William H. Barker

Bowdoin College
Brunswick, Maine

Books:  Continuous Symmetry: From Euclid to Klein (AMA, 2007)

________ Harmonic Analysis on Reductive Groups

(NOTE: A conference on Harmonic Analysis on Reductive Groups was held at Bowdoin College in Brunswick, Maine from July 31 to August 11, 1989. The stated goal of the conference was to explore recent advances in harmonic analysis on both real and p-adic groups. It was the first conference since the AMS Summer Sym­posium on Harmonic Analysis on Homogeneous Spaces, held at Williamstown, Massachusetts in 1972, to cover local harmonic analysis on reductive groups in such detail and to such an extent. While the Williamstown conference was longer (three weeks) and somewhat broader (nilpotent groups, solvable groups, as well as semisimple and reductive groups), the structure and timeliness of the two meetings was remarkably similar. The program of the Bowdoin Conference consisted of two parts. First, there were six major lecture series, each consisting of several talks addressing those topics in harmonic analysis on real and p-adic groups which were the focus of intensive research during the previous decade. These lectures began at an introductory level and advanced to the current state of research. Sec­ond, there was a series of single lectures in which the speakers presented an overview of their latest research.

_____ Lp harmonic analysis on SL(2,R)

• • Laplacian of the indicator
  • Non-analytic smooth function
  • Schwartz space
  • Bump function
  • Schwartz–Bruhat function
  • Plancherel inversion formula

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Most recent email: Friday, 7 February 2020

Dear Prof. Dr. William H. Barker:

My work in 1972 focused on continuity, symmetry, and harmony. I was attempting to define what I thought would entail “a moment of perfection” within our quantum universe. By 1980, after working with an array of distinguished scholars in Boston, Cambridge (USA), and Paris, I went back to work within a business that I had started in 1971. From a little service bureau, we soon had a software business with well over 100 employees. My first opportunity to attempt to dig back into it all back was in 2011. I was helping a nephew with his high school geometry classes when we went inside the tetrahedron — https://81018.com/tot/ — and then its octahedron, step-by-step, deeper and deeper by dividing all the edges by 2 and connecting those new vertices. Within 45 steps we were within particle physics. In 67 additional steps, we were within the Planck scale. By multiplying those classroom objects by 2, in 90 steps we were out to the approximate age and size of the universe. Instead of base-10 like Kees Boeke (1957), we used base-2, we had an inherent geometry, and we went from the Planck units to the current time.

It was an unusual, albeit, rather idiosyncratic chart of 202 notations: https://81018.com/chart/

Prima facie, do you see any merit to such a chart?

I will continue my readings of your work, Continuous Symmetry: From Euclid to Klein (AMA, 2007) and Harmonic Analysis on Reductive Groups (Springer, 1991) in hopes that you might have some guiding thoughts for  this rather idiosyncratic chart of the universe.  Thank you.

Most sincerely,

Bruce

PS. In 1746 our family settled in Bremen, Maine. Bowdoin had always been on my list of schools to consider, but in 1965 the call for voter registration in the South won the day.  I always think of you all on my way out of Freeport and as we go through Brunswick. -BEC

First email: August 2, 2016 3:20 PM

Dear Prof. Dr. William H. Barker:

My grandmother lived up the road a ways (Bremen…Damariscotta, then out to 1A and the coast). Often Dad would stop at Valerie’s in Ogunquit, my sister’s favorite restaurant; they shared the name. We’d fall quickly back to sleep as children for the final long slog up from Cambridge. For some magic reason, I would awake just as we were passing by Bowdoin. Bathed in the soft summer lights, I would secretly dream, “That’ll be my school.”

1965 came quickly and I marched off to the south to register voters, but Bowdoin always held that special place.

Today, I am delighted to find your book on continuous symmetries and remember my childhood once more. Images imprint the soul and make us who we are.

When and why is there spontaneous symmetry breaking?
Have you given it much thought?

So, I have discovered your work and I am grateful to now be taking a de facto course with you through your writing. And so I say, “Thank you!”

With warm regards,
Most sincerely,

Bruce

________________

How does one find your work:  https://en.wikipedia.org/wiki/Continuous_symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another.

Formalization
The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions of topological group, Lie group and group action. For most practical purposes continuous symmetry is modeled by a group action of a topological group.

One-parameter subgroups
The simplest motions follow a one-parameter subgroup of a Lie group, such as the Euclidean group of three-dimensional space. For example translation parallel to the x-axis by u units, as u varies, is a one-parameter group of motions. Rotation around the z-axis is also a one-parameter group.

Noether’s theorem
Continuous symmetry has a basic role in Noether’s theorem in theoretical physics, in the derivation of conservation laws from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of quantum field theory.

See also:

• Goldstone’s theorem
• Infinitesimal transformation
• Noether’s theorem
• Sophus Lie
• Motion (geometry)
• Circular symmetry

References:  William H. Barker, Roger Howe, Continuous Symmetry: from Euclid to Klein (2007)