I'm not sure why you were downvoted. Most of the academics I know hate it. They use it because certain journals require it, or their advisor makes them use it.
I use TeX. LaTeX also works, but the books are longer and less well written than Knuth's original TeXBook! :-)!
I love TeX -- it's one of my favorite and most important tools.
I have a Ph.D. in applied math, and IMHO TeX (or LaTeX) is just essential, call that more than ESSENTIAL for my work.
E.g., now I'm a "solo founder" of a startup, a Web site. The crucial core of the work is some original applied math I derived. So, yup, i typed it into TeX. So, as I wrote the corresponding software, I referred back to the TeX output of my core math -- worked great!
Without the math, the software would be impossible; one would just look at the screen and wonder what the heck to type. With the math, the software was just routine, essentially just trivial.
For typing material with a lot of math, I see no reasonable alternative to TeX or LaTeX.
I wrote my Ph.D. dissertation with word processing (thankfully!) but without TeX. What a pain. I could have included more math in the dissertation if I'd had TeX to do the word whacking. More generally, at one point in my career, I could easily have written and published a lot of original and tutorial applied math but didn't because of the difficulty of the math word whacking before TeX.
The last paper I published, some a bit wild mathematical statistics, was a good test for TeX -- some of my mathematical expressions in subscripts were a bit much, but TeX worked flawlessly!
If anyone is typing a lot of mathematical material and objects to TeX, then just encourage them to do the typing without TeX and see if they like that world better!
Computing is changing the world in major ways, some just astounding and/or astoundingly good; math is helping, a lot now and will more in the future; and TeX is just crucial for getting the math word whacking done. But Knuth knew that and did a great job.
So far, for the near and distant future of computing, TeX is one of the stronger pillars of civilization.
If it doesn't compromise your work, can you speak more of the path you took from a Ph.D. to startups/tech, and how your research allowed you to go down that path?
I'm a Ph.D. student in applied math as well, currently.
I tried a grad math department and didn't
like it: (1) In a course in real
analysis, early on the prof discussed some
set theory. The summer before I'd had an
NSF thing in axiomatic set theory, Suppes,
von Neumann, an appendix in Kelley, etc.
His first test had a problem, and at the
last minute I saw a solution and wrote it
down. He called me on the carpet -- nasty
guy. I apologized for using little omega
for its usual meaning without defining it,
and then he saw that my solution was
better than his and I was off the
carpet. Bummer. He was too quick to
cut me off at the knees. (2) Course was
in Kelley, General Topology. As a ugrad
senior, I'd lectured a prof once a week
and covered all of it except the last
chapter on compactness when I cut out to
finish my honors paper [The typing was so
hard that from rolling the carriage a half
step my left arm hurt for a year!] But the
course in grad school, same book, was
beneath me. I turned in a stack of solved
exercises and was a nice guy -- I didn't
submit any I'd done in ugrad. Waste of
time. (3) There was an abstract algebra
course from Herstein's book -- by then
nearly all beneath me. E.g., my ugrad
honors paper had been on group
representation theory which is heavy
linear algebra and abstract algebra stuff.
I solved some exercise in ring theory and
got sent to a full prof. The only thing
new in the course for me was Galois
theory, so I studied that some weekend and
took an oral exam for the course. Waste
of time.
I wanted the math for math-physics but
didn't see how to get that there.
Certainly not Galois theory. There were
some good ways but not with the courses
they put me in. The specs for the q-exams
were a disaster -- the faculty committee
had a political train wreck. Bummer.
I got recruited by the NBS&T in DC.
Getting to DC then was the land of milk
and honey for applied math. I got
married, and she went for her Ph.D.
We had a great time, good French cheese,
some quite good French wine, lots of
plays, concerts, trips to Shenandoah, etc.
I got into descriptive statistics,
multi-variate statistics, statistical
hypothesis testing, numerical linear
algebra, curve fitting, the fast Fourier
transform, second order stationary
stochastic processes, extrapolation, and
power spectral estimation, optimization,
the Navier-Stokes equations, did a lot of
catch up reading in the basics, a lot more
in linear algebra, multi-variate calculus,
e.g., exterior algebra, and more. Kept
busy. Had a great time. Also got into
computing in a fairly big way. Got some
nice items, e.g., two new cars, etc.
My favorite book on my bookshelf,
including for applied math, is J. Neveu,
Mathematical Foundations of the Calculus
of Probability.
Worked in industry and saw some problems
in combinatorial optimization,
deterministic optimal control, and
stochastic optimal control, identified a
problem in stochastic optimal control and
found an intuitive solution, applied to
grad school in applied math. Got into
Cornell, Brown, Princeton, and more.
Independently in my first summer did the
research for my dissertation in stochastic
optimal control. Had lots of delays
having to do with my wife and, then, our
budgeting. In a rush, wrote some
corresponding software in two months, much
of it over Xmas at wife's family farm, and
wrote and typed in the final dissertation
in six weeks, stood for orals, and got my
Ph.D.
During Ph.D., did work in military systems
analysis, some optimization, statistics,
and Monte Carlo -- wrote the corresponding
software.
The day my wife got her Ph.D. she was in a
clinical depression from the stress. To
help her get better, I took a job I didn't
want as a B-school prof in applied math
(also played a leadership role in campus
computing and did some consulting) but was
near her home family farm that I hoped
would help her. It didn't. I took a job
in AI at IBM's Watson lab and did some
optimization, mathematical statistics, and
AI. My wife never recovered from her
illness and died.
Then I became an entrepreneur.
I did some interesting work in two cases
of optimization; thus I found good
solutions to the customers' problems that
they believed could not be solved; that I
solved the problems scared them off. One
solution turned out to be just linear
programming on networks -- I was coding up
the W. Cunningham variation when the
customer ran away. The other problem was
just some Lagrangian relaxation; I got a
feasible solution within 0.025% of
optimality in 500 primal-dual iterations
in 900 seconds on a slow PC to a problem
in 0-1 integer linear programming with
40,000 constraints and 600,000 variables
-- scared the pants off the two top people
in the customer's company. They had tried
simulated annealing, failed, and concluded
that no one could solve their problem;
that I found a good solution, both the
math and the software, scared them off.
I looked into lots of stuff that didn't
work out.
Lesson: US national security, especially
around DC, was, maybe still is, really
eager for a lot in applied math --
optimization, stochastic processes, etc.
In wildly strong contrast, I've seen no
interest in business at all comparable,
not even in Silicon Valley. The US DoD is
often quite good at exploiting applied
math; in comparison,
business, in a word, sucks. The
flip side of that suckage is, in some
cases, an opportunity.
Lesson: Business is still organized like
a Henry Ford factory where the supervisor
knows more and the subordinates are there
to add routine muscle to the thinking of
the supervisor. Sooooo, US business just
HATES anyone who knows more than any of
the supervisors about anything relevant to
the business, and one can about count all
the good cases of applied mathematics in
business without taking shoes off.
Business CAN make good use of specialized
expertise and does with lawyers, licensed
engineers, and medical doctors. Each of
these, however, is usually outside the
usual organization chart pecking order, is
often from an outside service, in a
research division, in a staff slot off the
C-suite, etc. Each of these has a
profession that is crucial; applied math
doesn't. Bummer.
In business, an applied mathematician who
shows the company how to save 15% of the
operating costs is a lose-lose to the
C-suite: If the project flops, then it
was a waste, and anyone in the C-suite who
signed off on the budget has a black mark.
If the project is successful, everyone in
the C-suite feels that their job is at
risk from the guy who did the good
project. So, the C-suite sees any such
project as a lose-lose situation.
Nearly no one in US business got promoted
for doing an applied math project
successfully or got fired for not trying
an applied math project.
So, sure, to make money with applied math,
go into business, your own business, as
your own CEO, and own the business.
Now some of the opportunities are closely
related to the Internet -- take in data,
manipulate the data with some applied
math, maybe somewhat original and novel,
spit out valuable results. Then
monetize the results whatever way looks
best. Use the math as a crucial, core,
powerful, technological advantage, secret
sauce. Don't expect the customers/users
to see anything about the math -- just get
them results they will like a lot. Do the
other usual things when can -- viral
growth, network effects, lock in, good
publicity, own data, etc.
My software now is 100,000 lines of typing
with about 25,000 programming language
statements and the rest voluminous
comments. About 80,000 of the 100,000 are
for on-line, and the rest are for
off-line, occasional batch runs for some
of the data manipulations. There is some
light usage of SQL Server.
I am basing on Windows and the .NET
Framework. For the Web site, that is
Microsoft's IIS (Internet Information
Server -- handles the TCP/IP Web site
communications and much more leaving a
nice environment for my code for the
actual Web pages) and ASP.NET for the Web
page controls (single line text boxes,
links, check boxes, radio buttons, etc.).
My Web pages and my code for the pages is
just dirt simple -- Microsoft writes a
little JavaScript for me, and I have yet
to write a single line of it. There's no
use of Ajax, no pull downs, pop ups, roll
overs, overlays, icons, etc. The Web site
is also dirt simple, a seven year old who
knows no English and has only a cheap
smart phone dirt simple.
I wrote a little C code; am using some
open source C code, and otherwise wrote
all the code in Visual Basic .NET -- seems
fine to me. The important stuff is the
math I derived; given the math, the code
is routine, and Visual Basic .NET is well
up to the work. Since I don't like C
syntax, I don't like the syntax of C#.
Maybe someday I will convert to C#, but in
an important sense Visual Basic .NET to C#
is just converting to a different flavor
of syntactic sugar -- indeed, IIRC there
is a translator.
So, I'm an entrepreneur working to sell
the results of some math I derived.
So, to do such a thing, think of a problem
and a solution, write the code, sell the
results. Of course, problem selection is
a biggie. And want a problem that can
solve and with own applied math with a
better solution than available otherwise;
want the software not too much to write;
want the computing resources within what
is reasonable now or soon (possibly
considering the cloud); want the results
to be a must have instead of just a
nice to have for enough users/customers
times money per each to make some big
bucks.
If you are a solo founder, then you get to
avoid a common cause of failure -- founder
disputes. As a founder, you SHOULD
understand all the work, so if you are a
solo founder you will!
Don't expect any equity funders to write
you a check until you have revenue
significant and growing rapidly. Thus, as
a solo founder with revenue significant
and growing rapidly you won't accept their
check. No one in equity funding has yet
seen even 10 cents from applied math
research; you won't get back even laughs.
Ph.D. academics is really good at work
that is "new, correct, and significant".
Business just HATES anything really new or
significant and has no idea how to check
"correct". E.g., the information
technology VCs keep looking for
simplistic, empirical patterns and have no
idea how to evaluate anything new.
Really, their looking for such patterns is
likely also just a publicity scam;
instead, they want to invest money in a
business where accountants working for
their limited partners, who, if that is
possible, know even less about math, can
say that they made a good investment. In
an analogy, they want to buy a ticket on a
plane that has already reached nice
altitude and is climbing quickly. Maybe
the startup will take their money if there
are five founders, all exhausted, all with
all credit cards maxed out, and each with
a pregnant wife.
For a good applied mathematician -- with some
original, powerful, valuable work, good at
software, with a business with significant
revenue growing rapidly -- to report to a BoD
of business people, essentially anyone
else in business, is a bummer. E.g., at
an early BoD meeting you will outline an
applied math project for some nice
progress in the business, and about the
time you get to sufficient statistics, an
ergodic assumption, completeness of
Hilbert space, the polar decomposition,
something in NP-complete, or a martingale,
the board members will soil their clothes,
leave a smelly trail to the rest room,
and then run screaming from the building.
They will meet off-site, fire you, put the
business up for sale for the cash in the
bank, and be glad you are gone. Bummer.
So, go into business for yourself. Or,
don't expect anyone in business, who no
doubt knows next to nothing about math,
doesn't even remember sin' = cos, to
create a job suitable for your talents,
training, and business value in applied
math.
Heck, at one time in business, I saved a
major company just by formulating and
solving
y'(t) = k y(t) (b - y(t))
The BoD was thrilled, but I scared the
socks off the C-suite.
That was the third time. The first time I
wrote some software that pleased the BoD
and saved the company. The second time I
beat everyone in the C-suite at Nim -- I'd
read the algorithm in Courant and Robbins.
Scared the socks off the C-suite.
Applied math in business is a wide open
field -- nearly no one there now. So, you
will be alone. You can trust the solid
math you know, the solid, new proofs you
write, and a lot in software, but no one
will do anything but laugh until you have
the big bucks in the bank; then, since you
did something valuable they don't
understand and know they could not have
done, they will fear you and hate you;
they will all agree and may gang up on
you; they may attack you. The laughing
is not nearly new: Just read the Mother
Goose "The Little Red Hen"; that's still
the case.
There's a lot of good, foundational
applied math code out there for
optimization, statistics, etc. you might
be able to exploit. In computing,
operating systems, .NET etc., SQL etc. are
astounding and from free to usually quite
cheap.
Nearly no one in business can identify,
formulate, and solve even a problem that
is basically just linear programming --
the competence in applied math in US
business is, well, they forgot plane
geometry. To expect them to derive some
simple Lagrangian relaxation is asking for
hen's teeth.
As an applied mathematician in business,
you will be essentially alone out there,
in the nearly empty intersection of math
and business. If you are successful,
then you will necessarily also be
exceptional, and necessarily nearly
everyone who is exceptional is alone.
My first server is an AMD FX-8350, 64 bit
addressing, 8 cores, 4.0 GHz processor
clock, 32 GB ECC main memory, etc. That's
a lot of computing power for the money.
Fill that up doing something valuable, buy
20 more, fill those up, and sell out for
$1 billion or so. It's a heck of an
opportunity.
Thank you so very much, graycat. Incredibly helpful, and I appreciate the time you put into the discussion. I feel as if I need to read your post 3-4 times to absorb it all.
I am working in numerical methods for PDE so some of this was far afield but it does make sense that those are the areas ripe for opportunity.
