#### by Peter Haskell

**Haskell:** *Tell me a little about your growing up, particularly about your background and
family life in your early years.*

**Baum:** My parents had an apartment in a building on 106th Street near
Amsterdam Avenue in Manhattan. My earliest memory is of sitting on the
front steps of that apartment building and contemplating a nearby
candy store and an engineering project. The candy store interested me
because they sold a kind of chewing gum known as “Mandrake the
Magician” gum. This astounding gum changed colors as you chewed it.
The engineering project was the tearing down of the Amsterdam Avenue
El (“El” is the “elevated train”). I was fascinated by the big
machines and the amazing things they did.

My parents were intelligent and talented. My father, Mark Baum, was born in Sanok (in what was then Austria–Hungary and is now Poland) in 1903. When he was four years old, the family moved to Rszezów where his maternal grandfather was a successful businessman. He grew up in his grandfather’s imposing house surrounded by a large and prosperous extended family. Mark was in the first generation of Jewish boys to have a secular education. In the local gymnasium, each school day began with the Lord’s Prayer. The Catholic boys stood and recited the prayer. Each Jewish boy stood silently and defiantly with his arms crossed over his chest. World War I was a huge disaster which disrupted everything. After the war, Mark left for America. Traveling alone, he arrived in New York in 1920 at the age of seventeen.

My mother, Celia Frank Baum, was born in Schenectady, New York, in 1907. When she was one year old, the family moved to Brooklyn. The Rebbe of Tolna, i.e., one of the great rabbis of nineteenth century Eastern Europe, was her great-grandfather. Her parents had come to the USA after the failure of the 1905 attempt to overthrow the czar. She grew up in Brooklyn in extreme poverty. A neighborhood public library, partly funded by Andrew Carnegie, became a refuge for her from the problems and difficulties of her immigrant family.

My parents first met on the beach in Provincetown, Massachusetts, in the summer of 1932. The attraction between them was very strong and they immediately began their turbulent off-again-on-again relationship which continued until their deaths in 1997–98. My father was an artist. He began by painting landscapes and cityscapes. Very gradually, his painting style evolved into complete abstraction. He has paintings in many museums and private collections, including two in the permanent collection of the Metropolitan Museum of Art. (For more on my father’s paintings go to markbaumestate.com.) My mother started as a teacher in the New York public schools. She worked her way up and eventually became a professor of education in Brooklyn College. She liked to joke about her career saying, “Those who can, do. Those who can’t, teach. And those who can’t even teach become professors of education like me.”

My parents married in 1935. I was born in 1936 and my brother William (Billy) Baum was born in 1939.

I went through the public schools in New York, and then was an undergraduate at Harvard. After a year in France, I returned to the USA and enrolled in Princeton as a graduate student. My parents’ story and my personal story say something about the goodness and generosity of America. I hope that America will remain idealistic and good. At the present time (July, 2019) there is some possibility that this will not happen.

**Haskell:** *When did you first consider becoming a mathematician?*

**Baum:** In grade school I was dimly aware that mathematicians existed, but I
had no clear idea of what it was that mathematicians did. When I was
eight years old, I began to wonder about various mathematical
problems. One problem that interested me was the issue of calculating
the length of the diagonal of a rectangle from the lengths of the two
sides. I concluded that it was some sort of average of the lengths of
the two sides, although I didn’t come up with the precise formula.

In high school (Bronx High School of Science), I had a wonderful charismatic teacher, Julius Hlavaty, for tenth-grade geometry. I loved solving the geometry problems and working out the proofs of theorems. The proof of the Pythagorean theorem was specially satisfying to me as it definitively solved the rectangle problem I had thought about years earlier. Dr. Hlavaty was the coach of the math team and I was on this team. We competed against math teams from the other New York high schools. In tenth grade I decided that math was for me.

In twelfth grade I entered the Westinghouse Science Talent Search. I was one of forty finalists awarded a trip to Washington, DC, and a scholarship to be used against first-year college expenses.

**Haskell:** *When did you know that you would be a mathematician?*

**Baum:** In my Princeton Ph.D. thesis I solved a problem in algebraic topology
that had baffled the experts.
Armand Borel,
who was one of the experts
on the problem, advised me not to work on it. I was, however, able to
come up with a new approach and thus solve the problem. So at the time
it seemed clear to me that I would be a mathematician.

**Haskell:** *Did you have doubts after that?*

**Baum:** Alas, a few years after completing my Ph.D. thesis it became clear
that there was a gap in my proof. I had not made any false assertions,
but one somewhat technical point which had seemed obvious to me
required a proof. On the day that I realized there was a gap, I said
to my wife, “Pack your bags. They are going to kick me out of
Princeton.” That is exactly what happened. We moved to Providence and I
was on the faculty at Brown University.

Before leaving Princeton, I had attended a lecture by Raoul Bott. During his lecture I decided to abandon the problem of my thesis and work on the topics introduced in Bott’s very interesting talk. Once at Brown, I went regularly to Harvard. Bott and I worked on developing his ideas.

Meanwhile, various mathematicians worked on the project of completing the proof in my thesis. Several papers were published giving complete proofs. One of these papers was by my student Joel Wolf, based on his Brown University Ph.D. thesis. Joel went on to a long and distinguished career as an applied mathematician.

As anyone can imagine, there were moments of discouragement during all this. With the benefit of several decades of hindsight, it is now clear that my thesis was a breakthrough on a worthwhile problem. The work I did with Bott (foliation singularities) was certainly of equal interest to the work in my thesis.

**Haskell:** *Were role models an important part of envisioning yourself as
a mathematician? If so, how?*

**Baum:** I had a number of role models who were important to me. First, there
was my inspiring high school math teacher Julius Hlavaty. Then, while
an undergraduate at Harvard, there were
George Mackey
and
Oscar Zariski.
I loved Zariski’s Yiddish accent and his unfailing
cheerfulness. With Mackey, I admired his intense devotion to
mathematics. My role model when I was a graduate student at Princeton
was
Norman Steenrod.
Even now, I am still amazed by the attention and
guidance Steenrod gave to his students.

As a postdoc at Oxford, my role model was Michael Atiyah. The remarkable energy and imagination he brought to mathematics made a deep impression on me.

During my years at Brown, my role model was Raoul Bott. In addition to being a marvelous mathematician, Bott was a most entertaining and amusing personality. I really enjoyed interacting with him.

Several times I visited the Max Planck Institute in Bonn whose director was Friedrich Hirzebruch. I came to greatly admire and like Hirzebruch. For me, his Riemann–Roch theorem was an example of truly great mathematics. He was astoundingly nonegotistical and devoted great amounts of time and energy to helping young mathematicians.

With Alain Connes, I admired the daring and originality he brought to mathematics. The message from Alain always seemed to be “Go ahead. Don’t be inhibited. If it interests you, start thinking about it and see what will happen.” The artificial boundaries between the different fields in mathematics meant very little to Alain.

My parents were friendly with Wolfgang and Dorothy Fuchs. Wolfgang was a professor in the mathematics department of Cornell University. He and Dorothy had fled Germany when the Nazis came to power. Wolfgang encouraged me. He gave me some mathematics books. He and Dorothy gave me a beautiful tuxedo which had been custom made for Wolfgang when he was a student in England. I wore it at the formal dinners of Harvard’s Eliot House. When my nephew Aaron Baum was a freshman at Harvard, I gave the tuxedo to him.

**Haskell:** *After a couple of years as an assistant professor at
Princeton, you moved to Brown. How did you feel about that move at
the time? How did it turn out?*

**Baum:** When I arrived at Brown, I suddenly felt liberated. I was working with
Bott and it seemed that a whole wonderful world of pure mathematics
was opening up for me. The Brown math department made one superb
appointment after another. Within a few years of my arrival, the Brown
math department had become one of the best in the USA. We had a golden
age in which topology and algebraic geometry flourished at Brown. Bob
MacPherson’s seminar played a key role during those wonderful years.

At Brown, Bill Fulton and Bob MacPherson and I extended the Grothendieck–Riemann–Roch theorem to projective varieties which may be singular. Many mathematicians had tried to do this, but had not succeeded due to an incorrect overall view of what was involved. The correct point of view occurred to me one Rosh Hashanah when I was attending services at my local synagogue in Providence. Since then I have always said, “You see, there is no conflict between religion and mathematics.” This may or may not be true, but what without doubt is true is that Fulton and MacPherson are marvelous mathematicians and we made a substantial contribution to the development of Riemann–Roch.

During my time at Brown, after fifteen years of trying, I finally came
up with a simple explicit geometric definition of __\( K \)__-homology — i.e., the dual theory to __\( K \)__-theory.
Ron Douglas
and I worked together
to find the isomorphism between the BDF (Brown–Douglas–Fillmore)
functional analysis model of __\( K \)__-homology and my geometric model.

The basic idea of the functional analysis model is due to Atiyah and
was stated in his paper “Global theory of elliptic operators”
[e1].
Further development of the functional analysis model was done by BDF
and
G. Kasparov.
This model has the major advantage that it applies to
__\( C^* \)__-algebras which might not be commutative. The disadvantage of
this model is that it does not have an explicit clearly defined Chern
character. The theory, due to Alain Connes, of cyclic cohomology and
spectral triples, addresses this issue. Roughly speaking, any
sufficiently smooth object for Kasparov __\( K \)__-homology does have a Chern
character in cyclic cohomology. The geometric model of __\( K \)__-homology
comes equipped with an explicit Chern character defined in terms of
“classical” characteristic classes. Part of the Connes theory is that
this is consistent with the cyclic cohomology method.

At Brown, I began working with Alain Connes on the conjecture now known as the Baum–Connes conjecture. This conjecture is unusual for its breadth and for the many implications of its validity. There may be counterexamples lurking out there. So far no counterexamples — for discrete groups and locally compact groups — have been found. Over one hundred papers have been published which prove the conjecture for various classes of discrete groups and locally compact groups. In an expository paper, Vaughan Jones and Henri Moscovici describe the conjecture as “a powerful unifying principle.”

I had several excellent Ph.D. students at Brown. Among them were Peter Haskell, Hitoshi Moriyoshi, James Stormes and Joel Wolf. Hitoshi started his Ph.D. thesis at Brown and then continued at Penn State.

Unfortunately, golden ages do not last forever. Mathematicians at Brown began to receive outside offers. The administration of Brown did not take strong decisive action to preserve the math department. So it was time to move on. A number of universities were interested in me. I decided to take chance on Penn State.

**Haskell:**
*Talk about your time at Penn State.*

**Baum:** A remarkable person,
Richard Herman,
was head of the math department
at Penn State. He attracted excellent mathematicians to Penn State and
transformed the department into one of the leading USA math
departments. This was a spectacular achievement. When R. Herman became
head at Penn State, the department had a very low NRC ranking. In the
most recent NRC rankings, the Penn State math department ranked
seventh. In some ways, the rise of the Penn State math department was
even more astounding than what had happened at Brown. An important
difference was the much greater competence of the administration at
Penn State.

For myself, I found Penn State to be a really pleasant and
constructive place in which to live and work.
Nigel Higson
(one of the
first-rate mathematicians that Richard Herman had brought to Penn
State) organized a seminar that became my mathematical home. During my
time at Penn State, I continued to explore issues connected to the
Baum–Connes conjecture. Also — with
Anne-Marie Aubert,
Roger Plymen,
and
Maarten Solleveld — I began working on the representation theory
of
__\( p \)__-adic groups. With
Erik van Erp,
I worked on index theory for a
class of differential operators which are not elliptic.

Penn State promoted me from Professor to Distinguished Professor to Evan Pugh University Professor in Mathematics. I am grateful to Penn State for providing a positive environment in which I could develop the mathematics I was interested in.

**Haskell:**
*You were first known as an algebraic topologist. You did some work in
differential and algebraic geometry. Now you are primarily associated
with the operator algebra community. How did you come to follow such an
unusual professional trajectory?*

**Baum:**
Yes, I have followed an unusual trajectory. What happened was that one
thing led to another in a way that is not obvious.

One day in the autumn of 1963, at approximately noon, I decided to
leave my office in the Oxford Math Institute and do something about
lunch. As luck would have it, Michael Atiyah was also exiting the
building at that moment. We walked out together and proceeded together
for about 100 meters until we arrived at an intersection where we then
veered off in opposite directions. During the 100 meters, Atiyah
sketched out his functional analysis definition of __\( K \)__-homology. I
thought to myself (but did not say), “This is a lot of nonsense.
__\( K \)__-homology should be defined in terms of geometric cycles and
homologies of those cycles. I am going to find a geometric definition
of __\( K \)__-homology.”

During the next fifteen years, I would briefly visit Oxford every now
and then and describe to Atiyah my latest attempt at a geometric
definition of __\( K \)__-homology. His response was always the same. He would
listen for about two minutes and then say, “What you have done is
useless.” As time went by, however, I began to feel that I was
gradually homing in on a really good geometric definition of
__\( K \)__-homology. When I outlined one of my proposed definitions to Bob
MacPherson, he made a comment which led me to revise that
definition — and now I was certain that I had it. I returned to
Oxford. At the blackboard in Atiyah’s office, I presented what I
believed to be a correct and useful geometric model for __\( K \)__-homology. I
expected the usual “What you have done is useless.” Instead, an eerie
silence emanated from Atiyah. He broke the silence with “That is very
attractive and I want you to give a talk about that in my seminar
tomorrow.”

Once a useful geometric model for __\( K \)__-homology had been achieved, much then followed:

__\( K \)__-homology is the analog in topology of the Grothendieck group of coherent algebraic sheaves in algebraic geometry. This observation is the basis for the results on Riemann–Roch with Bill Fulton and Bob MacPherson.- The fact that the Dirac operator of a
Spin
__\( ^c \)__manifold determines an isomorphism between the geometric model of__\( K \)__-homology and__\( K \)__-homology defined via functional analysis led to the BC (Baum–Connes) conjecture. - V. Lafforgue
proved that BC is valid for reductive
__\( p \)__-adic groups. This indicated that there might be an interesting geometric structure in the smooth duals of such groups.__\begin{multline*} \text{Smooth dual of } G = \\ \{\text{Isomorphism classes of irreducible smooth representations of }G\}. \end{multline*}__The ABPS (Aubert–Baum–Plymen–Solleveld) conjecture asserts that there is such a geometric structure. This is relevant to the local Langlands conjecture. Specialists in the representation theory of__\( p \)__-adic groups at first dismissed ABPS as something that was “too good to be true.” ABPS has been proved for a very large class of reductive__\( p \)__-adic groups, including all the classical groups. - The isomorphism of geometric and functional analysis
__\( K \)__-homology turned out to be exactly what was needed (joint work with Erik van Erp) to solve the index problem for Fredholm operators in the Heisenberg calculus of contact manifolds. Many of these Fredholm operators are differential operators which are not elliptic. A number of excellent mathematicians had tried to solve this index problem. Apparently the heat equation method does not produce a solution.

**Haskell:** *I believe that you first met Alain Connes at the 1980 Kingston, Ontario,
operator algebras conference. What were your first impressions? Describe
the path that led to your collaboration on what is known as the
Baum–Connes conjecture. How have the formulation and status of the
conjecture evolved over the many years since it first appeared in a 1982
preprint?*

**Baum:** In the early summer of 1980 I drove to Stony Brook in my beat-up old
blue Dodge (my nickname for the car was “the blue wonder car”),
picked up Ron Douglas, and headed for Kingston. As Ron and I rolled
into Kingston, I said to Ron, “Except for you, I don’t know anybody at
this meeting. So I don’t have to go to any of the talks, and this is
going to be a great vacation for me.” However, I did go to many of the
lectures, including the talk by Ron. I had assumed that Ron would give
an exposition of what he and I had done on __\( K \)__-homology. Instead, his
talk centered on issues involving Clifford algebras. Thus it seemed to
me that it was up to me to give an exposition of our joint work. My
talk was scheduled for late in the afternoon of a day in the second
week of the meeting. I did not expect many people to attend the talk,
and very few did.

Alain (whom I did not know at the time) was one of the few attendees of my talk. That evening, after dinner, he lay in wait for me in the common area of the dormitory in which we were all staying. He came up to me and informed me that he and I (and his entourage) were going out to a nearby pub for drinks. The entourage consisted of Georges Skandalis, Pierre Julg, and Alain Valette. Over many glasses of beer, Alain began to explain his ideas to me and I struggled to understand. For the next few days, instead of going to talks, Alain and I would meet and I would try to understand what he was telling me.

My niece, Shona Baum, was returning from a trip to Europe and I had
promised my brother that I would meet her at the Boston airport. So I
left Kingston, met Shona, and then phoned Alain to tell him that I
would soon be back in Kingston. So, when I re-arrived in Kingston,
Alain continued to explain his beautiful idea that what Ron Douglas
and I had done was really just a beginning. A month later we continued
at IHES and I began to fully appreciate Alain’s point of view. He was
saying that given a __\( C^* \)__-algebra arising from geometry-topology,
there should be a formula for the __\( K \)__-theory of that __\( C^* \)__-algebra in
terms of the original geometry-topology which had produced the __\( C^* \)__-algebra.

During the next ten years, I sporadically visited Alain at IHES and we struggled to precisely formulate the conjecture now known as Baum–Connes. On my many visits to IHES, I usually would not go alone. My wife Barbara, some of our children, and sometimes my mother, would accompany me. A friendship developed between Alain and my mother. Despite the major age gap, there was an underlying rapport and understanding between them. There were many many good times during these years. Dinner parties given by Alain and his wife Dani were great fun. To begin, there was always champagne followed by a delicious dinner in the French style. For one of these occasions, my mother went to a hair stylist who dyed her hair a bright flaming red-orange. She made quite an entrance.

The Baum–Connes (BC) conjecture has a certain audacious quality to it. For locally compact groups and discrete groups, the conjecture had to be slightly reformulated once nonexact groups were proved to exist. This reformulation was achieved in joint work with Erik Guentner and Rufus Willett. So for locally compact groups and discrete groups no counterexample has been found. Granted the extreme generality of the conjecture and its many implications, it is certainly possible that eventually counterexamples will be discovered. If so, then the issue will be to determine the class of locally compact and discrete groups for which the conjecture is valid.

Various points of view are possible on BC. The one that I prefer is that BC is a generalization of the Atiyah–Singer index theorem. If the locally compact group under consideration is the trivial one-element group, then BC becomes Atiyah–Singer.

**Haskell:** *Why do* __\( K \)__*-theory and* __\( K \)__*-homology come up again and again in theorems that
connect different areas of mathematics?*

**Baum:** The combination __\( K \)__-homology-and-__\( K \)__-theory is simpler and more powerful
than the combination homology-and-cohomology of classical algebraic
topology. This is illustrated by
J. F. Adams’
solution to the vector
fields on spheres problem. Adams was getting partial results by
applying standard homology-and-cohomology. But his arguments were getting
more and more complicated as he was using secondary and tertiary
cohomology operations. Then Atiyah and Hirzebruch (guided by what
Grothendieck had done in algebraic geometry) brought __\( K \)__-theory into
topology. Adams applied __\( K \)__-theory to obtain his much-celebrated
solution of the vector field problem. In __\( K \)__-theory primary cohomology
operations sufficed. To understand the solar system, a model with the
earth at the center can be used — but everything gets more and more
complicated with cycles and epi-cycles and epi-epi-cycles, etc., etc. The
Copernican heliocentric model is simpler and more powerful.

__\( K \)__-homology-and-__\( K \)__-theory seems to capture in just the right way the
index-theoretic properties of many interesting mathematical and
physical problems. An example of this is the RR theorem of
Baum–Fulton–MacPherson. According to this theorem, a coherent
algebraic sheaf __\( \mathcal{F} \)__ on a complex projective algebraic variety
__\( X \)__ (__\( X \)__ may have singularities) determines an element in the
__\( K \)__-homology of the underlying locally compact Hausdorff space of __\( X \)__.
The index-theoretic properties of __\( \mathcal{F} \)__ are topologically
expressed by this __\( K \)__-homology element.

An extremely pleasant property of __\( K \)__-homology-and-__\( K \)__-theory is that it
extends very easily and naturally to noncommutative contexts. BC is
based on this. Within the noncommutative setting, G. Kasparov has
merged and unified __\( K \)__-homology and __\( K \)__-theory with his bivariant
__\( KK \)__-theory.

In graduate school we study homology-and-cohomology. Many mathematicians
are unaware that __\( K \)__-homology-and-__\( K \)__-theory is simpler and more
powerful. The foundation of __\( K \)__-homology-and-__\( K \)__-theory is Bott
periodicity. Thus Bott periodicity has emerged as central to much of
modern mathematics and theoretical physics.

**Haskell:**
*Both in one-on-one conversations and in talks to audiences, you have a
distinctive style of starting from the beginning and lingering on
definitions when other speakers would have hurried on to theorems. To
what extent is this style a communication tactic and to what extent does
it illustrate how you think about mathematics?*

**Baum:** It is primarily a communication tactic. To a limited extent, it is
also how I think about mathematics. In mathematics we try to make
precise definitions. Grothendieck believed that once really good
definitions have been formulated, proofs of theorems will then easily
and naturally follow. Raoul Bott used to say that if you have got it
right, mathematics is like being in rowboat and not rowing, “floating
down the river with the current.”

Unfortunately many math talks are only comprehensible to those specialists in the audience who already have a solid knowledge of the topics covered in the talk. This is a serious weakness in our subject. Mathematicians (especially young mathematicians) understand that they will be given little or no credit for good exposition. In my opinion, there is a legitimate place in mathematics for clear and well-organized exposition. It is worthwhile to note that some of the most eminent mathematicians write books and give talks in which the exposition is excellent. Over the years and decades, I have developed a point of view on index theory. This point of view has been elaborated in a series of papers with my coworker Erik van Erp [3], [2], [1]. I hope that Erik and I will write a book based on these papers. Of course, I also hope that in such a book the exposition will be OK.

**Haskell:** *Grothendieck’s influence on index theory seems pervasive. The
topological proof of the Atiyah–Singer index theorem is an example in
which the influence is explicitly acknowledged. The power of Gennadi
Kasparov’s* __\( KK \)__*-theory derives from using* __\( K \)__*-theoretic objects as
homomorphisms of* __\( K \)__*-theoretic groups is another example, which seems
further removed. Is it too simplistic to say that Grothendieck’s
contribution is the idea that (co-)homology theories are not just homes
for topological invariants but can play a direct role in geometric
constructions?*

**Baum:** Grothendieck’s influence on index theory stems from his Riemann–Roch
theorem (GRR). In order to state and prove GRR, Grothendieck
introduced and defined __\( K \)__-theory. So from
day one, __\( K \)__-theory has been
closely linked to index theory. I think that Grothendieck’s main
contributions to this area of mathematics are

- definition (within algebraic geometry) of
__\( K \)__-homology and__\( K \)__-theory; - functorial point of view.

Grothendieck aimed to give a proof of the Hirzebruch–Riemann–Roch theorem which would be functorial and very natural. He succeeded brilliantly. The analog for the Atiyah–Singer index theorem is much closer than was indicated in the “topological proof” of Atiyah and Singer. I have given several talks explaining this, and I hope to present it in a book.

The Kasparov __\( KK \)__-theory brings transversality into the
noncommutative setting. It is remarkable that such a geometric
concept (i.e., transversality) can be developed using functional
analysis.

**Haskell:** *In his* Celebratio *paper*,
Jonathan Rosenberg
*remarks that your
mathematical contributions show the influence of Alexander Grothendieck,
mediated perhaps by your work, with Bill Fulton and Bob MacPherson, on a
Riemann–Roch theorem for singular varieties. To what extent were you
conscious of Grothendieck’s influence? How did your Riemann–Roch work
influence your career trajectory?*

**Baum:** Bill Fulton and Bob MacPherson and I took Grothendieck’s proof of GRR
and revised it so as to make it even more natural and functorial. When
you do this, GRR becomes a theorem which applies to projective
varieties which may have singularities. Grothendieck was surely one of
the great mathematicians of the twentieth century. I am just one of a
vast number of mathematicians who were influenced by his inspiring
ideas.

**Haskell:** *You are almost certainly best known for the Baum–Connes conjecture. Why
has that conjecture captured the imagination of so many mathematicians?
Is it fair to say that the conjecture’s significance arises from the
questions about representation theory, rigidity statements in topology,
geometric group theory, and operator algebras that it draws attention
to, often by a process of using one area to cast light on another area?*

**Baum:**
As mentioned above, the Baum–Connes (BC) conjecture is unusual for its
breadth and for the many implications of its validity. When valid for
a locally compact or discrete group __\( G \)__ it implies some previously
stated conjectures: Novikov higher signature conjecture (topology of
manifolds), Gromov–Lawson–Rosenberg positive scalar curvature
conjecture (differential geometry), Kadison–Kaplansky idempotent
conjecture (__\( C^* \)__-algebras). In addition, BC suggests that for any Lie
group __\( G \)__ and for any reductive __\( p \)__-adic group __\( G \)__ there should be an
interesting geometric structure in the set of equivalence classes of
irreducible representations of __\( G \)__. So the conjecture is (within pure
mathematics) very interdisciplinary.

**Haskell:** *On the other hand, your conjecture has no applications outside of
mathematics (except perhaps to some very theoretical physics), has
little or no interaction with machine computation, and might be
dismissed by the majority of mathematicians as an example of the kind of
late-twentieth-century mathematics that, like the fine and liberal arts,
is of little significance in an increasingly practical age. How do you
respond to such assertions?*

**Baum:**
I respond by saying that we should not kill the goose who is laying
the golden eggs. Applications of science and mathematics are very
important. Some of these applications have given us technological
miracles and are truly wondrous. The obvious paradox is that in order
to have really good applications it is necessary to have a thriving
culture of pure science and mathematics. Einstein formulated his
theory of relativity in an effort to understand what is true about the
physical universe. For many years there were no practical applications
of Einstein’s theory. Now various engineering projects (e.g., digging
of the tunnel under the English Channel) do make relativistic
corrections. Number theory seemed to be the purest of the pure — a
subject that never ever would have any practical applications. Now
results from number theory are used in cryptography. The point is that
over and over again mathematics and science that at first
appear to
have no applications eventually
do have very useful and amazing
applications. A society that fails to vigorously support pure math and
science will, in the long run, lose out on applications.

There is, however, an even more basic issue. Applications are very important, but are not some all-important everything. What are the applications of Molière’s plays? What are the applications of Mozart’s musical compositions? What are the applications of Dylan Thomas’ poetry? Great art and music are life-enriching and should be valued and appreciated for what they are, and not in terms of some possible practical applications. The same is true for pure mathematics and science.