# Fall 2008

### The stable 4-genus of knots

#### Charles Livingston, Professor

#### Department of Mathematics, Indiana University

#### Friday, December 5th, Location TBA, at 2:30 pm

**Abstract:** The 4-genus of a knot K in S3, denoted g(K), is the minimum genus of a smoothly embedded orientable surface bounded by K in the 4-ball. This invariant is known to be subadditive: g(K # J) <= g(K) + g(J). It is known not to be multiplicative: there are knots with g(K) > 0 and g(2K) = 0. By considering the asymptotic behavior of g(nK), we can isolate some interesting problems concerning the 4-genus. In this talk I will describe the application of classical knot invariants to study the stable genus and then present further examples demonstrating phenomena that fall outside the realm of classical techniques.

### Play or not to play Lotteries?

#### Alexander Matros. Professor

#### Department of Economics, University of Pittsburgh

#### Tuesday, November 25th, AB-402 at 2:30 pm

**Abstract:** In both lotteries in the field, and contest games in controlled laboratory settings, total amounts spent in an attempt to win the prize are far in excess of the predictions assuming risk-neutral contestants interested only in their expected earnings. We report on experiments that show this overspending is persistent as the number of participants in the contest increases. Subjects as a group do not strategically reduce spending as group sizes increase, in contrast to the comparative statics theory provides. The lack of strategic response cannot be explained by learning direction theory, quantal response equilibrium, or level-k reasoning models, although logit quantal response equilibrium can fit the observed distribution of choices. Fits of these models imply behavior that is determined more randomly, measured relative to the financial incentives subjects face.

### A Model for Measurements of Lognormally Distributed Environmental Contaminants

#### Charles Davis, Principal Statistician

#### EnviroStat

#### Thursday, November 6th, AB-635 at 2:30 pm

**Abstract:** Concentrations of environmental contaminants are often assumed to have lognormal (LN) distributions, with empirical justification (skewed distributions, data assumed non-negative) perhaps augmented by appeal to a log-scale Central Limit Theorem. Decisions are made using measurements rather than unobservable actual concentrations, however, and distributions of measurements are not LN in general. At fixed concentrations the distributions of measurements are often normal. Moreover, if low-level measurements are unbiased, one unavoidably has negative values. The overall (marginal) distribution of measurements is thus a mixture; the model presented in this talk has five physically meaningful parameters. This reality is universally ignored in practice; rather, values less than a Reporting Limit (RL) are censored, being reported as simply "<RL", the negative values are never seen, and statisticians continue to write papers about and develop software for inference for left-censored LN distributions for environmental data.

There are two basic questions. First, how should one handle such a complicated model? The motivating application involves Upper Tolerance Limits (UTLs); i.e., upper 95% confidence limits for 95th percentiles. If the UTL is less than a Regulatory Criterion (RC), one rejects the null hypothesis that the facility is "dirty" in favor of the alternate hypothesis that it is "clean". The nonparametric approach requires at least n = 59 observations. One can estimate the five parameters by MLE, but where should one go from there? Alternately, is there a simple approximate treatment that does not behave too badly with respect to significance level? Or is there an appropriate Bayesian approach that one should use to overcome the computational complexity?

Then, how well or badly do conventional censored-data LN procedures perform under the more nearly correct mixed model? (What is the size of the Type III Error?) The answers to both questions are, of course, "it depends", in particular on the relative sizes of RC, RL, and the analytical variation present in low-level measurements. The answers may be somewhat counter-intuitive.

The motivating example involves facility surveys for beryllium contamination. The data presented are analyses of surface swipes using ICP-AES and ICP-MS.

### What is K-theory all about?

#### Bruce Blackadar, Professor

#### Department of Mathematics and Statistics, University of Nevada, Reno

#### Thursday, October 30th, LME-315 at 2:30 pm

**Abstract:** K-Theory began as a part of topology, and a rephrasing of the definitions led to the subject of algebraic K-theory. Topological and algebraic K-theory are both very important, but seem quite different.

Topological K-theory is the study of vector bundles, either real or complex. Not only do vector bundles arise naturally in many important settings, but topological K-theory has close and deep connections with ordinary homology and cohomology of spaces. It was discovered, primarily by Atiyah and Karoubi, that the study of vector bundles could be rephrased in terms of finitely generated projective modules over rings of continuous functions. This conversion led to algebraic K-theory, which can be described as ''linear algebra over rings.''

The operator algebra point of view unifies the two subjects and explains the connections, and there is now widespread agreement that the ''right'' way to approach most of topological K-theory is via operator algebras (or at least Banach algebras). Topological K-theory generalizes exactly to operator algebras and has led to spectacular advances in the understanding of the structure of operator algebras, as well as some remarkable applications of operator algebra K-theory to problems in topology and geometry.

In this talk, I will survey topological K-theory, the conversion to algebraic language, and the development and applications of K-theory for operator algebras. Most of the talk will be accessible to graduate students.

### A survey of knot invariants

#### Charles Livingston, Professor

#### Department of Mathematics, Indiana University

#### Thursday, October 9th, SLH-003 at 2:30 pm

**Abstract:** A knot invariant is simply a function that assigns to each knot in 3-space a value, usually numeric or algebraic. Basic examples include the crossing number and the Jones polynomial. Recent years have seen the introduction of many new invariants that are offering us deep insights into the nature of knotting. In this talk I will survey knot invariants of current research interest, describe surprising connections between these invariants, and mention some of their relationships to other areas of geometric topology. I will also present some of the long standing, easily stated, problems in knot theory that remain open.

### Ladies and Gentlemen...Introducing the Department of Mathematics & Statistics!

#### Thursday, September 16th, WRB-3006 at 2:30 pm

**Abstract:** In this special colloquium the Department will "introduce" itself, specifically to current and prospective graduate students. After some introductory comments, each of the departments' research/education "groups" will make a short presentation, in which they will describe (in very general terms) the research/teaching that their constituent members conduct. We enthusiastically invite student participation, this is a great opportunity to get to know your faculty and fellow students.

### Surfaces in sufficiently complicated 3-manifolds

#### Ryan Derby-Talbot, Professor

#### Department of Mathematics and Actuarial Science, The American University in Cairo

#### Wednesday, August 6th, AB-635 at 2:30 pm

**Abstract:** Consider a 3-manifold formed by gluing a (possibly disconnected) 3-manifold along two homeomorphic boundary components (for example, two knot complements glued along their boundaries, or (closed surface) x [0,1] glued along its two boundary components). We will explore the relationship between the topology of the glued-up 3-manifold and the complexity of the gluing map, in particular considering what surfaces can arise under "sufficiently complicated'' gluings. After highlighting several recent results in this area, we will turn our attention to the particular case that a connected manifold is glued along two tori.

### Metabelian SL(n,C) representations of knot groups

#### Hans Boden, Professor

#### Department of Mathematics and Statistics, McMaster University

#### Wednesday, August 6th, AB-206 at 11:00 am

**Abstract:** In this talk, which is a report on joint work with Stefan Friedl, I will explain why, for n prime (or more generally n a prime power), every irreducible metabelian SL(n,C) representation of a knot group factors through a finite group. It is a consequence that every such representation is conjugate to an SU(n) representation and that there are finitely many (up to conjugation). I will present a simple formula for this number in terms of the Alexander polynomial of the knot. These results are the natural n ≥ 2 generalization of results of Nagasato on metabelian SL(2,C) representations of knot groups (see math.GT/0610310).