Mathematics Colloquia

Spring 2004

Past Talks:

• Fall 2002 Talks
• Spring 2003 Talks
• Fall 2003 Talks

Feb. 6 (12 p.m.,LA5-265) Patrick Shanahan, Loyola Marymount.

An Introduction to Algebro-Geometric Invariants of Knots.

Thurston's geometrization conjecture has motivated much of the research in 3-manifold topology since the late 1970's. Recently G. Perelman has announced a proof of this conjecture, but the story does not end there. Even if this proof is verified, the largest and least understood class of 3-manifoldsis the hyperbolic 3-manifolds.A particularly tractable subclass of such manifolds are the hyperbolic knot complements. While many classical knot invariants are defined combinatorially, new invariants derived from the hyperbolic structure on a knot incorporate ideas from topology, group theory, geometry, analysis, algebraic geometry, and number theory. In this talk I will discuss some of the different algebro-geometric invariants of knots and some techniques to compute them.

Feb. 13 (12 p.m., LA5-265) Yana Mohanty, UCSD.

Slicing up Hyperbolic Tetrahedra: From the Infinite to the Finite.

The title of this talk refers to a construction of a 3/4-ideal hyperbolic tetrahedron out of ideal tetrahedra. Most of the talk will be addressing the background and context of this problem. Specifically, I will start with an introduction to hyperbolic geometry in 2 and 3 dimensions and discuss several types of hyperbolic tetrahedra. I will then discuss the motivation for the problem, which is related to the study of 3-manifolds and scissors congruences. There will be lots of pictures and very few equations!

Feb. 20 (12 p.m., LA5-265) Amber Rosin, Cal Poly Pomona.

Some Commutativity Theorems for Subweakly Periodic Rings.

A ring R is called periodic if for every x in R, x^m = xⁿ for distinct positive integers m and n, and weakly periodic if every x in R can be represented as a sum, x= a+b, with a nilpotent and b potent. It is known that a periodic ring is necessarily weakly periodic, but whether the converse is true is an open question. In the special case of commutative

rings or rings with nil commutator ideal the converse is known to hold. We will discuss a more general class of rings called subweakly periodic rings, with particular emphasis on conditions which imply that such rings are commutative or have a nil commutator ideal. Related results will also be established for weakly periodic (as well as periodic) rings.

Feb. 27 (12 p.m., LA5-265) Michael McCarthy, UCI.

Geometric Design of Mechanically Reachable Surfaces Using Polynomial Homotopy

This talk presents recent results in the kinematic synthesis of the articulated chains that form the structural skeleton of robot manipulators and spatial mechanisms. The task of the chain is defined as a set of spatial positions of an end-effector, and the goal is to determine the physical dimensions of all chains of particular type that can reach these positions.

An important class of chains has the property that they constrain a reference point in the end-effector on a specific algebraic surface, called the reachable surface of the chain. There are seven of these chains which have the associated reachable surfaces: the plane, sphere, right circular cylinder, right circular hyperboloid, elliptic cylinder, right circular torus and the general torus.

The algebraic equations of these surfaces combine with the specified spatial position to define a set of polynomials that are solved to determine the dimensions of the chain. This is termed "geometric design"of the chain. For the seven cases we consider the polynomial systems range in total degree from 32 for the plane to over 4 million for the circular torus. In order to solve these equations, we modified the polynomial homotopy software POLSYS-PLP, so that it would (i) use a start-system with a general linear structure, and (ii) run in a parallel computer environment. The result was a set of generic solutions to these polynomial systems, which range from 32 to over 42,000 roots.

Mar. 5 (double presentations, 12 p.m. & 2 p.m., LA5-265) Arthur Benjamin, Harvey Mudd College.

The Art of Mental Calculation. (12 p.m.)

Arthur Benjamin will demonstrate and explain the secrets of rapid mental calculation. Dr. Benjamin teaches mathematics at Harvey Mudd College and is one of the world's fastest "lightning calculators". He is the author of several books and has presented his mixture of math and magic to audiences all over the world.

Counting on Determinants. (2 p.m.)

We demonstrate how determinants solve many interesting combinatorial problems. Determinants count non-intersecting lattice paths, spanning trees, and permutations with specified descent points. Elegant proofs of these results are based on the definition of the determinant and occasionally the principle of inclusion-exclusion.

This talk is based on joint work with Naiomi Cameron of Occidental College.

Mar. 12 (12 p.m., LA5-265) Silvia Pascoli, UCLA.

Neutrino physics: recent discoveries and questions for the future.

Mar. 19 (12 p.m., LA5-265) Andrew Nestler, Santa Monica College.

The Simpsons Rule: Mathematical Morsels from The Simpsons.

Now in its 15th season, The Simpsons is an award-winning global pop culture phenomenon. But did you know that The Simpsons also contains over one hundred mathematical moments, with material ranging from arithmetic to geometry to calculus? There's even a resident mathematician/inventor, Professor Frink. Join us as we present some of our favorite mathematical excerpts from The Simpsons, and explore the related mathematical content, accuracy and pedagogical value.

This talk is based on joint work with Dr. Sarah J. Greenwald, Appalachian State University.

Mar. 26 (12 p.m., LA5-265) Jenny Switkes, Cal Poly Pomona.

How Fast Is Traffic Moving.

Drivers often base their perceptions of the average highway speed on the speeds of vehicles that pass them or are passed by them. This sampling of vehicle speeds usually results in a misperception of the average highway speed. A continuous probability model provides insight into this phenomenon and leads to some unexpected results..

Apr. 16 (12 p.m., LA5-265) John Alongi, Cal Poly SLO/Pomona College.

Recurrent Points, Indecomposable Sets and Test Functions.

Inspired by the work of Birkhoff, Morse, Smale and Conley, we present a topological perspective on recurrent points, indecomposable invariant sets and real-valued test functions for flows on topological spaces. Our goal is to separate the phase space of a flow into two invariant subsets, one in which every point is recurrent and another in which no point is recurrent. Furthermore, we seek to partition the collection of recurrent points into closed invariant subsets that are dynamically indecomposable. Finally, we develop test functions that are constant on indecomposable sets and decrease along orbits of points that are not recurrent. The presentation will culminate with a statement of Conley?s Fundamental Theorem of Dynamical Systems.

Apr. 23 (2 p.m., LA5-265) Masamichi Takesaki, UCLA.

The structure of a factor.

At the last talk I gave here at CSULB, I explained that operator algebras occur in many places in natural way. In this talk, I'll explain how von Neumann algebras look like.

Apr. 30 (12 p.m., LA5-265) Angelo Segalla, CSULB.

An Extension of Steiner's Deltoid Theorem.

Consider a triangle ABC. The locus of all those points P in its plane such that the perpendicular projections of P on the three sides of the triangle are collinear is the circumcircle of triangle ABC, a conic. The line of the projections is called the Wallace-Simson line and is the degenerate pedal triangle of P.

Among some of the geometric delights related to this theorem are the Nine-point circle, Morley's triangle, the Euler line, and Steiner's deltoid (which itself has some special properties in projective geometry; see the "Math Day at the Beach 2003" T-shirt). We will prove the above theorem straight away and demonstrate it dynamically in Sketchpad.

Then, the surprise: what if the projections from P to the three sides of the triangle are not orthogonal, but in arbitrary directions? This little known but interesting extension by Miguel de Guzman produces ... the other conic sections! We will provide the proof and another dynamic illustration (that is harder than the proof).

May 7 (12 p.m., LA5-265) Bernardo Abrego, CSU Northridge.

Constructing point-sets in general position with many geometric patterns.

In this talk we will show a method to construct finite point sets in the euclidean plane with the following properties: no three points in the set are on a line (the set is in general position), and there are many copies of a given pattern determined by the points in the set. We will show better constructions for some particular patterns, like the square and the equilateral triangle. The talk is elementary and assumes no prior knowledge other than simple notions of Euclidean Geometry.