Education is not the filling of a pail, it is the lighting of a fire. - W. B. Yeats

 

NEWS in the Department of Mathematics
Graduate Announcements

 

Eric Hanson - PhD defense
Date:  Tuesday, June 23, 2015
Time:  10:00 a.m.
Place:  Weber 201


Title: Algorithms in Numerical Algebraic Geometry and Applications

Advisor:  Dr. Dan Bates


Committee:
Dr. Chris Peterson
Dr. Renzo Cavalieri
Dr. Anthony Maciejewski


Abstract: The topics in this talk, while independent, are unified under the field of numerical algebraic geometry.  With ties to some of the oldest areas in mathematics, numerical algebraic geometry is relatively young as a field of study in its own right.  The field is concerned with the numerical approximation of the solution sets to systems of polynomial equations and the manipulation of the these sets.  Given a system of polynomial equations, the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions, as well as points on any positive-dimensional components of the solution set.  In a short time, the work done in numerical algebraic geometry has significantly pushed the boundary of what is computable. This talk will provide an introduction to numerical algebraic geometry and then discuss three algorithms/applications:  perturbed homotopies, exceptional sets and fiber products, and a numerical approach to finding unit distance embeddings of finite simple graphs.  A short description of each topic appears below:  

One of the most recent advances in numerical algebraic geometry is regeneration, an equation-by-equation homotopy method that is often more efficient then other approaches.  However, the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation.  We present an alternative to deflation in the form of perturbed homotopies for solving polynomial systems.  In particular, we propose first solving a perturbed version of the polynomial system, followed by a parameter homotopy to remove the perturbation.  The aim of this is two-fold.  First, such perturbed homotopies are sometimes more efficient than regular homotopies, though they can also be less efficient.  Second, a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions, including all singular isolated solutions.   This version of regeneration -- {\em perturbed regeneration} -- can decrease the efficiency of regeneration but increases its applicability. 

The second topic of this talk considers families of polynomial systems which depend on parameters.  There is a typical dimension for the variety defined by a system in the family; however, this dimension may jump for parameters in algebraic subsets of the parameter space. Sommese and Wampler exploited fiber products to give a numerical method for identifying these special parameter values.  We propose a refined numerical approach to fiber products, which uses recent advancements in numerical algebraic geometry, such as regeneration extension.  We show that this method is sometimes more efficient then known techniques.  In part, this gain in efficiency is due to the fact that regeneration extension allows the construction of the fiber product to be restricted to specified irreducible components.  This work is motivated by applications in Kinematics - the study of mechanisms.  As such we use an algebraic model of a two link arm to illustrate the algorithms we develop. 

The final topic is the identification of unit distance embeddings of finite simple graphs.  Given a graph $G(V,E)$, a unit distance embedding is a map $\phi$ from the vertex set $V$ into a metric space $M$ such that if $\{v_i,v_j\}\in E$ then the distance between $\phi(v_i)$ and $\phi(v_j)$ in $M$ is one.  Given $G$ we cast the question of the existence of a unit distance embedding in $\mathbb{R}^n$ as the question of the existence of a real solution to a system of polynomial equations.  As a consequence, we are able to develop theoretic algorithms for determining the existence of a unit distance embedding and for determining the smallest dimension of $\mathbb{R}^n$ for which a unit distance embedding of $G$ exists (that is, we determine the minimal embedding dimension of $G$).  We put these algorithms into practice using the methods of numerical algebraic geometry.  In particular, we consider unit distance embeddings of the Heawood Graph.  This is the smallest example of a point-line incidence graph of a finite projective plan.  In 1972, Chv\'{a}tal conjectured that point-line incidence graphs of finite projective planes do not have unit-distance embeddings into $\mathbb{R}^2$. In other words, Chv\'{a}tal conjectured that the minimal embedding dimension of any point-line incidence graph of a finite projective plane is at least $3$.  We disprove this conjecture, adding hundreds of counterexamples to the $11$ known counterexamples found by Gerbracht. 

 

Steve Ihde - PhD defense
 Date:  Wednesday, June 24, 2015
 Time:  2:00 p.m.
 Place:  Weber 201
 
Title: Preconditioning Polynomial Systems Using Macaulay Dual Spaces 
Advisor:  Dr. Dan Bates

Committee
Dr. Chris Peterson
Dr. Alexander Hulpke
Dr. Peter Young


Abstract:
Numerical Algebraic Geometry is concerned with using numerical methods to solve polynomial systems. The main tool from this area is the method of Homotopy Continuation. Sometimes, this method is not efficient due to a number of factors. This project concerns the preconditioning of polynomial systems in order to be better suited for homotopy continuation.  
 
In 2010, Hauenstein showed that computations could be done on the basis of a polynomial ideal by working in the Macaulay dual space. We will show that these computations can be used to find an H-basis, first introduced in 1916 by Macaulay. This H-basis will create a preconditioned set of homogeneous polynomials from a zero-dimensional system. We then improve upon these methods using the Closedness Subspace, first introduced by Zeng in 2009.

Background on homotopy continuation, h-bases, and the closedness subspace will be presented. This will lead to algorithms and examples of preconditioning for homotopy continuation.


Sofya Chepushtanova - PhD defense
Date: Thursday, June 25, 2015
Place: Weber, 201
Time: 10:00 a.m.


Title: Algorithms for Feature Selection and Pattern Recognition on Grassmann Manifolds

Advisor: Dr. Michael Kirby

Committee:
Dr. Chris Peterson
Dr. Dan Bates
Dr. Asa Ben-Hur

Abstact:  Three distinct application-driven research projects, inspired by ideas and topics from geometric data analysis, machine learning, and computational topology, are presented. We first consider feature selection based on sparse support vector machines (SSVMs) applied to hyperspectral band selection problem. A supervised embedded approach is proposed using the property of SSVMs to exhibit a model structure that includes a clearly identifiable gap between zero and non-zero feature vector weights that permits important spectral bands to be definitively selected in conjunction with the classification problem. Second, we describe an approach for performing set-to-set pattern recognition, via classification of data on embedded Grassmannians. A set of points from a given class characterizes the variability of the class information, so we propose organizing sets of data as subspaces on a Grassmann manifold. Multidimensional scaling finds Euclidean embeddings of the manifold, preserving or approximating the Grassmannian geometry based on a chosen distance measure. SSVMs are trained in the Euclidean space for classification and identification of optimal dimensions of embedded subspaces. Lastly, we consider an application of persistent homology to hyperspectral data analysis on the Grassmann manifolds. Persistent homology (PH) is a method of topological data analysis (TDA) that can be applied to a data set to capture the persistence of topological structure. Using the Grassmannian framework affords a form of data compression while retaining pertinent data structure. In this setting it becomes feasible to analyze large volumes of hyperspectral data as the high computational cost of PH applied to the original data space is greatly reduced.


Mike Mikucki - PhD Defense
Date:  Friday, June 26, 2015
Time:  1:00 p.m.
Place:  Weber 201


Title: Electromechanical and curvature driven molecular flows for lipid membranes

Advisor: Dr. Yongcheng Zhou

Committee
Dr. Simon Tavener
Dr. James Liu
Dr. Ashok Prasad


Abstract: Lipid membranes play a crucial role in sustaining life, appearing ubiquitously in biology.  Defining and computing the forces on lipid membranes is critical to understanding how these living systems operate.  Recently, much attention has been given to curvature generating and curvature sensing properties of proteins within vesicle membranes.  That is, certain proteins prefer regions of specific curvature and naturally flow to these regions. 


In this talk, we define the relevant mechanical, electrostatic, and curvature dependent energies of vesicle lipid membranes.  We numerically compute various membrane configurations which minimize these energies in a phase field model using Fourier spectral methods.  Next, we develop the mathematical theory for curvature driven diffusion along these membranes, which handles a variable diffusion coefficient.  This is done in a consistent framework to allow for the coupling of membrane shape with the curvature driven surface diffusion.  Finally, numerical results are presented which capture the curvature preference.