Department of Mathematics and Computer Science

Faculty Seminars

2012 - 2013

 Swords 302

Fall 2012

Monday, September 10, at 3:00 PM

Speaker: John Little

Title:  Continua of central configurations in the n-body problem with a negative mass

Abstract:  A configuration of bodies in the Newtonian n-body problem is said to be a 

     central configuration if the gravitational acceleration vector of each body is directed 

     toward the center of mass of the configuration and the proportionality factors between 

     the accelerations and the displacements from the center of mass are all equal.  Central 

     configurations are of interest in celestial mechanics mainly because they give some of the 

     very few sets of initial conditions for which explicit solutions of the equations of motion 

     are known.  Two central configurations are considered to be equivalent if there is a 

     combination of rigid motions (translations and rotations) and scaling that

     maps one to the other.   One of the main open questions about central configurations 

     is: Given a set of (positive) masses, is the set of equivalence classes of 

     central configurations finite, or can there be a positive-dimensional family for

     some collection of masses?  For n = 3, 4 bodies, it is known that the set of equivalence 

     classes is finite for all collections of positive masses.  However, several years ago,

     Gareth Roberts of our department produced a 5-body example in the plane, containing 

     one negative mass, where the set of equivalence classes of central configurations is a 

     curve ("a continuum").   While negative masses do not occur in nature (as far 

     as we are aware, at least!), there are analogous situations involving charged particles 

     or Helmholtz vortices where negative charge or vortex strength values naturally occur

     and analogs of central configurations are studied.  So this situation is of interest

     as a measure of the possible complexity of solutions of the central configuration equations.

     In this talk, based on work of one of the research groups at the PURE Math 2012 REU 

     program at the University of Hawai'i at Hilo, we will present a construction that generalizes 

     Roberts' example to produce similar continua of central configurations with one negative 

     mass in all even dimensions  k  = 4  and larger.   The construction relies mainly on

     properties of regular polyhedra and their symmetry groups.   All necessary background

     on celestial mechanics will be presented in the talk, so this should be quite self-contained.  

Monday, September 17, at 3:00 PM

Speaker: Gareth Roberts

Title: Relative Equilibria in the Four-Vortex Problem with Two Pairs of  Equal Vorticities

Abstract: We examine in detail the relative equilibria in the four-vortex 

problem where two pairs of vortices have equal strength, that is, Gamma_1 

= Gamma_2 = 1 and Gamma_3 = Gamma_4 = m where m is a nonzero real 

parameter. One main result is that for m > 0, the convex configurations 

all contain a line of symmetry, forming a rhombus or an isosceles 

trapezoid. The rhombus solutions exist for all m but the isosceles 

trapezoid case exists only when m is positive. In fact, there exist 

asymmetric convex configurations when m < 0.

In contrast to the Newtonian four-body problem with two equal pairs of 

masses, where the symmetry of all convex central configurations is 

unproven, the equations in the vortex case are easier to handle, allowing 

for a complete classification of all solutions. Precise counts on the 

number and type of solutions (equivalence classes) for different values of 

m, as well as a description of some of the bifurcations that occur, are 

provided. Our techniques involve a combination of analysis and modern and 

computational algebraic geometry.

This is joint work with Marshall Hampton and Manuele Santoprete.

Monday, October 15, at 3:00 PM

Speaker:  Silvia Jimenez, WPI   

Title: Nonlinear Neutral Inclusions: Assemblages of Spheres and Ellipsoids.

Abstract: If a neutral inclusion is inserted in a matrix containing a uniform applied electric field,

it does not disturb the field outside the inclusion. The well known Hashin coated sphere is an

example of a neutral coated inclusion. In this talk, we consider the problem of constructing neutral

inclusions from nonlinear materials. In particular, we discuss assemblages of coated spheres and ellipsoids.

Monday, October 22, at 3:00 PM

Speaker: Thomas E. Cecil

Title: A Geometric Version of Reeb’s Theorem

Abstract: Reeb’s Theorem in topology states that if Mⁿ is a compact n-dimensional manifold and there exists 

a Morse function φ on Mⁿ with only two critical points, then Mⁿ is homeomorphic to an n-sphere Sⁿ . In this

talk we prove the following geometric analogue of Reeb’s Theorem due to K. Nomizu and L. Rodriguez:

Theorem: If Mⁿ is a compact hypersurface in Rⁿ⁺¹ such that every Morse function of the form L_p(x) = |p − x|², p ∈ Rⁿ⁺¹, 

has only two critical points on Mⁿ, then Mⁿ is a metric sphere in Rⁿ⁺¹.

Monday, November 12, at 3:00 PM

Speaker: Bill Martin, WPI

Title: Perspectives on association schemes

Abstract: Association schemes arose in the statistical work of Bose and Nair in 1939 and have been re-invented in various forms in quite an array of mathematical subdisciplines. This combinatorial subject is particularly attractive due to the fact that an association scheme can either be viewed as a set of graphs or as a set of matrices, and each viewpoint informs the other.

An association scheme is an ordered pair (X, R) where R = {R₀, . . . , R_d} is a partition of X × X into d + 1 binary relations satisfying

• the identity relation id_X = {(a, a) | a ∈ X} belongs to R (we call this relation R₀); 

• for any R_i in R its transpose R_i^⊤ = {(b,a) | (a,b) ∈ R_i} belongs to R as well; 

• for any i, j and k in {0,...,d} there exists an integer p^k_ij such that R_i(a)∩R_j^⊤(b) has size p^k_ij whenever (a,b) ∈ R_k where we write R_i(a) = {x ∈ X | (a,x) ∈ R_i}

If we then define the 01-adjacency matrix A_i of relation R_i in the obvious way, we have A₀ = I and A_iA_j = ∑_k p^k_ijA_k and more. So the d + 1 adjacency matrices thus obtained form a vector space basis for a (d+1)-dimensional matrix algebra which is also closed under entrywise multiplication.Without going into details, we aim to indicate how the eigenspace structure of this algebra plays a role in various areas of mathematics.

As time permits, we will touch on the character theory of finite groups, distance-regular graphs, error-correcting codes, finite geometry, the statistical design of experiments, spher- ical codes and designs and their connections to lattices and root systems, spin models and knot invariants, and perhaps more. A low priority for me in this talk is to include a few of my own recent results on Q-polynomial association schemes, which have become an area of intense interest in the last decade.