Department of Mathematics and Statistics

Shoemaker Lecture Series

These lectures are supported by the Richard Shoemaker Fund.

Some Past Lectures

Fall 2015

Our distinguished speaker for the Fall 2015 Shoemaker Lecture Series was Prof. Amie Wilkinson from the Mathematics Department of the University of Chicago.

Prof. Wilkinson received her PhD from the University of California, Berkeley in 1995 under the supervision of Prof. Charles Pugh.

She was the recipient of a National Science Foundation Postdoctoral Fellowship at Harvard and has given American Mathematical Society invited plenary addresses at Salt Lake City (2002), Rio de Janeiro (2007) and San Francisco (2010).

She was also an invited speaker at the International Congress of Mathematics in Hyderabad, India (Section on Differential Equations and Dynamical Systems). In 2009, together with Ch. Bonatti and S. Crovisier she solved the $C^1$ case of the 12th problem in the Smale's list of mathematical problems for the 21st century.

In 2011 she was the recipient of the Satter Prize in Mathematics of the American Mathematical Society and she is a Fellow of the American Mathematical Society. She is a member of several editorial boards of highly prestigious journals and in recent years she has been collaborating with with Artur Avila, the first Brazilian Fields Medalist.

Her research is in smooth dynamical systems and ergodic theory (for more information about her research interests, see her webpage: Although her research lies firmly in the area of pure mathematics, she has been exploring recently the connections between smooth dynamics and the physics of particle accelerator design. This has culminated in an interdisciplinary NSF grant with members of the Univeristy of Chicago physical sciences community, starting Fall 2015.

Amie Wilkinson (University of Chicago) Sep 23-25, 2015

The Ergodic Hypothesis and Beyond

Lecture I: "The General Case"

Abstract: The celebrated Ergodic Theorems of George Birkhoff and von Neumann in the 1930's gave rise to a mathematical formulation of Boltzmann's Ergodic Hypothesis in thermodynamics. This reformulated hypothesis has been described by a variety of authors as the conjecture that ergodicity -- a form of randomness of orbit distributions -- should be "the general case" in conservative dynamics. I will discuss remarkable discoveries in the intervening century that show why such a hypothesis must be false in its most restrictive formulation but still survives in some contexts. In the end, I will begin to tackle the question, "When is ergodicity and other chaotic behavior the general case?"

Lecture II: Robust mechanisms for chaos, I: Geometry and the birth of stable ergodicity

Abstract: The first general, robust mechanism for ergodicity was developed by E. Hopf in the 1930's in the context of Riemannian geometry. Loosely put, Hopf showed that for a negatively curved, compact surface, the "typical" infinite geodesic fills the manifold in a very uniform way, a property called equidistribution. I will discuss Hopf's basic idea in both topological and measure-theoretic settings and how it has developed into a widely applicable mechanism for chaotic behavior in smooth dynamics.

Lecture III: Robust mechanisms for chaos, II: Stable ergodicity and partial hyperbolicity

Abstract: Kolmogorov introduced in the 1950's a robust mechanism for non-ergodicity, which is now known as the KAM phenomenon (named for Kologorov, Arnol'd and Moser). A current, pressing problem in smooth dynamics is to understand the interplay between KAM and Hopf phenomena in specific classes of dynamical systems. I will describe a class of dynamical systems, called the partially hyperbolic systems, in which the two phenomena can in some sense be combined. I'll also explain recent results that give strong evidence for the truth of a modified ergodic hypothesis in this setting, known as the Pugh-Shub stable ergodicity conjecture.

Spring 2015

Stephen Bell (Purdue University) Apr 8-10, 2015

The new improved Riemann Mapping Theorem

In this series of three lectures, I will describe my mathematical life as a journey that always seems to bring me back to the Riemann Mapping Theorem of classical complex analysis. The lectures will be aimed at graduate students in mathematics who have taken a course on complex analysis.

Lecture I: A surefire way to find new results about old things

Abstract: In this first talk, I will tell the story of how my PhD thesis advisor, Norberto Kerzman, and his mentor, Eli Stein, discovered a new property of the centuries old Cauchy integral and how it has influenced the way I think about complex analysis. I have tried to use the Kerzman-Stein modus operandi in my own research, and once in a while, it has led me to find shiny new things in moldy corners of the basement of complex analysis.

Lecture II: Bergman coordinates, quadrature domains, and Riemann Mapping Theorems

The unit disc in the plane is called a quadrature domain because the average of an analytic function over the disc with respect to area measure yields the value of the function at the origin. That is and is not as special as it sounds.

Lecture III: Complexity in complex analysis and Khavinson-Shapiro conjectures

Abstract: How much computational effort does it take to find classical objects of complex analysis like the Poisson kernel? I will explain my quest to get my hands on these objects that involves new ways of looking at the Riemann Mapping Theorem. Solving the Dirichlet problem can be as easy as the method of partial fractions from freshman calculus!

Fall 2013

Simon Brendle (Stanford University) Sep 11-13, 2013

Abstract: A central theme in geometry is the study of manifolds and their curvature. In this lecture series, we will discuss how techniques involving partial differential equations have shed light on several longstanding problems in global differential geometry. In particular, we will discuss the Ricci flow approach to the Sphere Theorem, as well as applications to minimal surfaces.

Lecture I: Partial Differential Equations in Geometry - Minimal Surfaces in the Three-Sphere and Lawson's Conjecture

Lecture II: Partial Differential Equations in Geometry - The Yamabe Problem in Conformal Geometry

Lecture III: Partial Differential Equations in Geometry - Hamilton's Ricci Flow and the Sphere Theorem

Spring 2013

Robert Lund (Clemson University) Feb 13-15, 2013

Lecture I: Multiple Changepoint Detection

Abstract: This talk presents a method to estimate the number of changepoint times and their locations in time-ordered data sequences. A penalized likelihood objective function is developed from minimum description length information theory principles.

Optimizing the objective function yields estimates of the changepoint numbers and location times. Our model penalty is based on the types of model parameters and where the changepoint(s) lie, but not the total number of model parameters (such as classical AIC and BIC methods). Specifically, changepoints that occur relatively closely are penalized more heavily.

Our methods allow for autocorrelation in the observations and general mean shifts at each changepoint time. A genetic algorithm, which is an intelligent random walk search, is developed to rapidly optimize the penalized likelihood. Several applications to climatic time series are given.

Lecture II: A New Way of Modeling Integer Count Time Series

Abstract: This talk proposes a new but simple method of modeling stationary time series of integer counts. Previous work has focused on thinning methods and classical time series autoregressive moving-average (ARMA) difference equations; in contrast, our methods bypass ARMA tactics altogether by using a stationary renewal process to generate a correlated sequence of Bernoulli trials.

By superpositioning independent copies of such processes, stationary series with binomial, Poisson, geometric, or any other discrete marginal distribution are easily constructed. Excursions into multivariate count series and models with periodic features are considered.

The models are naturally parsimonious, can have negative autocorrelations, and can be fitted via one-step-ahead linear prediction techniques for stationary series. As examples, count models with binomial marginal distributions are fitted to observed counts of rainy/precipitation days in consecutive weeks at Key West, Florida, and Coldfoot, Alaska.

Lecture III: Periodic Time Series

Abstract: This talk overviews modeling and inference procedures for time series data with periodic means and autocovariances. Such series arise in environmetrics, meteorology, astronomy, engineering, economics, health, and ecology.

The class of periodic autoregressive moving-average (PARMA) models is introduced to describe series with second-order periodicities. PARMA models are compared and contrasted to seasonal autoregressive moving-average models.

Testing for the presence of second order periodicities, asymptotic properties of PARMA parameter estimators, and parsimonious PARMA modeling are central issues. Several environmental applications of the developed methods are given.

Fall 2012

Duong Phong (Columbia University) Nov 14-16, 2012

Lecture I: Monge-Ampere Equations I

Examples, notions of generalized solutions and a priori estimates of Monge-Ampere equations

Lecture II: Monge-Ampere Equations II

Basic existence and regularity results of Monge-Ampere equations

Lecture III: Monge-Ampere Equations III

Monge-Ampere equations in complex function theory and Kahler geometry

Spring 2012

Martin Golubitsky (Ohio State University) Mar 28-30, 2012

Lecture I: Patterns Patterns Everywhere

Abstract. Regular patterns appear all around us: from vast geological formations to the ripples in a vibrating coffee cup, from the gaits of trotting horses to lapping tongues of flames, and even in visual hallucinations. The mathematical notion of symmetry is a key to understanding how and why these patterns form. This lecture will show some of these fascinating patterns and explain how mathematical symmetry enters the picture.

Lecture II: Pattern Formation and Symmetry-Breaking

Abstract. The equivariant branching lemma (EBL) allows us to find certain equilibria of symmetric differential equations by completing algebraic calculations. The transition from Couette flow to Taylor vortices in the Taylor-Couette experiment is one of the basic pattern forming transitions in fluid mechanics. The first half of this talk will introduce the lemma and show how it predicts the spatial characteristics of Taylor vortices from the symmetries present in the experimental apparatus. The second half will apply EBL to the study of geometric visual hallucinations, which originated with Cowan and Ermentrout.

Lecture III: Networks and Synchrony

Abstract. This talk will focus on patterns of synchrony (balanced colorings) in networks of systems of differential equations and their associated quotient networks. Examples that illustrate the relationship and differences with symmetry will be given. These ideas will be applied to a general model for rivalry introduced by Hugh Wilson.

Spring 2011

Siqu Fu (Rutgers University - Camden) Apr 13-15, 2011

The $\overline{\partial}$-Neumann Laplacian and the kernel functions

The main theme of this lecture series is how analysis interacts with geometry in several complex variables. Our main focus is the interplay between spectral behavior of the $\overline{\partial}$-Neumann Laplacian and geometry of the underlying domain. In particular, we are interested in positivity and pure discreteness of the spectrum, as well as distribution of eigenvalues. We will also talk about applications of $\overline{\partial}$-techniques to the study of kernel functions.

Lecture I: Positivity of the $\overline{\partial}$-Neumann Laplacian

In this lecture, we first review relevant spectral results for the classical Dirichlet Laplacian and contrast them with those of the $\overline{\partial}$-Neumann Laplacian. We then show how one can determine pseudoconvexity of a bounded domain in $\mathbb{C}^n$ via positivity of the $\overline{\partial}$-Neumann Laplacian. This talk should be accessible to graduate students.

Lecture II: Eigenvalue distribution of the $\overline{\partial}$-Neumann Laplacian

In this lecture, we continue our discussion of spectral theory of the $\overline{\partial}$-Neumann Laplacian. Here the focus is on pure discreteness of the spectrum and asymptotic distribution of eigenvalues. We explain how this is related to problems in quantum mechanics and how the compactly supported wavelets constructed by Dauchechies and Lemarie-Meyer play a role in the subject. Part of this talk is based on previous joint work with E. Straube and with M. Christ.)

Lecture III: Comparison of the Bergman and Szegö kernels

The Bergman and Szegö kernels are two important reproducing kernels in complex analysis. The Bergman kernel is related to the $\overline{\partial}$-Neumann Laplacian on the domain while the Szegö kernel is related to the Kohn Laplacian on the boundary. What are the relations between these two kernels? This was a question posted by Eli Stein in his book published in 1972. In this talk, we discuss boundary behavior of the quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domain in $\mathbb{C}^n$. Our analysis depends on the Hörmander type weighted $L^2$-estimates for the $\overline{\partial}$-operator by Demailly and Berndtsson. Also essential to our analysis is Blocki's estimate for the pluricomplex Green function on hyperconvex domains. This talk is based on joint work with Boyong Chen.

Leonard Scott (University of Virginia) Apr 27-29, 2011

Lecture I: Lie groups, algebraic groups, and finite groups

I will start with some simple examples, explaining the differences between Lie groups, algebraic groups and finite groups of Lie type. The interrelationship of these groups and their representations , especially between algebraic groups and finite groups of Lie type, will be a main theme of these lectures.

I will then concentrate on the birth of the modern theory of algebraic groups, as it sprang from the theory of Lie groups, and eventually found a major raison d'etre in the theory of finite groups. Very roughly, this early period starts in the early 40's and ends in the late 50's, with applications in to the construction of new finite groups continuing into the early 60's. In a lucky break of history, the Odd Order paper in finite group theory was published just afterward, in 1963, initiating an intense interest in finite groups, especially in classifying the simple ones. Remarkably, when the classification was achieved, some forty years later, all but 26 of the finite simple groups, and the alternating groups, could be found in the simple finite groups constructed from algebraic groups prior to the 1963 Odd Order paper, and could essentially be viewed as finite groups of Lie type.

The main focus of these lectures is less on groups themselves, however, than their representations, especially linear representations (though I will also discuss the nonlinear case). For Lie groups and Lie algebras there is the famous "highest weight" theory of Cartan for the irreducible representation, which, together with Weylâs character formula and completely reducibility, gives a strong understanding of all finite dimensional representations. I will discuss this theory, and try to say why physicists, after the inception of quantum mechanics, became so fascinated with it. As time permits, I will try to introduce the theory of irreducible representations for algebraic and finite groups, the topic of the next lecture.

Lecture II: Irreducible representations for algebraic and finite groups

The highest weight theory of Cartan is a theory for representations of simple or semisimple Lie groups and algebras, thus acknowledging the important place of the simple groups, and the plausibility of reductions of representation-theoretic questions to them. We will think similarly regarding algebraic groups and finite groups, thus "applying" their respective classifications.

It first turns out that there is a highest weight theory for algebraic groups, completely paralleling Cartan's theory for the Lie groups case. But there is no complete reducibility over fields of positive characteristic $p$, and the generalization of Weyl's character formula to such algebraic groups is, despite some progress, still a difficult problem, currently solved only for when $p$ is very large compared to a suitable measure of the size of the group.

However, in spite of the difficulties in the algebraic groups case, the reduction from these groups to the finite groups related to them (certain subgroups) is essentially perfect! All the the irreducible representations of these "finite groups of Lie type" come by restriction from irreducible representations of the ambient algebraic group, at least in the natural characteristic This is the content of a 1963 theorem of Steinberg, which received much attention beginning in the late 70's as the classification of the finite simple groups - with the finite groups of Lie type as the main output - began seriously appearing on the horizon.

After stating Steinberg's theorem, I will focus on the Lusztig conjecture, which gives a characteristic $p$ analog of Weylâs character formula, and I will discuss the incredibly rich theory which has led to what progress there is. So far, the main cases where this conjecture is proved may be traced ultimately back to geometry, and the theory of perverse sheaves (invented in the early 80's). I hope to give a very brief introduction to these objects, at least explaining to what perverse sheaves some standard representations correspond.

Lecture III: Homological considerations for representations of algebraic and finite groups

As mentioned in the second lecture, there is no complete reducibility for representations of algebraic groups in characteristic $p$. This means it is very difficult to understand general finite dimensional representations just from the irreducible ones. The first step is to understand the group of extensions Ext${}_1(L,L')$ between two irreducible modules $L$, $L'$, and issues involving more complicated modules can lead to higher extension groups. The analogous questions regarding finite groups are even more difficult, but there is at least an asymptotic theory, called "generic cohomology" which, for fixed algebraic groups modules, and sufficiently large finite subgroups of Lie type, describes cohomology or Ext groups of the latter in terms of the corresponding Ext or cohomology for (slightly modified) modules for the ambient algebraic group.

1-cohomology is deserving of some special attention because of its connection with maximal subgroup theory for finite groups. This may give me a chance to talk briefly about irreducible representations of finite groups of Lie type in crosscharacteristic, and about a conjecture of Bob Guralnick, on 1-cohomology with irreducible coefficients, that can be formulated for all finite groups.

The character formula of the previous lecture (the Lusztig conjecture) has homological consequences, some even relevant to the Guralnick conjecture above and related matters. I will discuss these, as well as the relevance of homological issues to establishing the formula. I will discuss quasi-hereditary algebras and Koszul algebras, and I will also briefly discuss quantum groups (which will also make a brief appearance in the second lecture). Finally, as time permits, I will discuss some questions about the structure of representation modules only party of a homological nature, such as Weyl module filtrations, and the role in positive characteristic of the Lie algebra, and its projective modules.

Spring 2010

Lijian Yang (Michigan State University) Mar 17-19, 2010

Applications of Splines in Statistics

Lecture I: Polynomial spline confidence bands for regression curves

Abstract. In this first of 3 talks, I will introduce the concept of confidence band as an extension of confidence interval. I will then describe asymptotic confidence bands for a nonparametric regression function, using piecewise constant and piecewise linear spline estimation, respectively. The confidence bands have the same width order as the Nadaraya-Watson bands of Hardle (1989), and the local polynomial bands of Xia (1998) and Claeskens and Van Keilegom (2003). Simulation experiments corroborate the asymptotic theory. The linear spline band has been used to identify an appropriate polynomial trend for fossil data. This talk is based on Wang, J. and Yang, L. (2009) Polynomial spline confidence bands for regression curves. Statistica Sinica 19 (1), 325-342.

Lecture II: Spline confidence bands for variance function

Abstract. In this second talk, I will highlight the concept of oracle efficiency and its importance in variance estimation. Asymptotic confidence bands are obtained for possibly heteroscedastic variance functions, using piecewise constant and piecewise linear spline estimation, respectively. The variance estimation is as efficient as an infeasible estimator when the conditional mean function is known, and the widths of the confidence bands are of optimal order. Simulation experiments provide strong evidence that corroborates the asymptotic theory while the computing is extremely fast. A slower bootstrap band is also proposed, with much higher accuracy. As illustrations, the bootstrap spline band has been applied to test for heteroscedasticity in fossil data and in motorcycle data. This talk is based on Song, Q. and Yang, L. (2009) Spline confidence bands for variance function. Journal of Nonparametric Statistics 21 (5), 589-609.

Lecture III: A simultaneous confidence band for sparse longitudinal regression

Abstract. In this last talk, I will present some latest, not yet published works on sparse longitudinal regression. Functional data analysis has received considerable recent attention and a number of successful applications have been reported. In this paper, asymptotically simultaneous confidence bands are obtained for the mean function of the functional regression model, using piecewise constant spline estimation. Simulation experiments corroborate the asymptotic theory. The confidence band procedure is illustrated by analyzing the CD4 cell counts of HIV infected patients. This talk is based on Ma, S., Yang, L. and Carroll, R. (2010) A simultaneous confidence band for sparse longitudinal regression.

Spring 2009

Robert Hardt (Rice University) Mar 2-4, 2009

Some Old Geometric Variational Problems in New Contexts

Lecture I: Modeling Soap Films and Soap Bubbles

Abstract. Soap films have fascinated mathematicians and physicists for hundreds of years with works by many famous people such as Lagrange, Riemann, Gauss, Weierstrass, Schwarz, Douglas, and Rado. The blind Belgian physicist Plateau formulated the problem of finding a surface of minimal area having a given spatial curve as boundary. Many different precise definitions have been given for surface, area, and boundary. We will discuss some of the interesting pictures, discoveries, properties, and some questions that are still open about these soap films.

Lecture II: Functions of Bounded Variation

Abstract. A function $f$ from an interval $[a,b]$ to a metric space $X$ has "bounded variation" if there is a bound on the expressions $dist\left(f(t_0),f(t_1)\right) + \cdots + dist\left(f(t_{n-1}),f(t_n)\right)$ independent on the choice of the partition $a=t_0<t_1<\cdots <t_n=b$. There is a simple generalization to functions of $n$ variables obtained by considering restrictions to lines in the domain. In case $X=\mathbb{R}$, such a BV function has a distribution gradient which decomposes into a classical pointwise-defined gradient, a jump part, and a third "Cantor function" part. BV functions have played a big role in many of the developments in higher dimensional Plateau problems. More recently, with $n=2$, there have been numerous applications to geometric calculus of variations, providing a class for minimizers and flows in image recognition and processing. They enjoy a certain approximation by Lipschitz functions, and their generic level sets and jump sets enjoy rectifiability properties of being strongly approximable by manifolds.

Lecture III: Rectifiable and Flat G-Chains in a Metric Space

Abstract. Rectifiability and compactness properties for Euclidean-space chains having coefficients in a finite group G were studied by W. Fleming (1966). This allowed for the modeling of unorientable least-area surfaces including a minimal Mobius band in 3-space. These properties were optimally extended by Brian White (1999) to any complete normed abelian group which contains no nonconstant Lipschitz curves. The new proofs of basic theorems from Geometric Measure Theory involved slicing to reduce to questions about 0 dimensional chains (which are finite or countable sums of weighted point masses). Independently L. Ambrosio and B. Kirchheim (2000) also generalized some basic rectifiability theorems of Federer and Fleming to currents in a general metric space. Our current work with T. De Pauw shares features and results with all these works, includes new definitions of flat G-chains in a metric space, and a proof that such a chain is determined by its 0 dimensional slices.

Spring 2007

Daniel K. Nakano (University of Georgia) Apr 25-27, 2007

Representation theory emerged about 100 years ago with the pioneering work of Frobenius and Schur, and has become a central area of mathematics because of its connections to combinatorics, algebraic geometry, number theory, and applications to physics. Cohomology theories were developed throughout the 20th century by topologists to construct algebraic invariants for the investigation of manifolds and topological spaces. Cohomology was also defined for algebraic structures like groups and Lie algebras to determine ways in which their representations can be glued together.

The purpose of the first two lectures will be to demonstrate how cohomology theories for algebraic structures can be used to reintroduce ambient geometric structures into the picture. Some of these geometric objects like varieties consisting of nilpotent matrices arise very naturally from the celebrated study of quantum groups.

The third lecture will address issues concerning integrating modern mathematical research in the training of undergraduates and graduate students with specific models that have been successfully introduced through the VIGRE program at the University of Georgia.

Lecture I: Why cohomology?

Lecture II: Hidden geometric structures in representation theory.

Lecture III: Integrating mathematical research and education.

Spring 2006

Ian Anderson (Utah State University) Apr 25-28, 2006

Symbolic Methods for Differential Geomerty, Lie Groups and Differential Equations

The theory of Lie groups and Lie algebras originated in Lie's (rhymes with see) work on the integration of ordinary differential equations. Professor Anderson will describe his recent work and how he is building a database of Lie algebras. In relativity and quantum theory Lie algebras and Lie groups are of fundamental importance and examples from general relativity will be presented. Professor Anderson will also talk about his software system VESSIOT that interfaces the MAPLE system used at UT. The first lecture is intended for a general audience and will have numerous computer demonstrations and will be of interest to anyone who has taken calculus. In the third lecture he will reconsider Lie's original program in the light of the currently available software.

Lecture I: What Lie and Einstein might have done with MAPLE.

Lecture II: Low Dimensional Lie Algebras

Lecture III: MAPLE and VESSIOT software demonstration

Lecture IV: From Lie Algebras to Lie Groups and the Symbolic Integration of Differential Equations

Spring 2005

Rafael de la Llave (University of Texas) Apr 4-8, 2005

Periodic and quasi-periodic solutions

Lecture I: Variational problems with symmetry

In this lecture we present an introduction to the variational methods to prove existence of periodic and quasiperiodic orbits in models of statistical mechanics. This is an elementary approach to some cases of Aubry-Mather theory.

Lecture II: Minimal surfaces in periodic media

In this lecture we extend the elementary approach mentioned before to deal with minimal hypersurfaces of codimension one in periodic media.

Lecture III: Smooth quasi-periodic solutions

In this lecture we present an introduction to the theory of Kolmogorov-Arnold and Moser on persistence of smooth quasiperiodic solutions. We will discuss some applications to persistence of extended systems.

Fall 2004

Arun Ram (University of Wisconsin) Nov 1-3, 2004

Lecture I: What is Combinatorial Representation Theory?

When I started in graduate school I complained to one of my professors that there were no pictures in my first year algebra course. Personally, if I wanted to be able to do algebra I had to, somehow, change that. Well, $\ldots$ this is algebra in pictures.

Lecture II: Strings and colors

"Symmetry is the way things have to be", Jane Siberry 1985

This talk is a survey about symmetry groups and their generalizations--symmetric groups, crystallographic groups, reflection groups, Hecke algebras, braid groups, affine braid groups, and affine Hecke algebras. In my head, these are all in terms of string diagrams with decorations on the strings. My goal is to show you what these objects are and why it unifies the subject to think of them this way.

Lecture III: Thinking about teaching mathematics

My last $e^\pi$ years of teaching have stimulated me to think hard about teaching mathematics and come up with some outlandish, but effective, ideas. I'd like to discuss some of these ideas, give you my reasoning, and see what you think. The wonderful thing, for me, is that in rethinking this I've learned beautiful and fascinating mathematics that blows my mind. I'd like to tell you about this wonderful mathematics. Mathematically, this is a talk about the definition of the real numbers and the function $x^y$ which was alluded to in the first sentence of this abstract.

Spring 2003

George Leitman (Univ. of Calif. - Berkeley) Apr 2-4, 2003

Lecture I: On a Transformation-Based Direct Method of Optimization

Lecture II: An Application of the Direct Method to a Class of Differential Games

Lecture III: On Vibration Control Employing the Derivative of the System State

Fall 2002

Edward Formanek (Penn State University) Oct 1-4, 2002

Lecture I: Rational Function Fields

Abstract. Let $K$ be a field, let $R = K[x_1 , \ldots , x_n ]$ be a polynomial ring over $K$ in indeterminates $x_1 , \ldots , x_n$ , and let $Q(R) = K(x_1 , \ldots , x_n )$ be the field of quotients of $K[x_1 , \ldots , x_n ]$. A field $L$ containing $K$ is called a rational function field over $K$ if it is isomorphic to $K(x_1 , \ldots , x_n )$ for some positive integer $n$.

$K$-subalgebras of $R$ are rarely polynomial rings over $K$, but there are some interesting situations when they are. One such, which goes back to Newton, is the subring $S$ of symmetric elements, meaning elements of $R$ which are left fixed by the action of $S_n$, the symmetric group on letters, which acts on $R$ by fixing $K$ and permuting $x_1 , \ldots , x_n$. In this case $S$ is a polynomial ring over $K$ in the n elementary symmetric functions $e_1 , e_2 , \ldots , e_n$. An equivalent statement is that every symmetric function of $x_1 , \ldots , x_n$ is uniquely expressible as a polynomial in $e_1 , \ldots , e_n$.

It is also true that $K$-subfields of $Q(R)$ may not be rational function fields over $K$. The first examples with $K$ algebraically closed only appeared in 1971. In general, it is difficult to decide whether a given subfield of a rational function field is a rational function field.

This lecture will survey some results about $K$-subalgebras of $R$ and $K$-subfields of $Q(R)$, as well as open problems.

Lecture II: Noether's Problem

Abstract. In 1913 E. Noether posed the following question: Suppose that a finite group $G$ acts as $K$-automorphisms of $S = K(x_1 , \ldots , x_n )$ by permuting the variables. Is the fixed field $$ S^G = \left\{r \in S \mathrm{ | } g(r) = r \mathrm{ for all } g \in G \right\} $$ a rational function field over $K$?

Although Noether's question has a positive answer in some cases, it has a negative answer in general. This lecture will survey some results on her question and outline the proof of a theorem proved independently by Swan and Voskresenskii:

Theorem. If $p = 47$ (or many other primes) and $G = \mathbb{Z}/p\mathbb{Z}$ acts on $S = \mathbb{Q}(x_1 , \ldots , x_p )$ by permuting the variables cyclically, then $S^G$ is not a rational function field over $\mathbb{Q}$.

The proof depends on the fact that $\mathbb{Z}[\eta]$ is not a unique factorization domain when $\eta$ is a primitive 46th root of unity.

Lecture III: The Ring of Generic Matrices

Abstract. Let $K$ be a field, and let $\left\{ x_{ij} (r) \mathrm{ | } 1 \le i, j \le n, r = 1, 2, \ldots \right\}$ be independent commuting indeterminates over $K$. The matrices $X_r = (x_{ij} (r))$ are called $n \times n$ generic matrices over $K$. The $K$-subalgebra of $M_n (K[x_{ij} (r)])$ they generate is called the ring of generic matrices. It is a noncommutative ring without zero divisors, and it has a division ring of fractions which is called the generic division ring.

This lecture will survey some of the applications of the ring of generic matrices and the generic division ring in noncommutative ring theory, and it will discuss a major open problem:

Is the center of the generic division ring generated by $r \ge 2$ generic matrices a rational function field over $K$?

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