### Aliquot Cycles for Elliptic Curves with Complex Multiplication

*Tom Morrell, Department of Mathematics, Washington University in St. Louis*

March 21, 2013 - 3:00 pm to 4:00 pm*Location: Cupples I, Room 6 | Host: Prof. Matthew Kerr *

*A Senior Honors Thesis Presentation.*

Abstract: We review the history of elliptic curves and show thatis possible to form a group law using the points on an elliptic curve over some field L. We review various methods for computing the order of this group when L is finite, including the complex multiplication method. We then define and examine the properties of elliptic pairs, lists, and cycles, which are related to the notions of amicable pairs and aliquot cycles for elliptic curves, defined by Silverman and Stange. We then use the properties of elliptic pairs to prove that aliquot cycles of length greater than two exist for elliptic curves with complex multiplication, contrary to an assertion of Silverman and Stange, proving that such cycles only occur for elliptic curves of j-invariant equal to zero, and they always have length six. We explore the connection between elliptic pairs and several other conjectures, and propose limitations on the lengths of elliptic lists.

### Life-Cycle Investment and Consumption Retirement Strategy via Markov Chain Monte Carlo with R

*Qing Maggie Liu, Department of Mathematics, Washington University in St. Louis*

March 29, 2013 - 3:00 pm to 4:00 pm*Location: Cupples I, Room 199 | Host: Prof. Renato Feres *

*A Senior Honors Thesis Presentation.*

Abstract: This thesis develops a simulation-based approach to a maximum expected utility (MEU) portfolio allocation problems between investment and consumption. Traditional way to solve a maximum utility is gradient-based approach, which performance is highly dependent on the analytical properties of the utility function, such as convexity, boundedness, and smoothness. However, in portfolio problems, the expected utility is generally not analytical available. MEU requires computation of expected utility and its optimization over the decision variable. And in this type of problems, the complexity of the boundary conditions is another problem. A simulation-based methos avoids the calculation of derivatives and also allows for functional optimization. The algorithm combines MCMC with the insights of simulated annealing and evolutionary Monte Carlo. It can exploit conjugate utility functions and latent variables in the relevant predictive density for efficient simulation. We begin this methodology with a portfolio problem with estimation risk and CRRA utility.

### Nonholonomic Diffusion

*Edward Bryden, Department of Mathematics, Washington University in St. Louis*

March 29, 2013 - 12:00 pm to 1:00 pm*Location: Cupples I, Room 6 | Host: Prof. Renato Feres *

*A Senior Honors Thesis Presentation.*

Abstract: The great variety of mechanical systems can be sorted into two broad classes. The first class is comprised of the so called holonomic mechanical systems; the second class consists of every other mechanical system, and they are called nonholonomic mechanical systems. The differing behaviors between the two systems have at their mathematical heart the condition of integrability and involutivity. Holonomic systems are represented by a configuration manifold and a tangent sub-bundle which is involutive. On the other hand the tangent sub-bundle of a nonholonomic system, naturally, fails this condition. One can introduce random motion into a mechanical system by constructing a diffusion process on its configuration manifold. This has been fairly extensively researched for holonomic systems. However, comparatively little work of the same variety has been done on nonholonomic systems. Introducing random motion into a mechanical system even so simple as the bi-planar bicycle produces some very interesting behavior. One might expect that, due to the simple and symmetric nature of the bi-planar bicycle, if one were to look only at the motion of the center of mass, one would observe standard Brownian motion on the plane. However, the diffusion actually demonstrated is not rotationally symmetric. In fact, the overall shape of the diffusion is directly dependent on two parameters, those being the distance between the two wheels, denoted as , and the radius of the wheels themselves, denoted r. More precisely, the shape of the diffusion is dependent on the ratio r/ and r. As the ratio r/ tends to infinity the diffusion reduces to pure Brownian motion on the plane, and as the ratio approaches infinity the diffusion reduces to 1 dimensional Brownian motion. Another interesting phenomenon which arose in the bi-planar bicycle after randomness was added was diffusion in the transversal direction. In the purely smooth case such motion would be impossible. This may turn out to be a general feature of nonholonomic mechanical systems.

### Lie groups, their representations, and the Theorem of Peter and Weyl

*Niko Kesten, Department of Mathematics, Washington University in St. Louis*

March 27, 2013 - 4:00 pm to 5:00 pm*Location: Cupples I, Room 6 | Host: Prof. Xiang Tang *

*A Senior Honors Thesis Presentation.*

Abstract: We begin with some basics about differential geometry and Lie groups. We then move on to representations of them, in particular finite-dimensional representations of compact groups. Finally, we talk about the Theorem of Peter and Weyl as well as some generalizations and applications.

### Applications of Mixed Effects Models: An Analysis of fMRI data in Anesthesiology

*Wenshuai Ye, Department of Mathematics, Washington University in St. Louis*

March 27, 2013 - 10:00 am to 11:00 am*Location: Cupples I, Room 6 | Host: Prof. Nan Lin *

*A Senior Honors Thesis Presentation.*

Abstract: Functional Magnetic Resonance Imaging(fMRI) is a procedure that measures brain activity by detecting associated changes in blood flow. Anesthesiologists are interested in learning the change in brain functional activity within networks linking different cortical areas under different sevoflurane concentrations by studying the so-called resting-state fMRI data. In this thesis, we propose a mixed effects vector autoregressive model analysis for such data. This model captures the condition-specific connectivity(fixed effect), subject-specific connectivity(random effect), and run-specific connectivity nested within conditions(random effect). We may simplify the model as we implement it in SAS based on the efficiency and feasibility.

### On a Small Non-Shellable Lie Algebra

*Ari Tenzer, Department of Mathematics, Washington University in St. Louis*

March 26, 2013 - 2:00 pm to 3:00 pm*Location: Cupples I, Room 6 | Host: Prof. John Shareshian *

*A Senior Honors Thesis Presentation.*

Abstract: Shellability is a property of simplicial complexes that indicates whether the complex can be built up in a "nice" way from its constituent facets. Shellability has been found to have some interesting implications in unexpected places. Specifically, it relates to the order complex associated with the subgroup lattice (which is by definition a partially ordered set), with the partial order relation being set inclusion. The order complex associated with a poset is the simplicial complex created when the elements of the poset are considered vertices, and the totally ordered subsets are considered faces. It has been shown that a finite group is solvable if and only if the order complex associated with its subgroup lattice is shellable. It is therefore of interest to consider the implications of shellability as applied to other types of mathematical structures, such as Lie algebras. As of yet, little is known about the implications of shellability as applied to Lie algebras, and there are few known examples of nonshellable Lie algebras. To that end, in this paper we demonstrate the nonshellability of a small (dimension 6) Lie algebra. To strengthen the correspondence with the result in group theory, the Lie algebra has been chosen to be simple and the underlying field is Z_2.