friday, February 13, 2015 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 202
- Speaker
- Andrzej Sweich – GeorgiaTech – swiech@math.gatech.edu
I will give a series of elementary lectures presenting basic
regularity theory of second order HJB equations. I will introduce the notion of viscosity
solution and I will
discuss basic techniques, including probabilistic techniques and
representation formulas.
Regularity results will be discussed in three cases: degenerate
elliptic/parabolic,
weakly nondegenerate, and uniformly elliptic/parabolic.
Ultra sub-Gaussian random vectors and Khinchine type inequalities
- Series
- Stochastics Seminar
- Time
- Thursday, February 12, 2015 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Piotr Nayar – IMA, Minneapolis – panayar@umn.edu
We define the class of ultra sub-Gaussian random vectors and
derive optimal comparison of even moments of linear combinations of such
vectors in the case of the Euclidean norm. In particular, we get optimal
constants in the classical Khinchine inequality. This is a joint work with
Krzysztof Oleszkiewicz.
Projections of probability distributions: A measure-theoretic Dvoretzky theorem.
- Series
- School of Mathematics Colloquium
- Time
- Thursday, February 12, 2015 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Elizabeth Meckes – Case Western Reserve University
Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed:the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A natural question is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss a result showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!). The talk will not assume much background beyond basic probability and analysis; in particular, no prior knowledge of Dvoretzky's theorem is needed.
Alexander and Markov Theorems
- Series
- Geometry Topology Student Seminar
- Time
- Wednesday, February 11, 2015 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Alan Diaz – Georgia Tech
Dynamics of the Lorentzian constant mean curvature flow near some special solutions
- Series
- Job Candidate Talk
- Time
- Tuesday, February 10, 2015 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Willie Wong – EPFL (Switzerland) – wongwwy@member.ams.org
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.
Two combinatorial applications of smooth numbers
- Series
- Combinatorics Seminar
- Time
- 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.
Holomorphic dynamics near a fixed point for maps tangent to the identity
- Series
- CDSNS Colloquium
- Time
- Tuesday, February 10, 2015 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Sara Lapan – Northwestern University – slapan@math.northwestern.edu
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.
Methods to solve Kohn-Sham equations for electron density
- Series
- Applied and Computational Mathematics Seminar
- Time
- 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.
Approximating C^0 foliations with contact structures
- Series
- Geometry Topology Seminar
- Time
- 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.