It is the end of the spring semester and almost all our graduating Ph.D. in Mathematics and Masters in Statistics students have now submitted their theses. See a detailed list of all Theses defended in 2012, below, following a few words from our graduating majors.
Jasmine Ng, Billiard Markov Operators and Second-Order Differential Operators.
"I study a class of Markov operators that arise from billiard dynamical systems," Jasmine summarizes of her research. "In addition to examining their convergence to second-order differential operators, I also try to approximate the spectrum of one in terms of the other. My research approximates Markov chains with PDEs, for which we have many more tools to study." Jasmine is headed for Irvine, California where she will provide mathematics content for ALEKS Corporation. Assessment and LEarning in Knowledge Spaces (ALEKS) is a Web-based, artificially intelligent assessment and learning system.
Marina Dombrovskaya, Quotients of Subgroup Lattices of Finite Abelian p-groups.
"My research area is Algebraic Combinatorics. In particular, I am interested in the study of properties of quotients of subgroup lattices of finite abelian p-groups under various actions." Asked about the ramifications of her research in the math world, Marina answered: "A part of my research is an extension of application of Fundamental Theorem of Projective Geometry to subgroup lattices of elementary abelian groups. Another part was inspired by classification of subgroup lattice of finite abelian p-groups in combinatorial terms done by L. Butler. In the future, I would like to investigate these actions on subgroup lattices of finite non-abelian p-groups and classify certain classes of non-abelian p-group in combinatorial terms."
Sara Gharahbeigi, Mumford regularity for general rational curves on hypersurfaces.
"My research is on rational curves on hypersurfaces. There are lots of studies that are going on about the geometry of the parametrizing space of rational curves of a given degree on a hypersurface. There are applications of this topic in physics, especially in string theory and Gromov-Witten theory." When questioned about her plans, Sara answered that she will be a Postdoctoral Lecturer at the University of Missouri next year.
Jeffrey Langford, Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions.
Asked for a description of his research interests in layman terms, Jeff put it thus: "I have been working in partial differential equations. Specifically, I study how to arrange the input data so that solutions are 'biggest'. My research aims to answer physically motivated questions such as the following: Why are your heating vents located under your windows? Where should you place heating vents to use the least amount of energy (create the greatest average temperature)?" Jeff will work to further answer these questions in his position as an Assistant Professor of Mathematics at Drake University.
The other proposed thesis in mathematics and statistics this year were:
Drew Lewis, Coordinates Arising from Affine Fibrations
Ben Manning, Composite MRA wavelets
Haylee Lee, Bayesian Inference of bidimensional regression
Josh Brady, Analysis of the Navier-Stokes-αβ equations
All 2012 Theses:
Bayesian inference of bidimensional regression
Haylee Lee , Department of Mathematics, Washington University in St. Louis
August 13, 2012
9:00 am - 11:00 am
Thesis Defense for Master of Arts in Statistics
Abstract: This study is motivated by estimating the spatial transformation when object's positions are reconstructed from memory. Bidimensional regression is typically used for estimating the transformation between two maps. We will develop Bayesian inference to extend the bidimensional regression to incorporate many memorized maps from one physical map.
Location: Cupples I, Room 6
Host: Prof. Nan Lin
Analysis of the Navier-Stokes-αβ equations
Josh Brady, Department of Mathematics, Washington University in St. Louis May 23, 2012
9:00 am - 11:00 am
Abstract: One of the outstanding problems in mathematics has been to prove (or disprove) well-posedness of the 3-dimensional Navier-Stokes equations. Because of these and other difficulties, many turbulence models have been proposed. In this talk we investigate such a model: the Navier-Stokes-αβ equations developed by Fried and Gurtin. We begin by presenting well-posedness results for the Navier-Stokes-αβ equations. Then we will discuss the ideas of determining nodes and modes, and the sufficient conditions for a set of nodes or modes to be determining. Finally we will relate these results to similar results that exist for the 2-dimensional Navier-Stokes equations.
Location: Eads, Room 116
Hosts: Profs. Renato Feres (Washington University in St. Louis) & Eliot Fried (McGill University)
Composite MRA wavelets
Ben Manning, Department of Mathematics, Washington University in St. Louis
April 26, 2012
1:00 pm - 2:30 pm
Abstract: We will look at composite MRA wavelets and present a foundation for studying these types of wavelets. We will produce examples of compactly supported composite MRA wavelets with desirable properties.
Location: Cupples I, Room 199
Hosts: Profs. Guido Weiss & Ed Wilson
Billiard Markov Operators and Second-Order Differential Operators
Jasmine Ng, Department of Mathematics, Washington University in St. Louis
April 20, 2012
1:30 pm - 3:30 pm
Abstract: We will consider a class of Markov operators that arise from billiard dynamical systems. In addition to discussing results about their convergence to second-order differential operators, we will approximate the spectrum of one in terms of the other.
Location: Cupples I, Room 199
Host: Prof. Renato Feres
Mumford regularity for general rational curves on hypersurfaces
Sara Gharahbeigi, Department of Mathematics, Washington University in St. Louis
April 20, 2012
10:00 am - 12:00 pm
Abstract: We show that for a general smooth rational curve on a general hypersurface of degree d ? N in P N, N ? 3 , the restriction map of global sections is of maximal rank, and therefore the regularity index of such curves is as small as possible.
Location: Cupples I, Room 199
Hosts: Profs. Roya Beheshti & Mohan Kumar
Quotients of Subgroup Lattices of Finite Abelian p-groups
Marina Dombrovskaya, Department of Mathematics, Washington University in St. Louis
April 19, 2012
11:30 am - 1:00 pm
Abstract: Let G be a finite abelian p-group of type λ. It is well-known that the lattice L(p) of subgroups of G is the order-theoretic p-analogue of the chain product [0,λ]. However, any surjection φ : L(p) → [0,λ] with order analogue properties does not respect group automorphisms. We are interested in L, the quotient lattice of L(p) under the action of a Sylow p-subgroup of the automorphism group of G. This quotient lattice is particularly interesting since it respects group automorphisms, has the property that the size of an orbit of the action is a power of p, and is closely related to the product of chains [0,λ]. We will discuss combinatorial properties of L as well as interesting properties of quotients of L(p) under the actions of lattice automorphisms and lattice automorphisms induced by group automorphisms that arise in the course of studying L.
Location: Cupples I, Room 199
Host: Prof. John Shareshian
Coordinates Arising from Affine Fibrations
Drew Lewis, Department of Mathematics, Washington University in St. Louis
April 16, 2012
2:30 pm - 4:00 pm
Abstract: A coordinate is a member of a minimal generating set of a polynomial ring. A central question in the study of polynomial rings is: given a polynomial, when is it a coordinate? One version of the Dolgachev-Weisfeiler conjecture asserts that polynomials arising from affine fibrations are coordinates. We will discuss such polynomials, including showing many of them to be coordinates. We will relate this to a well known class of examples called the Venereau polynomials; in particular, we show the second Venereau polynomial to be a coordinate.
Location: Cupples I, Rom 199
Host: Prof. David Wright
Comparison Theorems in Elliptic Partial Differential Equations with Neumann Boundary Conditions
Jeffrey Langford, Department of Mathematics, Washington University in St. Louis
April 13, 2012
8:00 am - 10:00 am
Abstract: For a partial differential equation (PDE) with Neumann boundary conditions, we ask how the input data should be arranged so that the solution has maximal Lp norms and oscillation. We begin our discussion with a physically motivated (open) problem and proceed to discuss the history of comparison theorems in PDEs with Dirichlet boundary conditions. We continue by discussing my comparison results that impose Neumann boundary conditions, and use these results to arrive at a conjecture to the original physically motivated problem.
Location: Cupples I, Room 199
Hosts: Profs. Al Baerstein (Washington University in St. Louis) & Richard Laugesen (University of Urbana, Illinois)
Looking to contact a past graduate? Visit our Recent Ph.D.'s page ⇨
Download a Spring 2012 Mathematics Theses from WUSTL's Open Scholarship⇨
—Math news, stories, videos, and interviews by Marie C. Taris, http://www.math.wustl.edu/marietaris/math.html⇨
Liked this story? See also June 2011 News story 2011 Theses in Mathematics and Statistics⇨ and 2013 Theses in Mathematics and Statistics⇨