2006-07 Colloquia talks

Abstract: Polynomials are the easiest function to work with. If we have a space curve parametrized by polynomials which is an embedding of R in R^3, then its one point compactification will be a smooth embedding of the unit circle in the unit three sphere which is nothing but a knot in the classical sense. On the other hand it can be easily seen that if we take an open knot K, then, up to equivalence, we can find a polynomial embedding from R to R^3 that can represent K. In this talk we will discuss how we can obtain a nice polynomial representation for any given knot type with the polynomials of lowest possible degree.

Abstract: A topological knot is a closed loop in three-dimensional space, or in the three-dimensional sphere. A knot is called slice if it is the boundary of a disk in the four-dimensional ball. I will illustrate this with simple examples and explain how this leads to the definition of a group called the knot concordance group (first defined in the 1960's by Fox and Milnor). I will describe some long-standing open questions about this group and some recent progress by Paolo Lisca using Donaldson's theorem and definite quadratic forms.

Abstract: Some general results on the rate of convergence of rational approximants to analytic functions will be discussed. In particularly, we tell about the well known (1/9)-problem of R. Varga and about a sharp version of this problem - the A. Magnus conjecture.

Abstract: We deal here with a class of integral transformations with respect to parameters of hypergeometric functions or the index transforms. In particular, we treat the familiar Olevskii transform, which is associated with the Gauss hypergeometric function as a kernel. It involves, in turn, as particular cases, the Mehler-Fock type transforms, which are used in the mathematical theory of elasticity. An analog of the Plancherel theorem is proved. It gives that the Olevskii transform is an isometric isomorphism between two weighted L₂-spaces. More examples of such an isomorphism are exhibited for the Mehler-Fock type transforms.

Abstract: Polynomials in one variable are one of the foundations of all higher algebra. The basic results about the ring of polynomials in one variable (the existence of greatest common divisors, the structure of ideals, etc.) all follow from the polynomial division algorithm.

A natural question is: what happens in more than one variable? Multivariable generalizations of the division algorithm are actually not too difficult to produce. Here, one divides a polynomial g by several polynomials f₁, ... ,f_s in n variables. This theory has both algebraic and geometric aspects. The algebra involves the ideal generated by the polynomials, while the geometry enters by considering the solutions of the equations f₁ = ... = f_s = 0.

In general, multivariable division doesn't behave as well as the one-variable version. The key observation made by Buchberger is that we can get better behavior by using a special type of basis for the ideal called a Gröbner basis. These allow one to manipulate ideals in an algorithmic fashion and they are computable via algorithms developed by Buchberger. One of the main applications of Gröbner bases is solving systems of polynomial equations. We will see that by using a lexicographic Gröbner basis, a system of equations can be transformed into a simpler system in which the variables are successively eliminated. This is similar to the row-reduced equations which occur in linear algebra, and the solution strategy is also similar. A less naive approach will also be sketched, in which Gröbner bases furnish an analog of the companion matrix of a polynomial in one variable.

Abstract: Let \(l\) be an odd prime, let \(n\ge 1\) be an integer, and let \(\zeta_n\) denote a primitive \(l^n\)-th root of unity. Consider the number field \(K={\mathbb Q}(\zeta_n)\) with ring of integers \(R={\mathbb Z}[\zeta_n]\). We have \((l)=(\lambda_n)^{l^{n-1}(l-1)}\), where \(\lambda_n=1-\zeta_n\). Let \(U\) denote the group of cyclotomic units in \(R\).

For each \(j\) with \(0\le j\le l^{n-1}\), there is an ideal of \(R\) of the form \((\lambda_n)^{(l^{n-1}-j)(l-1)}\). Put \({\overline R}_j=R/(\lambda_n)^{(l^{n-1}-j)(l-1)}\), and let \(\sigma_j:\ R\rightarrow {\overline R}_j\) denote the canonical surjection. Let \({\overline R}_j^*\) denote the group of units in \({\overline R}_j\). Since \(\sigma_j(U)\) is a subgroup of \({\overline R}_j^*\), we can consider the quotient group \[{\overline R}_j^*/\sigma_j(U).\] This quotient group, which can be identified with the Hopf-Swan subgroup of the class group of a Hopf order, has important applications to Galois module theory.

In this talk we show how to compute this group using the program GAP.

Abstract: Pick interpolation and its generalization the commutant lifting theorem have been a source of inspiration for more than a century, remaining vibrant because of applications to engineering and other areas of mathematics and because of a series of major advances.

Following an overview of the classical Pick interpolation and commutant lifting, we will discuss some recent developments in the area.

Abstract: Cauchy's theorem states that a finite group contains an element of prime order for every prime that divides the order of the group. Since the exponent of a group is the least common multiple of the orders of all its elements, this can be reformulated by saying that a prime that divides the order of a group also divides its exponent. It was an open conjecture by P. Etingof and S. Gelaki that this result, in this formulation, holds also for semisimple Hopf algebras. In this talk, we present a proof of this conjecture, which is joint work with Y. Kashina and Y. Zhu.

Abstract: It is well known that the equations of motion of three point vortices in an ideal (= inviscid, incompressible) fluid in a plane comprise a Hamiltonian dynamical system that is completely integrable in the sense of Liouville and Arnold. Accordingly the dynamics are quite regular in the sense that the orbits lie on invariant tori, on which they are quasiperiodic or periodic. For certain types of sufficiently small non-integrable Hamiltonian perturbations of such three vortex dynamics, many regular features of the dynamics are preserved, and it is often the case that chaotic regimes start to appear as the size of the perturbation increases.

In this talk we shall discuss recent results on the nature of such non-integrable perturbations, and their effect on the dynamics of the system. After giving a brief introduction to the mathematical preliminaries including such notions as Liouville-Arnold integrability of Hamiltonian dynamical systems and Kolmogorov-Arnold-Moser (KAM) Theory, we discuss some new theorems and applications concerning the dynamics of perturbed three vortex dynamics. A rather general class of perturbations satisfying the hypotheses of our theorems is identified, and it is shown that this class includes three vortex motion in a half-plane, three coaxial vortex ring dynamics, and the restricted four vortex problem in which a fourth vortex of arbitrarily small strength is assumed. Some proofs, which require an interesting mix of new and old techniques from topology and analysis such as the Brouwer Fixed Point Theorem, a relatively new extension of the Poincaré-Birkhoff Fixed Point Theorem, and a recent extension of KAM Theory will be sketched. Our results, and also the limitations of the regularity they predict (signaled by transitions to chaotic dynamical regimes), will be demonstrated with several numerical experiments on half-plane, coaxial vortex ring, and restricted four vortex perturbations. We shall also discuss open problems associated with this work, possible future related research directions, and some interesting applications.

Abstract: We construct a family of groups having two-dimensional Dehn functions which are a tower of exponentials of arbitrary height.

Abstract: We will discuss recent progress in the following question: "When does a crossing change preserve the isotopy class of a knot?" If time permits, we will also consider simultaneous crossing changes of knots and discuss knot invariants that detect them.

Abstract: Modular representation theory studies actions of various objects, for example finite groups, on vector spaces over a field of positive characteristic. The simplest example is an action of the cyclic group Z/p on a vector space. Such an action is described by a single matrix which, in turn, is classified by its Jordan form. Hence, we say that we completely understand representations of a cyclic group. Unfortunately, this is where our understanding stops. Most of the other finite groups have so called "wild representation type" which in plain English means one cannot even hope to classify their representations.

Not being discouraged, we try to understand some of them. I shall describe some of these attempts using the group Z/p x Z/p as my case study. Jordan forms will make an appearance again, and I shall explain how we take the most advantage of the fact that we do understand the case of a cyclic group.

This is a joint work with Jon Carlson, Eric Friedlander and Andrei Suslin.

Abstract: Fix an integer \(d\geq 2\). We study the number of ways a nonnegative integer \(m\) can be written as a sum of \(d\) squares. To do this define \[\Box_{d,m}=| \{(x_1,\ldots, x_d)\in\mathbb{Z}^d\;|\;m=x_1^2+\ldots+x_d^2\}|.\] It is not hard to see that \(\Box_{d,m}\) is the coefficient of \(q^m\) in \(\Box(q)^d\) where\begin{equation}\Box(q) =\sum_{j\in\mathbb{Z}}q^{j^2}.\end{equation}

In 1829 C.G.J. Jacobi obtained identities which determine the numbers \(\Box_{d,m}\) for \(d=2\), \(4\), and \(6\). For example when \(d=2\) we have \begin{equation}\Box(q)^2 =1-4\sum_{{\stackrel{j,k=1}{}}}^\infty (-1)^{k}q^{j(2k-1)}.\end{equation} This says that for \(m\geq 2,\) \(\Box_{2,m}\) equals 4 times the difference between the number of divisors of \(m\) congruent to \(1\mbox{ mod }4\), and the number of divisors of \(m\) congruent to \(-1\mbox{ mod }4\).

After giving some examples of Lie algebras and Lie superalgebras, we explain how Jacobi's identity mentioned above and others can be obtained from the representation theory of affine Lie superalgebras. The main results are based on the paper "Integrable highest weight modules over affine superalgebras and number theory," by V.G. Kac and M. Wakimoto.

This talk is accessible to undergraduates.

Abstract: We discuss several asymptotic results on the powers of a square matrix associated with SVD (singular value decomposition), QR decomposition (Gram-Schmidt process), Cholesky decomposition, and some more if time permits.

Abstract: In this talk we consider a variant of the traditional non-circular model for crossover designs. Instead of assuming that each treatment applied to an experimental unit imparts the same carryover effect regardless of the treatment applied to the next period on the same unit, we consider the model which assumes two types of carryover effects that extend from a period to the next period. One type is called self carryover effect when a treatment is followed by itself in the next period on the same unit and the other type is called mixed carryover effect when a treatment is followed by any other treatment in the next period. Such models are useful in sensory trials. Efficient estimation and testing of the direct treatment effects (imparted by the treatment itself on the experimental unit of application) as well as the carryover effects under different models are of interest to the practitioners from application and model building point of view and have been addressed by many researchers. In the present article the problem of identification of optimal designs for the estimation of the mixed carryover effects has been taken up. It is shown that under the self and mixed carryover model generalised Patterson's balanced designs (termed also totally balanced designs in the literature), which are known to be universally optimal for the estimation of the direct treatment effects are also universally optimal for the estimation of the mixed carryover effects provided that the number of periods exceeds two. Patterson's balanced design does not exist for two periods. In the case of two periods, for the self and mixed carryover model, universal optimal designs are also identified.

Abstract: Given data from L experiments or observational studies initially believed to be similar, it is desired to estimate the mean corresponding to an experiment or observational study, E, of particular interest. It is often profitable to use the data from related studies to sharpen the estimate corresponding to experiment E. However, it is essential that all of the data that are combined be concordant with the data from E. Our method uses the observed data to determine the nature and amount of the pooling of the data. We show the efficacy of the method by proving an asymptotic result about the posterior probability function associated with all partitions of the L experiment means into subsets, and by carrying out a numerical investigation. We illustrate this methodology by analyzing a data set from six clinical trials that studied the effect of using aspirin following a myocardial infarction.

Abstract: In the classical theory of dynamical systems which deals with diffeomorphisms and smooth flows on compact manifolds hyperbolic behavior is known to imply stability of the global topological orbit structure under small perturbations of the system, called structural stability. However, differentiable orbit structure is never stable. Furthermore, even topological stability has to be qualified in the continuous time case where time change must be allowed. And full hyperbolic structure is necessary for structural stability.

It is quite remarkable that for actions of higher rank abelian groups much stronger rigidity phenomena appear. First, there is global rigidity of differentiable orbit structure for standard examples of actions with global hyperbolic behavior, such as commuting hyperbolic automorphisms of a torus, or Weyl chamber flows (which are higher rank counterparts of geodesic flows on symmetric spaces of negative curvature). Notice that for R^k actions with k>1 only linear time changes are allowed. Rigidity for these actions was established in the mid-1990ies in a series of papers joint with M. Guysinsky and R. Spatzier. The central idea of the method is proving regularity of the structural stability maps by building invariant geometric structures for perturbed actions and showing that the topological conjugacy must intertwine the invariant structures for perturbed and unperturbed actions.

More recently jointly with D. Damjanovic we showed that similar differentiable rigidity takes place for several classes of partially hyperbolic actions where there is no structural stability in the rank one case and hence the previous method is totally useless. Instead we developed two complementary methods which may even be more interesting that the results they produce.

One method is based on linearization of the conjugacy equation, solving the linearized problem with tame estimates (based on vanishing of the obstructions due to higher rank) and using KAM (Kolmogorov-Arnold-Moser) type iteration scheme to construct a converging sequence of approximate conjugacies.

The other method is based on translating the conjugacy problem to a cohomology problem over the perturbed action and using description of generators and relations in classical split Lie groups to solve those equations.

All necessary notions and constructions will be explained during the talk.

Abstract: Let G be a reductive algebraic group, and let W be its Weyl group. (For example, if G = GL(n), then W is the symmetric group of degree n.) A recurring theme in representation theory is the fact that many deep ideas and sophisticated structures attached to G are accessible via fairly elementary calculations in terms of W. Weyl groups themselves are fairly well-understood - they are all crystallographic finite Coxeter groups, which have been studied since at least the 1930's - so this means we can really "get our hands on" abstract things like perverse sheaves on the unipotent variety of G.

Now, suppose we start with a group W that is not a Weyl group of anything, but is close: perhaps a non-crystallographic Coxeter group, or even a complex reflection group. Many representation-theoretic calculations still make sense, and the results have some shocking properties (various compatibility, integrality, and positivity conditions that are all explained by G in the Weyl group case). It looks as though we are studying the representation theory and geometry of "nonexistent" algebraic groups! I will discuss various results in this vein, in particular for the case where W is a dihedral group. This is joint work with A.-M. Aubert.

Abstract: This will be an expository talk on the history of the Cauchy transform - starting with the classical Cauchy integral formula and proceeding to more modern results.

Abstract: James Clerk Maxwell is remembered by generations of physics students for his profound insights about electricity and magnetism, summarized in a series of daunting equations. Maxwell also left behind a collection of serious toys and a string of commendably funny poems. We take a look at Maxwell's toys and humor, finding clues about his modes of thought.

Abstract: Research involved in the translational regulation of suspected genes involved in cancer has come to the forefront in recent years. There is mounting evidence that post-transcriptional and translational controls mediated by various regulatory molecules, such as RNA binding proteins and non-coding miRNAs, is critically important. Translational control mediated by p53 is important for cell cycle control and chemosensitivity. Recent studies from our laboratories have shown that non-coding miRNAs play key roles in colorectal cancer by mediating the translational rate of their target mRNAs, with some strongly suspected of being transcriptionally regulated by tumor suppressor gene p53. Due to the critical role of miRNA in gene regulation and malignant progression, it is essential to improve our understanding of the molecular basis of miRNAs regulated by the p53 tumor suppressor gene. We studied the possible interaction between p53 and miRNAs in regulating gene expression using human colon cancer HCT116 (wt-p53) and HCT116 (null-p53) cell lines. The effect of p53 on the expression of miRNAs was investigated using miRNA expression array analysis. Our investigation indicated that the expression levels of a number of miRNAs were affected by wt-p53. Down regulation of wt-p53 via siRNA abolished the effect of wt-p53 in regulating miRNAs in HCT116 (wt-p53) cells. Global sequence analysis revealed that over 46% of the 326 miRNA putative promoters contain potential p53 binding sites, suggesting that some of these miRNAs were potentially regulated directly by wt-p53. In addition, the expression levels of steady state total mRNAs and actively translated mRNA transcripts were quantified by high density microarray gene expression analysis. The results indicated that nearly 200 cellular mRNA transcripts were regulated at the post-transcriptional level, and sequence analysis revealed that some of these mRNAs may be potential targets of miRNAs. Some of these miRNAs have been shown to be associated with clinical outcome of colon cancer treatment. We believe the comprehensive analysis of global gene regulation at multiple levels will provide critical information for cancer biology and for novel target discovery and development.

Abstract: The distortion of a curve is the supremum, taken over distinct pairs of points of the curve, of the ratio of arc length to spatial distance between the points. Gromov asked in 1981 whether a curve in every knot type can be constructed with distortion less than a universal constant C. Answering Gromov's question seems to require the construction of lower bounds on the distortion of knots in terms of some topological invariant. I will talk about why this has been difficult as well as new results that may help in constructing invariants that can be used to provide lower bounds.

Abstract: see below

Abstract:

• What does "a 95% chance" mean?
• Is hindsight 20/20, or is it 50/50?
• How can 400 voters tell you what 2,000,000 are thinking about Sen. Lieberman?
• What do statisticians do in polling, agriculture, medical treatment development? (What does it take to be a statistician ... do you have what it takes?)
• What is a statistical distribution, how can it quantify benefits of taking Aricept for Alzheimerís?
• How much confidence do we have in statistical answers?

Statistics today is all of the above ... the science of decision-making under uncertainty. Virtually every field of human endeavor utilizes statistical methods. These methods tell us how to plan experiments or simulations, what data to obtain, and how to use it to make decisions: How to predict elections. How to analyze climate data to determine if temperature is increasing or merely within a range of natural historical variation. How to quantify the benefits of taking a medication such as Aricept vs. taking a supplement such as Alpha-Lipoic Acid for Alzheimerís disease. How to determine how far beavers migrate. One question always is: "How much confidence do we have in our answer?" The Bootstrap Method (BM) draws samples at random with replacement from the data at hand to assess confidence in the answer. It is widely used. But the BM can have problems. For example, the BM will never yield observations larger than the largest in our dataset. The Generalized Bootstrap (GB) was introduced to deal with these drawbacks. We will discuss: what the Bootstrap does, and how it can fail; what the Generalized Bootstrap does to remedy the failure; comparisons; open problems. Examples involving beaver migration; also data mining, water pollution, and wheat seed infection as time allows.

This lecture will be accessible to students!

Abstract: In this talk, we will show that the quantization condition posed by Gukov is true for the SL(2,C) character variety, by using the regulator map of Beilinson-Deligne on a curve. Furthermore, we will give a generalized volume conjecture from the motivic perspective.

Abstract: The finite graphs considered here are termed as slightly irregular if one or two vertices have different degrees than all others. We give constructive and combinatorial proofs to decide why certain families of slightly irregular graphs have no planar representation and why certain families have such planar representations. Several non-existence results for infinite families as well as for specific graphs are given, for example the nonexistence of the graphs with n=11 and degree sequence (4,5,5,..,5) and n=13 and degree sequence (6,5,5,...,5) are shown.

A maxplanar (triangulation) graph is termed as slightly irregular if the difference in the maximum degree and minimum degree does not exceed 1. We discuss an algorithm to construct such graphs for all values of n.

Abstract: Many areas of modern mathematics and the sciences explore the actions of groups on other mathematical objects. In particular, one might be interested in groups acting on vector spaces. This allows for group elements to be "represented" as matrices. In this talk we will use some easy examples to introduce the audience to the various flavors of the representation theory of finite groups. No mathematical background beyond some basic knowledge of linear algebra is needed in order to enjoy the presentation.

Abstract: Formal languages are studied in linguistics, computer science, mathematics, and elsewhere. Here we look at one connection to formal languages in algebra, namely the word problem for groups and monoids. The talk will give an overview of formal languages, grammars, automata, and their application to the word problem. My goal is to take an informal approach, avoiding a lot of the technical definitions, illustrating the ideas mainly by way of examples.

Abstract: Hyperbolic systems have been one of the main objects of study in smooth dynamics. Exponential expansion and contraction in these systems produces chaotic behavior with complex and stable orbit structure. We will discuss rigidity properties exhibited by a single hyperbolic diffeomorphism and compare them to the properties of actions by several commuting diffeomorphisms.