Spring 2007

AB 110  Undergraduate Seminar Series

The Euler number, Platonic solids and vector fields

Prof. Stanislav Jabuka

Dept of Mathematics & Statistics, University of Nevada

Thursday, 25 January at 4:00

Abstract: The talk will introduce the Euler number for surfaces and compute it for many examples. The Euler number is one of the simplest instances of a "topological invariant" - a tool used in much of the progress in topology in recent decades. After introducing the Euler number, we will look at some of its many applications. The first one will be the derivation of a complete list of the classical Platonic and Archimedean solids (known already to ancient Greek geometers). We will also look at some of their generalizations to higher genera. The second application will be related to vector fields on surfaces. The Euler number is a very effective obstruction to the existence of a non-vanishing vector field, a fact that in the context of surfaces has a number of interesting and amusing consequences. The final portion of the talk will explain how the Euler number can be defined for all topological spaces. Except for this final portion, the talk will be accessible to anyone having taken calculus.

The Banach-Tarski Paradox

Bruce Blackadar, Professor

Dept of Mathematics & Statistics, University of Nevada

Wednesday, 31 January at 5:30

Abstract: The Banach-Tarski Paradox has been called "the most surprising result in theoretical mathematics." Banach-Tarski Theorem The unit ball in R3 can be decomposed into a finite number of pieces (5 will do) which can be rigidly moved to fit together to form two disjoint balls of radius 1. The "pieces" are very complicated -- dense in the ball; the Axiom of Choice is even needed to define them. A variation of the Banach-Tarski Paradox is often popularly stated: "A ball the size of a pea can be cut into finitely many pieces which can be reassembled to form a ball the size of the sun." This result is called a "paradox" since it seems to fly in the face of intuition or common sense. However, there is nothing paradoxical about it; it can be done using careful application of sound and fairly elementary mathematics. In this talk, we will explain all the components and steps used in the construction, including a discussion of the Axiom of Choice and a generalization of the Schröder-Bernstein Theorem of set theory. Everyone is welcome to come; there is enough good mathematics here to challenge any of us. But the talk will be specifically designed for undergraduates. To understand virtually everything in the talk, you will only need to know:

  • Basic facts about Euclidean space Rn, n ≤ 3.
  • Basic facts about complex numbers.
  • Set and function notation and terminology.
  • Countable and uncountable sets.
  • The definition of a group.
  • Equivalence relations.
  • Some knowledge of linear algebra and abstract algebra will be helpful but not essential. Together, Math 283, 330, and 331 will be more than enough background.

lecture notes (pdf file)

Structures of stable and related stochastic processes

Jan Rosinski, Professor

Department of Mathematics, University of Tennessee, Knoxville

Thursday, 1 February at 2:30

Abstract: In 1918, after experimenting with ideal vacuum tubes, Schottsky reported two different types of random noise observed: 'Wärmeeffekt,' the continuously fluctuating thermal white noise, and, 'Schroteffekt,' discontinuous in nature and conveyed by discrete pulses shot noise. Diffusive-type Gaussian noise and pulse-type Lévy noise became building blocks for large classes of stochastic processes. Gaussian, stable, and related stochastic processes can be obtained by applying deterministic integral kernels to a random noise. Knowledge of such kernels provides intrinsic information on how such processes are built from a basic stochastic noise, which is crucial for their analysis, modeling and simulation.

From mathematical point of view, representation theories of stationary stable and of tempered stable processes build upon a connection between probability and infinite ergodic theory. It turns out that such processes are determined by equivalent classes of deterministic flows. This establishes a fundamental relationship between stochastic and deterministic dynamical systems. From a practical point of view, such theory leads to a huge class of stochastic models with well-understood structures and gives intuitive explanation for long range dependence predominant in the classes of stable and related processes. In this talk we will provide historical introduction to the subject and some recent results on tempered stable processes.

Climate Change in the Western Great Basin: Physical Basis and Current Status

Kelly Redmond, Professor

Deputy Director

Western Regional Climate Center, Desert Research Institute

Thursday, 8 February at 2:30

Abstract: The climate change issue has many more dimensions that just the effects of greenhouse gases. A variety of physical forcings influenced by human activity contribute to different degrees, though a few dominate. The inherent fluctuation that characterizes weather and climate greatly complicates the detection of change and the attribution of causes. A variety of lines of evidence are pointing toward a conclusion that the projected changes are beginning to be seen, and indeed in retrospect may have been under way without our awareness for the last decade or two. Recent trends and anomalies are much more pronounced in the western part of the United States than in the rest of the country. Ultimately, this issue requires an understanding of earth processes from the bottom of the ocean to the outer fringes of the earth's atmosphere. In many ways, climate change is unlike any other problem we have faced before. Uncertainty is pervasive: in basic understanding, in observations, in projections, and in how to react. Methods that address this uncertainty and that seek to characterize some of the underlying probability distributions are becoming increasingly common. The problem is rich with challenge, but is not insoluble. This talk should be accessible to both graduate and undergraduate students.

Groups and Geometry

Valentin Deaconu, Professor

Dept of Mathematics & Statistics, University of Nevada

Thursday, 15 March at 2:30

Abstract: An influential research program and manifesto was published in 1872 by the German mathematician Felix Klein, known as the Erlangen Program. This proposed that group theory, an algebraic approach that encapsulates the idea of symmetry, was the correct way of organizing geometrical knowledge. For example, plane Euclidean geometry is associated with the group of transformation of the plane that preserve the Euclidean distance. Plane projective geometry is associated with the group of projective transformations. Plane hyperbolic geometry can be associated with the group of projective transformations that map the unit circle onto itself.

In this talk, I will illustrate with many examples some connections between group theory and geometry. I will discuss concepts like group actions on a space, discrete subgroups of isometries, Moebius geometry, complex linear fractional transformations and Fuchsian groups. Basic knowledge about groups, linear algebra and some elementary geometry are sufficient to understand the talk.

Noncommutative geometry and self-adjoint operators

Jerome Kaminker, Professor

Dept of Mathematics, IUPUI and UC Davis

Thursday, 5 April at 2:30

Abstract: Noncommutative geometry, in the sense of Alain Connes, is based on the idea of replacing the commutative algebra of functions on a space with a noncommutative algebra (which might not have any underlying space). One may then transfers tools from geometry and topology to this new setting. In particular, K-theory, an abelian group associated to a space, very smoothly translates to the noncommutative setting. The talk will survey some of the basic ideas of noncommutative geometry and discuss various ways of viewing K-theory. One of the main applications of the subject is to the study of elliptic differential operators--both their index theory and spectral theory. I will describe some applications of the theory to the problem of continuously splitting a Hilbert space into positive and negative eigenspaces for a parametrized family of self-adjoint operators--an issue that arises in physics. This leads to a notion of higher spectral flow. This is joint work with Ron Douglas.

Semiparametric Marginal Mean Models for Multiple-Type Recurrent Events

Hao Liu, Professor

Division of Biostatistics

Dan L. Duncan Cancer Center, Baylor College of Medicine

Thursday, 12 April at 2:30

Abstract: In many longitudinal medical studies, a patient could experience series of events of distinct types during the follow-up. Examples of so-called recurrent events of multiple types include repeated episodes of squamous cell carcinoma and basal cell carcinoma, several types of skeletal-related recurrent events among patients with cancer metastatic to bone, and recurrent wheezing and cough among asthmas patients. In this talk, we consider semiparametric regression models for marginal means of two types of recurrent events. The estimation procedure was derived from a full likelihood function where a gamma frailty models the correlation between the two types of recurrent events. Large sample properties were established that allows for robust inference. Consistency and weak convergence were proved by modern empirical process method. Small sample properties were studied by computer simulations, which showed that our estimators were relatively efficient. We illustrate our method by the data of recurrent wheezing and cough in a clinical trial on bronchial asthmas.

Ribbonlength of Torus Knots

Thomas Mattman, Professor

Department of Mathematics and Statistics, California State University at Chico

Thursday, 19 April at 2:30

Abstract: (Joint work with Kennedy, Raya, and Tating). A regular pentagon can be formed by tying a knot in a strip of paper. We generalize this construction and show that every regular polygon of at least seven sides can be formed from a single strip of paper. Using these constructions, we estimate the Ribbonlength, or length to width ratio, of certain torus knots. This allows us to give bounds on the constants c_1 and c_2 that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c_1 C(K) \leq R(K) \leq c_2 C(K).

Four-manifolds, knots, and quadratic forms

Brendan Owens, Professor

Department of Mathematics, Louisiana State University

Thursday, 26 April at 2:30

ABSTRACT: Integral quadratic forms have been studied for over 3000 years and arise in many different contexts. In topology, the intersection pairing of a four-dimensional manifold is such a form. Fundamental theorems of Donaldson and others on the intersection pairings of smooth four-manifolds enable many interesting topological problems to be translated into questions about quadratic forms. I will describe some problems in knot theory that can be addressed in this manner. A notable example is the recent proof of the slice-ribbon conjecture for two-bridge knots due to Paolo Lisca; I will discuss this result and some related problems.