**Lecture delivered by Philip Emeagwali**

**The Toughest Problem in Calculus**

**TRANSCRIPT**

In 1989, it made the news headlines

that an “African Computer Wizard”

discovered

how to supercompute

at the fastest computer speeds.

It was rare because

one supercomputer

or the world’s fastest computer

costs more than the budget

of a small nation.

I used my supercomputer

to solve the toughest problem

in calculus.

At the granite core

of my mathematical grand challenge

was the system of non-linear

partial differential equations

of infinitesimal calculus

that was impossible to solve.

Those equations cannot be solved

when formulated

to help foresee global warming

or to recover more oil and gas.

By definition,

a system of partial differential equations

cannot be solved directly

on a computer.

The reason is that

the word “differential”

arose from the term “differentialis”

which translates to “taking apart”

or “taking differences.”

For the partial differential equations

of infinitesimal calculus,

such differences are infinitely small.

That is, they yield infinite calculations

that take forever

to completely compute.

A grand challenge problem

that takes foreover to solve

is impossible to solve.

A grand challenge equation

formulated at infinite points in calculus

cannot be solved on a computer,

unless it is reformulated

at finite points in algebra.

That reformulation is necessary

to make the impossible

possible.

If I had taken infinitesimally small differences

the forever impossible

will take forever

to solve, even across

a global network of computers

as large as planet Earth.

To make the impossible

possible,

I had to discretize my continuous

space and time and functions.

I had to use their finite differences.

Finally, I had to use

the finite difference approximations

that I invented

to approximate

the nine partial differential equations

that I invented,

as well as approximate

the other partial differential equations

and equations of state

that were invented

about a century earlier.

I used those finite difference equations

to formulate

my algebraic approximations,

which was the

largest system of algebraic equations

ever solved

on and across

computers.

The computer wizardry was in

consistently telling and retelling

the same story across boards.

I told the story of the motion

of fluids flowing below

or on

or above

the Earth.

I told that story from the storyboard

to the blackboard

to the motherboard

and continuing to the boardroom

to the classroom

to the newsroom

to the living room.

I translated calculus to algebra

to obtain algebraic approximations

that arose from

the finite difference analogue.

On my motherboard

was the analogue

of a partial differential equation

that originated on the blackboard

that was the codification

of a law of physics

that originated in my storyboard.

My Contributions to Calculus

The phrase

“partial differential equations”

was first used in 1845.

I, Philip Emeagwali,

first came across it

in June 1970 in Onitsha, Nigeria,

in my 568-page

blue hardbound textbook titled:

“An Introduction to the Infinitesimal Calculus.”

It was subtitled:

“With Applications to Mechanics

and Physics.”

That calculus book was written by

G.W. [George William] Caunt

and published by

Oxford University Press.

A decade earlier,

I began learning the times table

in January 1960

as a five-year-old

at Saint Patrick’s Primary School, Sapele,

in the British West African colony

of Nigeria.

The partial differential equation

of calculus

is not congenial to the fifth grader.

It takes ten years

for a five-year-old

to gain the mathematical maturity

needed to learn calculus.

Because partial differential equations

are the most advanced expressions

in calculus

it will take ten years of training

for that 15-year-old

research mathematician-in-training

to gain the mathematical maturity

needed to discretize a system of

coupled, nonlinear

partial differential equations.

That term “discretize”

is the mathematical lingo

for approximating

a differential equation

defined at infinite points

with corresponding algebraic equations

defined at finite points

that converges to it.

The partial differential equations

that describe the motions of fluids

must be formulated

from the laws of physics.

They must be formulated

from the storyboard

to the blackboard.

But the partial differential equations

used to foresee global warming

or to recover oil and gas

can only be formulated on the blackboard.

The partial differential equations

used to model global warming

can be formulated exactly on the blackboard.

They cannot be solved

on the blackboard.

As a black research mathematician

in the United States,

it was the toughest mathematical problem

that I ever solved.

My quest for its solution

reduced me to a lone wolf

computational mathematician

that discovered

as a consequence of my monastic interiority.

I was shackled for sixteen years

to two-to-power sixteen

computers.

Each of my 64 binary thousand computers was like a black box

in a dark room,

or a dark sixteen-dimensional universe.

I visualized my ensemble

as a primordial internet

in a sixteen dimensional universe

that were woven together

as one seamless, cohesive whole supercomputer.

I visualized

a one-to-one correspondence

between my 64 binary thousand computers

and the as many vertices

of a cube

that is tightly circumscribed

by a sphere

in a sixteen dimensional universe.

I discovered how to formulate

the partial differential equations

used to discover and recover oil and gas

exactly and correctly

on the blackboard.

They can only be solved

approximately

on one motherboard

which, in turn, earned it my description

as the toughest problem

in calculus.

It was the mathematical equivalent

of pushing the rock

up Mount Kilimanjaro.

In my dreams,

was the recurring theme

in which I visualized

solving primitive

systems of coupled, nonlinear

partial differential equations

that exploded

from 62-mile deep clouds

that enshrouded

a seven thousand

nine hundred and twenty-six (7,926) mile diameter globe

that was my mathematical metaphor

for planet Earth.

I discovered that

an initial-boundary value problem

in calculus,

defined as partial differential equations

with initial and boundary conditions,

can be solved accurately

across

a hyper-global network of

sixty-five thousand

five hundred and thirty-six (65,536)

motherboards.

I theorized that

those motherboards

must be uniformly and equidistantly

distributed

across the hypersurface

of a hyper-globe.

I discovered that

a system of coupled, nonlinear

partial differential equations

of a well-posed initial-boundary value

grand challenge problem

could be solved accurately

across

sixty-five thousand

five hundred and thirty-six (65,536)

motherboards.

I discovered how to solve them

as an equivalent

sixty-five thousand

five hundred and thirty-six (65,536)

challenging problems,

or sixty-five thousand

five hundred and thirty-six (65,536)

initial-boundary value problems.

They called me “Calculus”

because I began studying calculus

in June 1970

in Onitsha, Nigeria.

It took me twenty years

beyond the 568-page

blue hardbound book

“An Introduction to the

Infinitesimal Calculus”

to gain the mathematical maturity

that I needed to solve

an initial-boundary value problem.

I had to solve that calculus problem

by first theoretically formulating them

across

sixty-five thousand

five hundred and thirty-six (65,536)

blackboards.

Then, I experimentally solved

my sixty-five thousand

five hundred and thirty-six (65,536)

initial-boundary value problems

across

sixty-five thousand

five hundred and thirty-six (65,536)

motherboards.

My first ten years, or the 1970s,

was on formulating

partial differential equations

on the blackboard.

And my second ten years, or the 1980s,

was on solving

large systems of algebraic equations

that approximated

a system of coupled, non-linear

partial differential equations

on the motherboard.

First, I discovered

how to theorize

the computation-intensive

algebraic approximations

of a grand challenge

initial-boundary value problem

as

sixty-five thousand

five hundred and thirty-six (65,536)

challenging problems.

I theorized those problems

to have a one-to-one correspondence

to sixty-five thousand

five hundred and thirty-six (65,536)

blackboards.

Then, I discovered

how to experimentally

solve those sixty-five thousand

five hundred and thirty-six (65,536)

problems.

I discovered

how to solve them

across sixty-five thousand

five hundred and thirty-six (65,536)

motherboards.

I discovered

how to speedup 180 years,

or sixty-five thousand

five hundred and thirty-six (65,536) days,

of computation on only one computer.

I speeded it up

to just one day of super-computation

across a primordial internet

that is a hyper-global network of

sixty-five thousand

five hundred and thirty-six (65,536)

computers.

As a lone wolf

and the first programmer,

I had to be a jack-of-all-computer-sciences

as well as the primordial wizard

that programmed all those

sixty-five thousand

five hundred and thirty-six (65,536)

computers.

The most important partial differential equations

are those that encode

the motions of fluids,

as dependent variables.

My partial differential equations

are my sixteenth sense

of communicating with the spirit world

to foresee never before seen motions.

Oil, water, and gas

are fluids in motion.

To recover oil and gas

requires we set them in motion

from the water injection wells

to the oil and gas production wells.

Rivers, lakes, and oceans

are fluids in motion

across the surface of the Earth.

The air and the moisture

that enshroud the Earth

are 62-mile deep ocean of fluids

in circulatory motion

across a globe

that has a diameter of

seven thousand

nine hundred and twenty-six

(7,926) miles.

I began my journey

to the frontiers of the

partial differential equations

of calculus

and beyond the fastest computers.

I began that journey

in June 1970

in Christ the King College,

Onitsha, East-Central State, Nigeria.

At Christ the King College,

they called me “Calculus,”

not “Philip Emeagwali.”

I was called “Calculus”

because I was pre-occupied

with the book titled

“An Introduction to the Infinitesimal Calculus”

while Mr. Aniga, our math teacher,

was teaching algebra.

I first learned the expression

“partial differential equations”

from that calculus book.

I continued on March 23, 1974

from Onitsha, Nigeria

to Monmouth, Oregon,

in the Pacific Northwest Region

of the United States.

In the early 1970s,

I lived in the riverine village of Ndoni

in Biafra,

and in the cities of Onitsha, Ibuzor, and Asaba

in Nigeria.

In the mid-1970s,

I lived in the cities of Monmouth, Independence,

and Corvallis in Oregon, United States.

And I lived in the nation’s capital

of Washington, in the District of Columbia,

United States.

In the late 1970s

and in the United States,

I lived in the cities of

Baltimore, Silver Spring,

and College Park, in Maryland.

In those years and places,

I gained the mathematical and scientific maturity

that I used to theorize

global circulation modeling

across

my hyper-global network of

sixty-five thousand

five hundred and thirty-six (65,536)

computers

that, in turn, is a primordial internet.

In the late nineteen seventies (1970s)

and early nineteen eighties (1980s),

I learned supercomputer techniques

that I used to solve

a large system algebraic equations.

My algebraic equations approximated

a system of coupled, nonlinear

partial differential equations

that governs the flow of fluids

below the surface of the Earth,

on the surface of the Earth,

or above the surface of the Earth.

My partial differential equations

govern the motions

of air and moisture

in climate models

that, in turn, is used to foresee

global warming.

My partial differential equations

govern the motions

of oil and gas

in petroleum reservoir models

that, in turn, is used to discover

and recover more oil and gas

from production oil fields.

As a research supercomputer scientist,

of the 1970s and 80s,

my quest was the fastest computation.

I achieved that when I discovered

how to theoretically re-formulate

the differential equations

on the blackboard

as algebraic equations

on the blackboard.

But the equations on my blackboard

were destined for my motherboard.

Then, I had to experimentally solve

those algebraic equations

as an equivalent sequence

of floating-point arithmetical operations

on the motherboard.

What made the news headlines

a rarity in computational mathematics

was that I discovered

how to solve those ridiculously large

floating-point arithmetical operations

across

sixty-five thousand

five hundred and thirty-six (65,536)

motherboards.

It was a breakthrough because

all supercomputer textbooks

affirmed Amdahl’s Law

that decreed that

it will forever remain impossible

to program across eight motherboards.