The History of Analysis Seminar took place in the Fall of 1996, 1997, and 1998, at the Math Department of UC Berkeley. It featured weekly talks by professors, post-docs, and students. The seminar was organized by Alex Gottlieb, Tom Hadfield, and Henry Helson.

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*Objective*

Our goals were to make analysis seem like a smaller subject by placing some of its modern branches in developmental perspective, and also to highlight the unity of analysis and its role as a science.

In 1996 the speakers were: Henry Helson, Angus Taylor, Craig Evans, Brian Tvedt, Trent Walker, Paul Chernoff, Donald Sarason, Alex Gottlieb, Morris Hirsch, Dylan Selegue, John Neu, and Lucien Le Cam. Unfortunately, the abstracts of those talks have been lost forever.

The abstracts from 1997 and 1998 appear below.

(Alex G would like to thank the speakers, and everyone else who participated.)

#### September 16, 1997

### Elie Cartan, Fundamentalist Extraordinaire

#### by Greg Anderson

No one doubts that Cartan made important contributions to mathematics, but few modern sources accurately gauge his widespread influence. Along with a biographical sketch, we'll present a laundry list of (now ubiquitous) concepts due originally to Cartan and a few Fundamental Theorems to go with them.

Disclaimer: The speaker is not a historian, merely a fierce admirer of Cartan.

#### September 23, 1997

### Bourbaki --- Progress or Disaster ?

#### by Lucien LeCam

Bourbaki was born out of the piteous state of French mathematics in the mid-Thirties. He set himself a very ambitious task, namely to write a treatise giving , in its first part, a solid foundation to the rest of the treatise and even to most of modern mathematics.

For reasons of clarity and economy of space, this forced a certain "abstract style, the intent of which has been wildly misunderstood.

Besides recounting some anecdotes about Bourbaki, I shall venture an opinion about his success or lack of it and try to refute the attack by Arnold that Bourbaki is responsible for the Chernobyl disaster.

#### September 30, 1997

### Noncommutative Geometry

#### by Vaughan Jones

I will give an account of Connes' development of noncommutative geometry as I saw it happen.

#### October 7, 1997

### Irreversibility from Boltzmann to the Present Day (in that order)

#### by Alex Gottlieb

Boltzmann's Equation and H-Theorem. The Critique of Irreversibility. Brownian Motion. Master Equations. The BBGKY Hierarchy. Irreversibility and Information. Theorems about the Increase of Entropy.

#### October 15, 1997

### A Historical Sketch of the Notion of Compactness

#### by Manya Raman

Come and find out lots of juicy facts about the history of compactness. What were some of the motivating problems that inspired the definition? Why was the term "compact" chosen and who chose it? What's the difference between compact, sequentially compact, countably compact, quasi-compact, relatively compact, and pseudocompact? Who really contributed to the development of the notion and who just happened to produce the right result at the right time? What's the difference between nets and filters anyway-- and what do either have to do with compactness? How is compactness used in logic? in algebraic geometry?

#### October 21, 1997

### K-theory for Operator Algebras

#### by Tom Hadfield

I'll try to outline the development of K-theory from roughly 1960 to 1985 -- especially with regard to operator algebras.

#### October 28, 1997

### The Canonical Anticommutation Relations

#### by Bill Arveson

The canonical anticommutation relations and their real cousins, Clifford algebras, have occupied a central position in differential geometry, group representations, and quantum physics. In this talk I want to emphasize their role in operator theory. In the mid-seventies, Joe Taylor initiated a very original approach to defining the joint spectrum of a commuting n-tuple of operators. The Taylor spectrum consists of all points in complex n-space for which the homology of a certain complex is nontrivial. It is now understood to be the "correct" spectrum for multivariable operator theory.

I'll discuss the basic properties of the anticommutation relations in finite dimensions (starting at the beginning) and bring out their role in the construction of the Taylor spectrum.

#### Tuesday, November 4, 1997

### Some Reflections on the History of Unitary Representations of Locally Compact Groups

#### by Calvin Moore

I will pick up the story in 1939, a turning point in the development of unitary representation theory, when the first systematic advances were made in dealing with unitary representations of groups that were neither abelian nor compact. I will briefly discuss the pre-1939 background and then follow the development of several important themes as they evolve over several decades following 1939.

#### Tuesday, November 11, 1997

### The Legacy of John von Neumann

#### by Peter Alexander

Perhaps the greatest mathematician of our century, this amazing man's penetrating mind helped shape our current world in many profound ways. We will look at some of the highlights of his work in set theory, lattice theory, economics, ergodic theory, and of course what he viewed as his greatest achievement --- the development of operator algebras.

#### Tuesday, November 18, 1997

### B* = C*

#### by William Bade

In the early days of Banach algebra theory an abstract object emerged --- the B*-algebra. The obvious example was a closed, adjoint-closed subalgebra of the bounded operators on a Hilbert space (called a C*-algebra). Gelfand and Naimark in 1943 proved that every B*-algebra is a C*-algebra, but they needed an extra axiom. This talk is about the fight to get rid of that axiom and the pretty math that was needed.

#### Tuesday, December 2, 1997

### A Hundred Years of Analysis: A Personal Reminiscence

#### by Paul Halmos

The talk offers an eagle's eye view of the articles on analysis that the speaker has dared to inflict on the mathematical public during the last hundred years. It consists of something like ten little "lecturettes" about (1) gambling systems, (2) isomorphisms of measure spaces, (3) ergodic transformations, (4) subnormal operators, (5) commutators, (6) unilateral shifts, (7) invariant subspaces, (8) quasi-triangular operators, (9) perturbations of diagonal operators, and (10) positive approximation.

#### Tuesday, December 9, 1997

### History of the Prime Number Theorem

#### by Paul Vojta

This talk will give a survey of the history of analytic number theory, with its primary emphasis on the prime number theorem, which shows that the number of primes less than n is asymptotic to n/log(n).

#### September 3, 1998

### Noncommutative Probability

#### by Michael Anshelevich

Noncommutative, or quantum, probability is a generalization of the classical probability theory whose objects are (possibly noncommutative) algebras of random variables/ observables and functionals on them, rather than probability spaces with measures. I will briefly indicate the origins of this approach in Quantum Mechanics. Then I'll give the definitions, and a number of examples. A more recent sub-theory of Noncommutative Probability, different but surprisingly parallel to the classical case, is Free Probability Theory. I'll give the very basic definitions from it. I'll finish by formulating Wigner's theorem on the convergence of random matrices to the semicircular distribution.

#### September 10, 1998

### Free Probability

#### by Michael Anshelevich

I'll indicate how Wigner's theorem is a consequence of the Free Central Limit Theorem, and prove it by simple diagrammatical techninques.

#### September 17, 1998

### An Aspect of Geometric Function Theory

#### by Shoshichi Kobayashi

A quick survey starting from the mid-nineteenth century to the present.

#### September 24, 1998

### Selected Theorems of Shizuo Kakutani

#### by Alex Gottlieb

Shizuo Kakutani is a great analyst of the 20th Century. He has done fundamental work in Banach spaces, ergodic theory, fixed-point theory, Brownian motion, measure theory, and other subjects. I will survey Kakutani's work, concentrating mainly on his surprising theorems about Brownian motion. I will also prove at least one of his theorems, probably the one about "equivalence of infinite probability measures."

#### October 1, 1998

### Rearrangement Inequalities

#### by Michael Goldberg

The symmetric-decreasing rearrangement of a function (in R^n) possesses two useful properties, namely it is symmetric and it decreases as a function of distance from the origin. Using little more than Fubini's Theorem and a little ingenuity, I plan to show how taking rearrangements yields useful inequalities in several familiar integral expressions. My primary focus will be a 1930 theorem of Riesz which appears, among other places, in a 1983 proof of some sharp constants for the Hardy-Littlewood-Sobolev inequality.

#### October 8, 1998

### Reconstruction from Projections: the Past, the Remote Past, and the Future

#### by Alberto Grunbaum

I will tell my own version of the history of what is known as the Radon inversion problem: reconstructing a function in R from its integrals along all (n-1) dimensional affine subspaces. I will make an effort to see how this problem has arisen in many different setups and how some modern biomedical imaging problems lead to radically more complex (nonlinear) and yet unsolved versions of this type of inversion problem.

#### October 15, 1998

### Removable Sets for Bounded Holomorphic Functions

#### by Michael Christ

A compact set *K *in the complex plane is said to be removable for bounded holomorphic functions, if for any open set *U *containing *K,* any bounded holomorphic function in *U minus K* continues to a function holomorphic in *U.*

Which *K *are removable? A characterization for all sets of finite one-dimensional Hausdorff measure was completed in 1977 by G David. The solution relies on diverse tools developed by many mathematicians over a long period, including potential theory, Calderon-Zygmund singular integrals, geometric measure theory, a version of the travelling salesman problem, Menger's notion of cuvature, and a magic identity. In this talk I will outline the main ideas and techniques.

#### October 22, 1998

### The *h* Principle

#### by Danny Calegari

Many problems in geometry involve the global solution of a system of partial differential equations that may be highly undetermined locally, but for which there may be highly nontrivial obstructions to choosing local solutions appropriately so that they patch together: problems like that of immersing one manifold in another, or finding an isometric map near a distance-decreasing one, or of finding a certain geometric structure on a manifold --- a foliation, or a contact structure , or a symplectic structure. Often there is a preliminary topological obstruction to finding a solution, but once this is overcome, the analytic provlems can be dealt with. Equations with this property are said to satisfy the h-principle, and their examples are very numerous. We will give a number of historically famous examples and probably say nothing at all about the proofs.

#### October 29, 1998

### Banach Algebras: The Early Years

#### by Irving Kaplansky

#### November 12, 1998

### A Survey of Type III Factors

#### by Erling Størmer

In the theory of von Neumann algebras, the main building blocks are those with trivial centers, called factors. Among them, the ones which were the last to be understood were those of type III. I'll give a survey of their theory and try to describe what they look like and who did what and when.

#### December 3, 1998

### Fatou's Thesis

#### by Don Sarason

The thesis of Pierre Fatou, published in 1906, was a path breaking work in which the new theory of Lebesgue integration was used to study harmonic and holomorphic functions in the unit disk. While Fatou may be better known nowadays for his later work on iteration, his thesis had a comparably large impact. This talk will describe the contents of the thesis, discuss some of the developments that ensued, and relay what I have been able to learn about Fatou himself.

#### December 10, 1998

### A Brief History of Quantum Groups

#### by Nicolai Reshetikhin

In this talk I will outline some ideas behind the development of the notion of "quantum groups" and some recent and less recent results in this direction.