In fact, I often feel pangs about not staying in academia. On the other hand, I've been able to carve out a life with a big dose of public intellectual -- for want of a better or less pretentious term -- so that's cool too. E.g., I'm repeatedly told that my blog is "required reading" for grad students in the relevant areas of computer science ...

I heard similar advice from people in many different lines of work including laws, finance, medicine, academia... usually they add it's not as interesting as it used to be.

It's a matter of compromise. I wish I could have the salary of banker, the freedom of a tenured professor and the prestige of a doctor.

Well, one disadvantage of academia is that you don't really get to choose where to live, even if your CV is quite good. Indeed, there are a number of academic couples that make crazy long-distance commuter marriages work because they weren't able to get jobs at the same institution (or in the same city).

There early in my career, once I sent some resumes and in two weeks went on seven interviews and got five offers.

Big topics were scientific engineering programming, numerical analysis, especially numerical linear algebra, standard applied statistics -- hypothesis testing, linear statistics, analysis of variance, curve fitting, etc. -- the fast Fourier transform, digital filtering, power spectral estimation, deterministic optimal control theory, linear and non-linear programming, discrete and continuous time stochastic processes, Monte-Carlo, numerical solutions of ordinary and partial differential equations, etc.

Some of these topics are part of mathematical EE.

At one time I my annual salary was six times what a new, high end Camaro cost. My wife got to pursue her Ph.D. in essentially mathematical sociology as part of her desire to save the world while we had big times at Thanksgiving and Christmas, shopped in Georgetown for high end French wine and cheese, were regulars with good tables at high end restaurants, went to lots of concerts and plays, got a good piano for her and a good violin for me, got boxes of fancy French pastries and sweet, sparkling dessert wines and on Saturday nights pigged out watching old movies, went to Shenandoah for vacations, had plenty of time for trips to families in Tennessee and Indiana, did some fancy cooking at home, etc.

Of course the money was coming almost entirely through the US Federal Government, mostly for US national security, e.g., via the US Navy, DARPA, etc. But, still, there was money via the DoE, NIST, and more.

I went ahead and got an applied math Ph.D. at Johns Hopkins, research on stochastic optimal control.

My startup has at its core some applied math I derived based on some advanced prerequisites I got mostly at Hopkins. The math is an advantage. If the startup works, then the math will have been the key -- except for the math and the basic business idea, it's all routine, that is, the math is the only thing really special.

Math can be used to say how to take data that is available and manipulate it to get data that is valuable. Now there is a lot of data and means to communicate, store, process, and display it, infrastructure software to make the rest much easier, ads to run to get revenue, etc.

Except possibly for coding theory and cryptography, I don't know if there are valuable applications of algebraic topology, algebraic geometry, algebraic number theory, commutative algebra, etc., but for the topics I mentioned it would seem that there is some practical value.

For some of what can be done with applications, there is a long dessert buffet from the Hahn-Banach theorem and more in D. Luenberger, Optimization by Vector Space Methods. One heck of a toolkit. Make just one good application and retire, make a donation to a university, and get your name on a building.

Here's another possibility: Attack integer linear programming problems in business. Focus on the problems that can attack with min cost flows on networks with integer arc capacities. Why? Because it is just linear programming; the simplex algorithm becomes really simple in that case; in practice the simplex algorithm is fast beyond belief on problems large beyond belief; and get integer programming for no extra effort -- if start with an integer basic feasible solution, then the simplex algorithm will maintain an integer solution and, if there is an optimal solution then will get an integer optimal solution. Can use this in various resource allocation problems. For more, use it as a linear approximation in nonlinear problems and for still more enhance it with some Lagrangian relaxation. Show that P = NP? Nope. Make money? Maybe.

The B-schools are basically forced by some standards efforts to teach some courses that are essentially applied math.

One might guess that the B-schools want to be professional or clinical but, instead, they tend to want to see themselves as applied social science, and social science has physics envy and wants to be mathematical. Their favorite math is linear statistics. So typically there's a lot of such statistics in the research at B-schools. And typically the B-school profs are not very good at math or statistics.

B-schools also want to dabble in computing -- if you can teach some courses in computing, even as electives, then you might get liked for that.

No doubt some B-schools would like someone who could teach courses in financial engineering, say, with Brownian motion, stochastic integration, Black-Scholes and generalizations -- basically the Brownian motion solution to the Dirichlet problem. So, if you like mathematical finance, then maybe teach it in a B-school.

DC may still be a good area for applied math: So, submit a Civil Service application and also send copies to the usual suspects -- NIST, various DoE labs, various DoD labs, various groups interested in economics or mathematical finance, that is, maybe the Federal Reserve (also try the regional Fed banks), the Social Security Administration, etc. If you want, try NSA, CIA, DIA, FBI, etc. -- no telling what they might want to do with some applied math. Also try the JHU/APL.

There have long been lots of Beltway Bandit shops because commonly Congress gives the departments of the Federal Government more money to spend than full time head count to spend it on so that a lot of the money and/or work goes through companies. E.g., Edward Snowden worked for Booz-Allen or some such.

For technical jobs around DC, it used to be that the job ads in The Washington Post were good places to look.

There is a chance that Google, Facebook, Yahoo and some others are interested in some projects that might be able to use some high end applied math. A problem may be that the other workers there are not very good at math and, really, make a mess out of the work, the opportunities, the positions, etc., and that can be bad for everyone.

It's not clear that everything in mathematical finance is a rat race. I doubt that James Simons ran a rat race shop.

Houston has some important slots for optimization, e.g., as in the work of Princeton Chem Eng prof Floudas. E.g., here's all the crude oil inputs we have available and the costs, and here's all the prices for all the possible refinery products. So, say what crude oil to buy and what products to sell to maximize earnings. That's an old problem, but likely people are still working on it.

The airlines have some very serious problems in fleet and crew scheduling, and first cut it's integer linear programming. But with uncertainties, if you want to handle them at all realistically, the formulations and solutions can become challenging in all respects. A better solution can show its value just on the computer and get taken seriously. For some airline spending $100 million a month just on jet fuel, saving a few percent can pay for some nice computing, work, etc.

Maybe look at what the operations research people are doing these days in scheduling for trucks, airplanes, cargo ships, etc.

No doubt Amazon, Wal-Mart, etc. have some good problems that are generalization of the old transportation problem -- the one Kantorovich got a Nobel prize for. Now we regard that problem as least cost flows on a network and use the simplex algorithm to solve it and where a basic feasible solution corresponds to a spanning tree of arcs on the network.

My view is that broadly now being handy with probability and probabilistic modeling of real situations is valuable, not that you can expect anyone else to know this so that, really, all you can sell are results, not the work.

E.g., with the Kolmogorov definition of a random variable, a function from a measurable space to the reals measurable with respect to the Borel subsets of the reals and a sigma algebra on the measurable space, go get a number, any number, and then regard it as the value of a random variable, that is, intuitively, one number among others that might have been observed. Now you know the first step in analyzing any given data -- call each number the value of a random variable. Then look for some assumptions, independence, orthogonality, or other relationships among the random variables. Keep the weak law of large numbers at your side -- that is, get some independent, identically distributed samples and take an average. There's more! Uh, the set of all real valued random variables X so that E[X^2] is finite forms a Hilbert space. The inner product is (X,Y) = E[XY]. So, yup, that completeness holds is a bit amazing, but it does and the proof is mostly just the Minkowski inequality from the inner product. Then in a Hilbert space we can do projections, and they are approximations, e.g., least squares.

Currently the Internet ad people and their ad targeting should be an opportunity to do some valuable applied math.

For what to study beyond the usual ugrad and Master's pure/applied math, I'd say the math sciences, the math of operations research or systems analysis or electronic engineering. So, linear algebra, numerical linear algebra, linear systems of various kinds, optimization (linear, integer linear, non-linear, network linear programming), etc.

My view is that soon computing will run aground due to running out of any additional utility from just their usual intuitive and heuristic approaches to solving practical problems and need math. But don't hold you breath.

Maybe you can have a good, long term career, say, long enough to have kids and get them through college, at some one organization. Maybe. Maybe for some DoD lab. Maybe.

Otherwise I have to suspect that in the end you will have to be a businessman where you think of the product/service, develop it, deliver it to your customers, and get the revenue. Then you get to use whatever math helps without trying to talk some non-mathematical manager into letting you try, and then you just deliver the results, not the math itself, to your customers.

In the short term, I'd suggest get a job for a salary, keep one foot in pure/applied math, keep the other foot in computing, try to make a career out of the job, but really expect to have to do, and, thus, to look for a way to do, a startup based on applied math and computing. There you just use the math as an advantage, maybe the crucial core -- still the rest that is more routine is also important.

For how "lucrative", essentially that's up to you! Broadly, can't really expect someone else to create a good job in applied math for you. Instead, you have to create the job you want and then take that job, do well with it, and build a career. You can do this starting a pizza shop; with math and computing, they should be advantages. E.g., with some abstract algebra applied to coding theory, A. Viterbi built Qualcomm.

If you know someone who knows someone, then maybe could get into some niche slot somewhere, maybe on the staff of a committee of Congress.

FedEx had candidate investor General Dynamics (GD), and GD had sent two of their guys, one aero engineer and one finance guy, to help FedEx. FedEx needed the GD investment.

At one point, the Board wanted some revenue projections. I wasn't asked to get involved and didn't want to, but no one had any ideas of how to do the projections except just draw a free hand line on a graph based on hopes, dreams, intentions, guesses, etc.

So, I thought: We know (1) the current revenue. We know (2) the revenue when the planned service to 90 US cities with 33 airplanes is full. So, the projections are roughly how interpolate between (1) and (2).

For how the interpolation will go, we have (3) current customers and (4) the rest of the customers we will have when we have (2) and are full.

Then say that the growth is from current customers (3) talking to the rest of the customers (4). That is, the rate of growth is directly proportional to the number of (happy) customers (3) talking to the rest of the customers we will have (4).

So, let t denote time in days with t = 0 corresponding to the present. Let the revenue at time t be y(t). Let the revenue from (2), being full, be b dollars a day. Then at time t, the growth rate is y'(t), that is, d/dt y(t). So, we have that

y'(t) = k y(t) ( b - y(t) )

How 'bout that! Maybe it's so simple it's almost just a joke, but the rest of the ideas floating around the office are much worse!

So, sure, it's an initial value problem for a first order, linear ordinary differential equation but, really, is so simple can just use freshman calculus directly and find the closed form solution, right, some exponentials I omit here.

So, on a Friday the Senior Vice President (SVP) for Planning and I got out some graph paper, picked a value for k, and drew the graph.

The next day, Saturday, I was in my office, and at noon I got a phone call asking if I knew about the revenue projections. Well, the SVP was traveling, and, yes, I knew. So, I was asked if I could come over to the HQ offices and explain. So, I got in my Camaro hot rod and drove over.

When I got there, people were unhappy. The two GD guys were standing in the hall with their bags packed.

I was led to the graph, given a time, and asked to reproduce the value on the graph at that time. So, I had brought my HP scientific calculator, punched the buttons, and got the value on the graph. I did that a few more times, and people started to be happy again.

It turned out that the Board meeting had been that Saturday morning; the graph had been presented; and the two GD guys had asked how it had been calculated.

Well, all the FedEx people there tried to figure out how the graph had been calculated and went on for a few hours without success, and then GD guys lost patience, got plane reservations back to Texas, went to their rented rooms and packed their bags, and as a last chance returned to the HQ offices to see if FedEx had an explanation for the graph. FedEx was about to die.

That's when I arrived and explained the graph.

The FedEx guys stayed, and FedEx was saved.

I never got thanked! Really my impression was that the rest of the top of FedEx had concluded that I'd been dangerous, maybe had 'hooked' the company.

And I'd never been invited to the Board meeting. So, I began to conclude that my contribution was so resented that the top of FedEx would rather see all of FedEx go under than let me help.

The promised FedEx stock was already 18 months late. I was in Memphis while my wife was in her Ph.D. program at Johns Hopkins near our home in Maryland. So, to heck with FedEx -- I went home and to Hopkins for my Ph.D.

So, that's an example of applied math in US business! E.g., I was the only one around who still knew freshman calculus! And, heck, I'd never even taken freshman calculus, had studied it on my own and started with sophomore calculus!

A broad lesson is, for anything very technical, have a tough time just making a contribution in the middle of a lot of operational and management people who forgot freshman calculus -- such people will feel out of control and resentful.

So, instead, for technical contributions, need an appropriate organizational structure, maybe be in a research division and out of the main line operations or management, maybe be a respected external vendor, maybe be in a profession complete with legal liability and licensing, etc. And charge a LOT.

Of course, broadly, the flip side of such technical incompetence, at the level of just freshman calculus, should be an opportunity!

There were several other cases where I did some good technical work on an important problem but resentment by others and competitive office politics kept me from getting credit or actually blocked my contribution. In the field of organizational behavior, such politics is called goal subordination, that is, someone finds it in their interest in the company to hurt the company, subordinate the good of the company to their own good.

Broadly, being a really good technical contributor among a lot of less technical people is a great way to create enemies who will sabotage your career, not friends who will help it.

So, net, usually have to make a technical contribution from outside the pecking order of the organization. That is, be a vendor.

Still better, just sell a product or service the customers will like but where the customers need not be aware of anything technical in the work.

I work for an extremely large defense contractor, and it never ceases to amaze me how many of my fellow engineers have forgotten the mathematics they were taught in high school! I know many of them took AP courses and learned calculus there, but they have completely and totally forgotten it.

That story about FedEx is hilarious!

For what companies to go to to get going on a career in such applied math, I can't think of any. Mostly people and their organizations would rather waste money than get involved with mathematicians doing work only mathematicians understand.

Again, if want to make math pay, just sell the results, usually as a vendor outside the customer.

If are inside the company, then likely no good work will go unpunished. Basically to be successful, you will need the strong personal support of the CEO and hopefully the Board -- literally. And the CEO will routinely have to go around and break 2 x 4s over the heads of recalcitrant middle managers to see if they are just asleep or really dead -- otherwise those middle managers can sabotage you with high determination and creativity.

E.g., twice I saved FedEx from going out of business and had much more that would have saved FedEx maybe tens of millions of dollars a year, but I got flack and push back and contempt and none of the promised stock.

Such are some of the challenges in getting stuff done. But successful people have been overcoming challenges, some much worse than I faced at FedEx, for many centuries.

Now that you know about the possibilities of some of the absurd challenges, don't be surprised. Really, one of the best rules is just to keep your good technical work essentially a secret until you have a solution that is just too darned good to refuse, and then pull back the curtain, let people see the results and also see that it's too late to fight with you.

Better still, just provide such results as a business and own the business. E.g., in some cases, just do the work for free, show the results, and then announce "You can have this for your real operations for a fee of only 15% of the savings."

Once there was an in-house research group that was successful. It was for WestVaco paper -- think specialty papers, e.g., coated papers for milk cartons. So, they had Ph.D. chem eng guys running their plants in jungles, etc., that is, where the trees were, but it was a family owned business with HQ in NYC.

The research shop was in Maryland, and I got an explanation from them: Each year Research went to NYC for their budget. Always the CEO, family guy, offered much more money for the Research shop than the shop was willing to accept (first-cut, good budget situation!).

Why? The rules: Research picks their own projects. Projects tend to run from a few months to a few years. One project in 10 gets to operations. When Research has a project ready for operations, they approach the relevant operational managers. In the first three years, savings are allocated half to Research and half to the operational unit. After three years, all the savings are allocated to the operational unit. With this accounting, Research was returning $3 to the company for each $1 in their budget. So, sure, HQ wanted to increase the budget, but Research didn't want to have to find ways to return $3 for each extra budget $1 so didn't want a bigger budget!

It is just crucial that the CEO is supporting such in-house innovation. The rules are also important.

Ah, while I was in grad school, to make enough money to support my wife and I while we both finished our Ph.D.s, I took a job at a DC area think tank working for the US Navy.

At one point our problem sponsor group wanted an analysis of a special, worrisome scenario of global nuclear war but limited to sea -- they wanted to know how long the US SSBN fleet could survive if it was just held in reserve and didn't shoot its missiles with nuke warheads. And they wanted the results in two weeks.

Gee, two weeks to model global nuclear war limited to sea! How generous! Good that they were not in a real hurry! Besides it was good timing since the day after the due date my wife had already gotten us reservations for a vacation at Shenandoah!

Well, there was a guy B. Koopman who in WWII had written a report OEG-56 on finding things at sea. So, given A and B wandering around at sea, when might they have an encounter? Well, Koopman took the area of the sea, the two velocities, and the detection radius and came up with an arrival rate for a Poisson process. Hmm ....

So, list the Red forces and Blue forces (the US SSBNs were part of the Blue forces), that is, list airplanes, helicopters, destroyers, battleships, attack submarines, etc., the speed of each, and the detection radius. Then have a table with a row for each Blue weapon type and a column for each Red weapon type and the detection radius and, if there was an encounter, the probability that one, the other, or both died.

Now tap lightly with an independence assumption (maybe not wildly unjustified), realize that a sum of independent Poisson processes is again a Poisson process with arrival rate the sum of the contributing arrival rates, and get for the state of the war a continuous time, discrete state space (very large from a combinatorial explosion) Markov process subordinated to a Poisson process. So, yes, there is a closed form solution as a matrix exponential for the transition probabilities from state to state, but that matrix exponential is absurdly large. Instead using Monte Carlo to run off, say, 500 sample paths was easy. I typed in the code.

Prof J. Keilson did a technical review. He said, "There's no way you can fathom that enormous state space.". I said, "At each point in time, say, 10 days into the war, the number of SSBNs remaining is a random variable. Moreover it is bounded so has finite expectation and finite variance. So, the law of large numbers applies. So, run off 500 independent, identically distributed samples, add them up, divide by 500, and get the expectation within a gnat's ass nearly all the time. Really the Monte Carlo puts the effort where the action is.". He agreed with me and passed my work.

The Navy got their results on time, and my wife got her vacation on time.

Later I was told that my work had been sold to a leading US intelligence agency. I could tell you which one, but then I'd have to ...!

I thought that the whole effort, which ignored so much maybe crucially important detail, was close to a joke, but, surprisingly, the output did look reasonable. Also, the two weeks was also so short it made the whole exercise next to absurd, another joke.

The group I was working for didn't trust me to get results on time so had another person working independently. Well, at the end of the two weeks, the other person had nothing at all. So, I'd won.

Office politics: One guy said "You write nice computer programs" and otherwise I got no thanks but just some jealousy, contempt, and silence. So, soon afterward when I'd gotten my Ph.D., I wanted to leave and did.

Again, no good deed need go unpunished.

Again, own the business, use math as an internal secret sauce and technological advantage, and sell just the results.

Again, thanks so much. One last question, what are you doing with your life right now? It seems like you have had a pretty broad career and I am interested in the end game. If you are retired, what is your lifestyle like and in particular how has your career impacted your life as a retiree.

Well, in part we aimed too high: My wife was brilliant, quite broadly -- yes, Valedictorian, Summa Cum Laude, Woodrow Wilson, PBK, piano, clarinet, voice, prizes in cooking, sewing, raising chickens. But her family had her try to be perfect and to dedicate herself to saving the world.

She wanted a Ph.D. in mathematical sociology to do social engineering of social change to save the world and, thus, get her praise, acceptance, and emotional and financial security. And those concerns filled her plate.

Well, at the roots there some anxiety, from nature and/or nurture, was involved, then some perfectionism and some fear of criticism from powerful, influential people -- net, call it a special case of social phobia. That brought stress which can bring depression. That slows the work and in her case caused more stress and more depression then clinical depression. She was in a clinical depression the day she got her Ph.D. She never recovered and, net, didn't make it.

Took me a while to reinvent and learn the basics of clinical psychology to understand what was going on and what to do about it. I did learn but nearly always too late.

In trying to have a stable job so that I could take care of her, I took a job at IBM's Watson lab, in what we called artificial intelligence (AI) to do monitoring and management of large server farms and networks.

Then IBM got sick and the lab phone book went from 4500 full time names down to 1000 or so. The guy who hired me, a big star who was deliberately ignored by the higher ups, left for greener pastures -- eventually ended up with a nice place in Malibu.

IBM Watson Research was run by a clique of people who stuck together and blocked out everyone else. At one point it appeared that the company HQ in Armonk tried to correct that situation.

Instead of or better than the AI, I had some ideas: Detecting a problem in a server farm or network is necessarily essentially a statistical hypothesis test with Type I (false alarm) and Type II (missed detection) errors. So, want hypothesis tests that keep down the rates of the errors.

Since there's a lot of data readily available and want to exploit it to keep down the rates of the errors, also want multi-variate inputs -- nearly all hypothesis tests have only univariate inputs. Also for such multi-variate data can't hope to know any of the probability distributions so need tests that are distribution-free.

I don't think there were any such tests, so I dreamed up some. I used some group theory, summed used the classic S. Ulam, guy on the left in

http://www-history.mcs.st-and.ac.uk/BigPictures/Ulam_Feynman...

result LeCam called tightness (see P. Billingsley, Convergence of Probability Measures), and was able to permit selecting false alarm rate in advance and then getting that rate exactly. Asking for all of the Neyman-Pearson result would take more data than we had any chance of having, but there was a somewhat useful sense in which my work gave asymptotically, for any selected false alarm rate, the highest possible detection rate with that false alarm rate.

I cooked up some synthetic data that was challenging -- the critical region was something like the red squares on a checkerboard in several dimensions. My work did fine.

I got some data from a cluster of computers at Allstate, wrote some prototype software, and confirmed the false alarm rate empirically. And I cooked up an algorithm to make the computations nicely fast (in part I used k-D trees -- reinvented those -- a few years earlier and k-D trees would have been mine).

Politics: A guy in the clique up there didn't like me. But it took two levels of management to fire someone, so he reorganized to put me under a wuss who would go along with firing me. They claimed my research was not publishable (reviewed by a guy in the clique who admitted he couldn't read my math but claimed to have found someone who could but found nothing wrong with my paper) and walked me out the door.

The next day the wuss was demoted out of management. Two weeks later the main nasty guy was moved down to have him under an additional level of management and given a six month performance plan which he failed. He was demoted out of management -- lost his corner office, budget, secretary, and 55 subordinates.

I got a PC and Knuth's TeX and submitted a paper on my research. Since the paper had some measure theory in it, e.g., Ulam's result, much of the computer science community couldn't read it. But the journal that offered to review the paper kept at it; apparently the editor in chief walked the paper around his campus, to a CS department to see if the problem was important and to a math department to see if the math was correct, and accepted the paper. He invited me to present at a conference he was running, but I didn't want to bother going. The paper was published. IBM was wrong.

So it appears that I have the world's only collection, and it's large, of statistical hypothesis tests that are both multi-variate and distribution-free, with some nice properties, with a fast algorithm, with some confirmation of the false alarm rate calculations from some real data, etc.

Asymptotically the critical region can be a multi-dimensional fractal. Nice.

People should my work. I did give a talk on the work at the main NASDAQ site in Trumbull, CT.

IBM didn't pay very well, and cost of living was high -- I'd always saved money, even in grad school, but I lost quite a lot of money working at IBM.

When I joined IBM, it'd just won the Nobel prize in physics two years in a row and had a long string of being "the most admired company in the world". Now I'd advise anyone just to stay the heck away, a long way away.

But being pushed out of IBM and age left me 100% permanently unemployable. I sent 1000+ resumes. Zip, zilch, zero.

If in computing, be sure by age 40, hopefully by age 35, to have a rock solid stable career and/or be wealthy. So, really about have to own your own business and make it successful.

For now, sure, I know some math and can stir up some more, and I know some computing and can learn more.

So, I've got a 1.8 GHz AMD single core processor, Windows XP Professional SP3 (I have an official copy of Windows 7 Professional on DVD but have seen no great reason to go to all the trouble to install it and rebuild all my software environment yet), three hard disks, 100 million files, .NET Framework 4.0, a good text editor (KEdit), Visual Basic .NET (comes with the .NET Framework), ASP.NET, ADO.NET, IIS (low level Web server), SQL Server Express, etc.

I thought, why not go after about 2/3rds of Internet search, the safe for work part served at best poorly by looking for keywords/phrases?

So, how to do that? Not with just routine software! And not with anything commonly talked about for search!

I stirred up some math, typed the theorems and proofs into Knuth's TeX, worked up a scalable architecture, and typed in the software.

Maybe I should have used Redis, but instead I thought that writing my own session state server would be faster than even understanding Redis. So, my session state server is single threaded ("Look, Ma, no concurrency problems"), for faster lookup rates is trivial to run as several instances as in sharding, and is just some TCP/IP sockets, some de/serialization of instances of my session state class, and uses two instances of a .NET collection class. Simple.

It's fast! A server for less than $1500 should be able to keep session state for an hour of inactivity for each user and do the session state work for sending 5000+ Web pages a second. Two standard racks should be able to handle session state for the world.

The rest of the software is also readily scalable also from just simple sharding.

Currently I have one bug in one Web page -- it's not handling session state just right! But I have the fix in code in another Web page and should copy it over today!

My interactive development environment is just KEdit with about 100 macros and some careful use of file system directories.

I'm using SQL Server only to record the data from the users and for the results of some batch computations; at one point I actually do make use of a transaction; the data for the searches is drawn from SQL Server with a batch program (run it maybe once a day); some solid state drives (write rarely, read thousands of times a second) should do wonders for the data for the searches.

I was about to fix the bug in the Web page but took some opportunities to gather some good initial data. The site will start focused and only slowly grow to be comprehensive -- right, at first do some things that don't scale and please some niche group of users a lot instead of trying to please 2+ billion users a little.

Currently the database has only some meaningless data I put in for first testing of the software. It's about time to load in some of the good initial data I have. Then give a critical review, go live, etc.

It's getting there.

All the work uniquely mine has been fast, fun, and easy, but the whole project has taken far, far, far too long. Why? I worked through about a cubic foot of books and 6000+ Web pages of documentation of Windows, .NET, Visual Basic .NET, ASP.NET, ADO.NET, SQL Server, etc.

The main problems: (1) Badly written, obscure documentation (worst bottleneck in the future of computing); (2) computer viruses; (3) SQL Server installation bugs destroying my boot partition requiring rebuilding starting with the XP DVD (barbed wire enema with an unanesthetized upper molar root canal procedure); (4) SQL Server management and administration (e.g., a week of throwing stuff against a wall to see what sticks just to get a SQL Server connection string that will let code for a server side Web page connect with SQL Server); (5) clean, smooth means of system backup and recovery (including for both user data and bootable partitions); (6) Sony DVD drives that quit for no good reason (and inability to buy more IDE DVD drives).

Good stuff: (1) KEdit and its macro language; (2) the scripting language Rexx (Microsoft's PowerShell may also be terrific but have yet to move to it); (3) NTFS (fantastic); (4) Visual Basic .NET design, functionality, speed of compilation, compiler error messages, minimal bugs (sweetheart language); (5) what ASP.NET does when it compiles a Visual Basic program (enough for a really nice IDE); (6) NTBACKUP (once understand how to use it, e.g., do have to ask to save "system state", whatever the heck that is, or the saved copy, restored, won't boot -- learn this and how to get around it the hard way, weeks of work); (7) XCOPY; (8) the tools to have server side Web page code write to a log file; (9) Firefox (except for virus vulnerabilities); (10) the classes in the .NET Framework (once learn how to learn about them and use them); (11) Adobe Acrobat (except for virus vulnerabilities); (12) the ability of XP to find device drivers and recognize new devices; (13) Microsoft's anti-virus Safety Scanner (if only from CP67/CMS and Multics, there should be no virus vulnerabilities, but since there are the safety scanner is terrific to have); (14) Knuth's TeX; (15) the Western Digital Passport Ultra 2 TB USB drive!

I'm not retired or retiring! Likely I'm still 100% unemployable at anything that would pay enough to let me keep a car going to commute to the job. Y Combinator and VCs want nothing to do with me.

But if I can get my Web site up to a search a second, on a server for about $2000, I will be in decent shape financially and on the way to organic growth for my business and much more.

Then I'll get a nice house, a building for some cars, at least one Corvette, visit the rest of the family still alive, take off two weeks to pig out on lobster in Maine, get some good grape juice from between Beaune and Dijon, do some cooking, give some dinner parties, go to concerts and operas, continue with the business, go to seminars on mathematical physics, etc.

I'll implement and deploy my server farm monitoring techniques and maybe spin it off as a separate business.

I have some guesses for some approaches to real AI and might try to implement those.

And I will get a kitty cat! We'll see!

Good luck on your work with math. Maybe what I typed in here will help; I wish I'd known all that when I started.

Hi graycat,

I'm an applied math & CS undergrad at Caltech and I love all of your posts, especially the ones about how data center scale computing really needs to embrace statistics. The FedEx stories are also excellent. I'm personally interested in high performance computing and machine learning, but I'm also interested in solving "real problems" and like how you seem to focus on the actual value of the applications. I love the feeling of engineering solutions to mathematical problems, and this seems to be something that you also enjoy.

I'd really like to hear more about your career, research, and also the startup that you're working on. I'd be very happy if you shoot me an email at eric@ericmart.in

Best, Eric

I really enjoy your posts and your writing style. Thanks much. I am currently working on some math problems as a part of a business that makes accessible to Indians who do not have access to structured banking services. Would love to discuss it with you, if you are inclined. My email is takenottie at google's mail. Thanks.

> Having started working mathematically recently it seems to me that even though I can propose a solution which isn't the best, they are usually better than what is currently available. Is that a common experience as a working mathematician.

Sure. For the politics, just report the savings. The stuff about the optimal solution, that is, its, say, moral or ethical value, is lost on business people. With some good common sense, the business people just want to know how much has been saved (or extra earned). If you can also have, say, a lower bound on the possible savings and get close to that, then you have in practice a stopping criterion. Or, what in practice is really difficult about the NP-hard problems is usually saving the very last possible tiny fraction of one penny and proving that did that. Instead, just concentrate on saving, say, the first $5 million a month on, say, the $100 million a month of jet fuel for an airline.

For a while in optimization, having the optimal solution was made a moral objective with anything that wasted even 0.0001 pennies regarded as a sin. Religious nonsense.

Really sloppy work is not good, but close approximation usually is plenty good.

For not the best, that's the other time I kept FedEx from going out of business: The founder, Fred W. Smith, spent the afternoon in his office trying to schedule the fleet, staggered out exhausted, said "We need a computer", and a college friend of mine heard that plea and called me.

Sure, digging in, I heard a lot about optimal solutions. But I knew that first just get something that helps, say, so can type in a schedule, have the software check it out for, say, planes being over loaded or having to fly too far, get the arrival times, etc., and print out all the details. So, I typed in 6000 lines of code to do that. Right, great circle distances are from just the law of cosines for spherical triangles!

Soon the Board wanted a schedule; biggie Board members doubted that the scheduling could even be done; biggie funding was being held up; the company was at risk.

So, I got a call from the same guy who called me to the Board meeting for the revenue projections, Roger Frock (he wrote a book about FedEx), and he and one short, easy evening he and I used my software to develop a schedule for the whole planned system of 90 cities and 33 airplanes. It printed out and looked nice.

At the next senior staff meeting, Fred's reaction to the printout was "Amazing document -- solved the most important problem facing Federal Express". Our same two guys from General Dynamics went over the schedule carefully and announced "It's a little tight in a few places but it's flyable". The Board was pleased; the funding was enabled. That was the first time I saved FedEx.

So, I got into how to do better scheduling. So, for one period from, say, 5 PM to midnight (again from 1 AM to, say, 8 AM) set up a table with one row for each city to be served, 90 cities, and one column for each candidate airplane tour from Memphis and back. So, may have some thousands of columns. Considering all fine details want to account for, and take expectations for random costs, find for each of the candidate tours the cost (first cut, assume that if a plane goes to a city, then it completely serves that city).

Then for the schedule, want to pick no more than 33 columns from the 1000s so that all 90 cities are served and total cost is minimized.

So, each column covers some cities, and want to pick columns that cover all 90 cities -- so have a set covering problem. Right, it's in NP-complete.

Well, that table is essentially the matrix in a 0-1 integer linear programming least cost optimization problem. There have one unknown variable for each of the columns, and the variable is 1 if use that column and 0 otherwise. If relax the 0-1 constraint, then have a fairly routine linear programming problem but will likely end up with some fractional airplanes allocated. Could use branch and bound to get to a good or optimal 0-1 solution.

Should have saved FedEx a huge bundle. Now, with software such as the IBM Optimization Subroutine Library (OSL), C-PLEX (now owned by IBM but from Robert Bixby at Rice in Dallas), etc., that would be a fun and likely fairly easy problem.

Alas, for proposing that work as my project, I got too much guff and flack back from too much of the top of FedEx, wasn't getting much thanks for the two times I'd saved the company, was not seeing the promised stock, was away from my home and wife, was in a company close to going belly up, saw a lot of dumb work, so went for my Ph.D.

At one time I wrote Fisher Black at GS and got a nice letter back from him, I still have, saying that he saw no opportunities for applied math on Wall Street!

I didn't hear about James Simons until much later.

Yep, this can be lucrative if you find a company with a genuine need for it. The company I work for has recently seized an opportunity to apply a mixed integer linear algorithm to a specific problem in the logistics industry, and I think it will soon become one of our most successful products.

Are you kidding me? This is huge; this is what prevented the financial crisis, as bad as it was, from turning into something much worse (it's "bad" for a bubble to pop, but it's worse for it to keep growing and pop later). This guy made the difference between billions of dollars going into houses that people couldn't afford and weren't worth the cost, and that same money going into productive investments. He probably contributed more to humanity than the rest of the list put together, with the possible exception of the other Goldman guy.

I know everyone thinks their own field is the most important, and I love academic maths, but goddam the snobbery towards the people who did something more directly useful with their talents is irritating here.

So, if you're a talented mathematician not working in that industry... it's reasonably likely you've made a conscious decision not to work there - perhaps because you don't agree that it's directly useful to society.

There's a reason why the financial industry has to pay the highest salaries. It's surely not out of generosity.

I kind of surprised how those things are frequently mixed together. The math is a science while various industry modelling is an application of the known/established tools/skills. It is like designing a drill vs. actually drilling a hole. (note: i have an MS in Math (3.8 GPA) from one of the top Russian schools)

They are one of few who can afford to pay higher, and give all kinds of perks. Everybody wants the best of breed to work for them, don't we? On the other side, while searching for a job, the lure of neat cashflow is a strong persuator for many.

Plus there is plenty of pretty clever people in banking crowd (maybe some of them don't use their talent to max or in a way they would find fulfilling, but that's another topic).

That said, I too dislike his tone about finance. Why? Why is it bad to work in one industry over another?

Baker? Sure, making cookies adds value to the world. Software? Yeah, writing programs adds something to the world. Holding onto money, lending some of it out, and taking a cut? Not so much.

That's slowly starting to happen in the financial industry, with robo-advisors and cheap index funds taking trillions of dollars out of the hands of overpaid Wall Street firms, but that process takes time. Hopefully we'll start to see the same thing with Lyft and Sidecar eating into Uber's nonsensical valuations.

When I pay him off, slowly, over time, and somewhat enrich him, nothing of value has been added to the world.

It's counterintuitive, maybe, but any sort of trade adds value to the world even though all that's happening is that goods and/or money are being moved from one place to another. What enables this is that different people have different preferences; in the case of your mortgage, it's differences in how you value future money relative to present money -- you want money now and having less money later is a cost you're willing to pay, whereas your mortgagee prefers to have more money later and doesn't mind having less now.

And so you make the trade -- in this case, you exchange a pile of money for some obligations to make future payments adding up to somewhat more money -- and both parties consider themselves better off than before. If that isn't adding value to the world, I don't know what is.

It's not a very glamorous kind of added value: it's less exciting than inventing a new kind of machine, or shaping raw materials into beautiful sculptures. But added value it is.

(Assuming, of course, that you and the mortgagee aren't badly wrong about your preferences. If it turns out that you're going to be spectacularly unable to repay the mortgage, so that you lose your home and the mortgagee loses the money, then maybe everyone loses. See, e.g., 2008. But on the whole it seems like the existence of mortgages is a considerable net benefit.)

Nothing new is created when ownership changes hands, as in the case of me possessing the money to buy a house, where before I didn't.

A change of ownership, or a new contract that two people have signed, or a promise to do something, none of that adds value to the world. The value is in the doing.

Andrew says: March 18, 2015 at 9:41 am

Nick:

You ask why I said “That’s too bad” after learning that Gregg is a vice president at Goldman Sachs.

Perhaps the best analogy is, what if you knew someone who was one of the top high school basketball prospects in the country, but instead of ending up in the NBA he made a career as a sports hustler, using trick bets to make money off suckers. This is not a perfect analogy, as this sort of sports betting is illegal, I guess, but the point is that it’s sad to see a great talent that is being used neither in its pure form or for scientific progress or the public good.

Hey, I hate the anti-bank sentiment as much as anybody, but you do realize that every asset they sold by Goldman required a buyer, right? So, it prevented nothing, just moved the risk onto someone other institution's balance sheet.

I wonder how their lives is in the financial industry. I imagine finance as a demanding and stressful environment, driven by big egos, commercial types. How do these math geniuses fit there? what level of freedom do they have in their job? in term of schedule, things they work on and so on...

The same is true of academia, from what I understand, just less money.

If anything I was surprised so many of them had done so well. I was on the UK squad for a few years and most of my friends from then seem to have ordinary jobs at companies you've never heard of (like me).

While the hedge fund landing zone is very typical of the math grifted grads, I expected a higher % in PHD/professor world then were listed.

If someone has a natural talent that they've developed into a very useful skill (mathematics, programming, selling vacuum cleaners, whatever) and they use that to get a cush 9 to 5 job, all the more power too them. Let them spend less time on politics, and more time on family, friends and hobbies. Who are we to say it was a bad decision to let someone else write the next great math paper?

If anything, this highlights that an aptitude for theoretical mathematics can be a launching point to a lot of different things, and that's refreshing. (I'd be less likely to encourage kids in math if Professorships were the only outcome)

There's a huge "Shit Happens" factor. One bad decision or major health problem, or just a loss of interest, and you're on a different path and it's hard as hell to fight your way back. Take that out 20 years, and you get the high fall-out rate. It's not that a lot of people fall off "the good track" for one reason; it's a bunch of reasons that affect small numbers of people.

Success is also hard to measure at an individual level. If you're 35 and making $130k as a full professor of mathematics, you're a star. If you're a consultant making $130k working specifically on what you want to work on, you're a success. If you're still a non-shot-caller making $130k at a Silicon Valley company, at 35, then you've failed. We don't really know how people are doing, though. The "web developer in Boston" might be able to pick his projects, or making a lot of money.

Unfortunately, I'm a guy with a very bad impostor sindrome, and a) whenever I see guys like him in worse/intellectually less challeging jobs than me I think I must be just lucky b) whenever I see guys like him I think I must be useless. :)

It's not clear to me that he would have contributed more to the world had he become some high-status academic.

Different goals lead to different outcomes, regardless of innate ability.

And I'm a big believer of luck and randomness. Since I've got a kid I wonder how do we even survive, life is so risky, starting from the birth itself.

Anyway, I think perspective gives you more happiness. At least it did for me.

Indeed. Like, for example, being born with a high IQ?

Would you mind explaining this a little bit more, please? I would imagine that significant or groundbreaking mathematical discoveries qualify as "great things" by someone's reckoning.

Most work in pure mathematics -- even outstandingly brilliant work done by outstandingly brilliant people -- is never useful outside pure mathematics. If you regard pure mathematics as very valuable in its own right, that's no problem. But if you care mostly about practical benefits, you might think that someone with the mental characteristics needed to win a Fields Medal would have done better to use that wonderful brain for something that benefits the world more.

(For my part, I'm glad that some people with wonderful brains use them to enrich pure mathematics, because I happen to love pure mathematics. But I think those more practically minded people have a point.)

Perhaps this is untrue on a timescale of decades or centuries.

The list is expansive if you care to look for it.

The question that's relevant here, though, is what fraction of pure mathematics is ever useful outside pure mathematics.

I think this would be difficult to measure. My guess: only a small fraction, but some of that turns out to be incredibly useful.

Myself, while I was not an Olympian in Math or Physics (the US fields some very competitive teams full of people who had better resources than me), I consider myself accomplished for what I had & the environment I grew in. I ultimately abandoned academia, in part because of the uncertainty, difficult work, and because my caring for friends won out - attempting to become wealthy enough to support friends in their endeavors, and giving them the resources I never had mattered more to me. I am perfectly happy being a lead developer, mentoring people to success & building products that help people's lives. I have no regrets.

It should also be mentioned that being someone who is supremely accomplished in competitions does not mean that the person is the smartest - there may be others smarter than him/her, and just did not have the fortune of growing up in an environment that could best foster that intelligence. Many of the smartest realize this, and don't hang their hats on any particular competition achievement.

Have you considered that he might be perfectly happy where he is as he can basically relax at his job as most of the tasks that come up , even the difficult ones must be a cakewalk for him.

Success has a very narrow definition these days.

Here's an interesting story : http://heuna.tumblr.com/post/2415073898/kim-ung-yong-a-child...

I can say exactly the same about me, sometimes feeling like the biggest idiot around, with serious brain damage (really, not joking). But then I realize that I managed when working 100% to increase my paycheck 20x compared to my first Java 100% job right after school, doing roughly same job (it's a long story). Many kids being brighter than me, they now have "meh" jobs at best. For success, initial talent is just one of many properties.

Anyway, what we should strive for is happiness, with oneself, with job, people and world around us. Rest are just details.

Why not very good? Because it looks like the students are given what, for the criterion, is poor direction and use of time and effort.

E.g., when I look at my education in math and its use in applications, most of the best of the education had nothing to do with anything like a contest, especially a contest in the early and mid teens. Instead, the best education was well selected, well presented material into the best work in math, e.g., via the best authors -- Birkhoff, Halmos, Rudin, Royden, Neveu, Tukey, Kelley, Suppes, Simmons, etc.

Or, my impression of encouraging kids in their early and mid teens to pursue math is to give them some enrichment material such as Pascal's triangle, some number theory, various puzzle problems, a lot of recreational math, etc. Why? Because mostly the people directing the efforts and selecting the materials are not very well educated in math.

In simple terms, to have kids do well on the criterion in the title is just to have the kids proceed along the main line -- get through the standard high school material in algebra and geometry, with some applications to high school chemistry and physics, and then move on to a fairly standard college major in math -- calculus, modern algebra, linear algebra, advanced calculus, ordinary differential equations, advanced calculus for applications, partial differential equations, point set topology, measure theory, functional analysis, probability, statistics, stochastic processes, etc.

Then that background promises to be good for a good answer 30 years later for the criterion "where are they now"?

Edit: Pascal's triangle??? Sorry but if you give some mopper Pascal's triangle they'll have already heard about it or invented it themself.

Good grief: Lots of people in middle and high school get pushed into math competitions such as the Math Olympiad but don't go on to be math majors in college and grad school.

My point is that the middle and high school Math Olympiad directions basically don't help people be good at math later in life or good at anything later in life. So, bingo, presto, people who were good at Math Olympiad show little or no good effect later in life.

But if want to study some math that has a chance of having a positive effect later in life, then follow what I outlined.

Net, sadly, Math Olympiad and other middle and high school math competitions are unpromising for doing any good later in life. Such middle and high school efforts would, could, and should be helpful but not by pursuing recreational math, etc.

Seems totally obvious to me -- sorry you don't agree.

Difference of opinion is what makes horse races.

> Good grief: Lots of people in middle and high school get pushed into math competitions such as the Math Olympiad but don't go on to be math majors in college and grad school.

This isn't true at all. Most people in high end math contests are just smart and do no preparation. They just accidentally scored high on the AMC. A lot do some practice for fun, because the contests are fun. Possible exceptions are a handful of people at Philips Exeter++ and maybe Thomas Jefferson High School, I'm not familiar with that dynamic. Contests like the ARML are oriented around making local friends more than anything. At the most local levels it's like, some high school teachers will corral their students into doing a contest at the nearby college one afternoon.

For example some friends of mine in the Philadelphia area thought it would be neat to enter the Harvard/MIT math contest (which is way more fun and difficult than the Temple School of Actuarial Sciences contest or Drexel's). It was the kids telling their parents they were doing this, and deciding to practice for it. I think the same happened with a group of students in Albany.

++and they were recruited after performing well on middle and elementary school math contests that parents are completely unaware of.

No, I did just the opposite: I said that others proposed such materials; I did not propose them but criticized such proposals. Again, for a high school student to be pushed into seeing lots of tricky things in Pascal's triangle, attacking number theory exercises, doing recreational math won't do the students much good in later life -- so as in the title of this thread, we won't expect to see many such students successful in math later in life. Instead, the efforts there in middle and high school will rarely come to anything.

Helping middle and high school students do better in math than what is in just the usual courses would, could, and should be doable and done, but the path is not recreational math, etc. but, instead, as I outlined, just proceeding with the main line of math education -- rush through the high school math of algebra and geometry, likely also trig, and then get a college calculus book and dig in.

E.g., the high school math teachers I had were eager to say that we "were not ready for calculus". Nonsense. It was the teachers who didn't know calculus.

I very much would have rushed ahead had I had some good guidance. Eventually I knew enough to rush ahead when I was a college freshman: I'd had four years of math, grades 9-12, at a relatively good high school but for my college freshman year went to a state school that was cheap and close enough for me to walk there and back. They wouldn't let me take calculus but pushed me into a course beneath what I'd already done in high school. So a girlfriend told me when the tests were, and I showed up for those. The teacher said I was the best math student he'd ever had -- in no very real sense was I his student!

I asked to be permitted to start calculus but was turned down. So I got a good book and dug in. I did well (in high school mostly I'd been teaching myself from the books and sleeping in class anyway). For my sophomore year I went to a good college with an excellent college math department, started on their sophomore calculus, from the same text used at Harvard, and did well.

I'm hot on this stuff about what middle and high schools do to good math students: My eighth grade arithmetic teacher gave me a D and took me aside one on one and fervently urged me never to take anymore math. In some significant ways, my father was good at education and just laughed.

That eighth grade teacher didn't have a clue about what math was: For the actual math, I'd done well in her class. What I did poorly on was multiplying two four digit numbers, and the reasons were (1) my handwriting was sloppy (common for boys), so sloppy I didn't get the columns lined up and then added wrong for the final answers, (2) my clerical ability was just awful (common for boys), (3) I understood the ideas right away but never worked on the clerical skills to get correct answers, (4) my parents never urged me to work to get right answers (my father had some high-end views of education and just wanted me to learn, especially out of interest, and didn't care at all about grades, and I did learn).

But in the ninth grade, the teacher saw right away that I was his best student, and he sent me to the state math tournament. In the summer after the 11th grade I was sent to an NSF math and physics program.

So, net, I was a good math student, was very interested in math, was eager to race ahead, but the school did a poor job helping me.

There is a little more to the story: When the SATs came back, I was called out of study hall to get my scores, from the same teacher I'd had in the sixth grade. Nice woman. Sweet. Ignorant! She, along with most of the rest of the teachers, thought I was a dunce. So, she read me the Verbal SAT score -- 538. She said "Very good". BS! It sucked! But she was sweet. She may have thought that I would get 250 or some such!

Then she looked at my Math SAT and hesitated. She looked worried. Confused. She said "There must be something wrong." Then she said the score was 752 and "That's uh, uh, very good.". Darned right it was good. But she was correct that there "was something wrong" and had been for 4, 6, 12 long, wasteful, painful years.

I had had no idea what the SATs were about and had made no effort to prepare. But I finished the Math SAT early, checked my answers, and still had 15 minutes. I'd very much like to know what the CEEB thought I'd missed. Maybe they scaled the scores and didn't give anything much higher.

My Math SAT was second out of a grad class of about 180, which is about right from the statistics, 2.5 sigma above the mean of 500. The #1 guy beat me by a few points. The best student in the grad class was a little behind me -- he went to MIT and burned out his fuses in his freshman year.

I was a good math student, had been eager to rush ahead in math, physics, and chemistry, but I'd had poor guidance. So, I'm hot on grades 6-12 continuing to give good math students bad advice. So, here I say, mostly, for the actual Math Olympiad competitions, waste of time. To do well with math, just go down the main road to a ugrad math major as I outlined. Simple. And to heck with, say, learning to play Nim in Courant and Robbins, What is Mathematics, about Pascal's triangle, etc.

Yes, I'd learned about Pascal's triangle. So when I was in grad school and had been pushed into a silly ugrad class in probability, the prof went all arrogant and emphasized that due to the factorials it can take some really long precision arithmetic to calculate even relatively ordinary binomial coefficients. BS: Just work in from one side of Pascal's triangle and never see a factorial. Indeed, I raised my hand and stated so in the lecture. I'd long since written software do to just that.

Next he did a sloppy job explaining the glory of the central limit theorem, and I dropped the course. But I was taking and grading in the Neveu measure theory level course at the same time, my first official course in probability, and did well. E.g., I was the only student who showed that there are no countably infinite sigma algebras.

The schools who want to send their good math students to Math Olympiads are likely also doing a poor job helping their students. Then, as in the title of this thread, in later life we mostly won't see much effect from such high school efforts.

Net, my hot button is that the schools in grades 6-12 don't know enough math to do well helping their students eager to race ahead in math.

> > Good grief: Lots of people in middle and high school get pushed into math competitions such as the Math Olympiad but don't go on to be math majors in college and grad school.

> This isn't true at all.

Of course it's true: E.g., you gave some examples of how high school teachers try to get some of their students into such competitions, i.e., "push" the students. That's lots of students, and in college math is not a very popular major. So, lots of middle and high school students in math competitions don't major in math in college, just as I claimed.

For more, you mentioned some activities at the college level; I was addressing the high school level. Sure, college students who do well on the Putnam are likely college math majors who are taking math very seriously. Again, I was addressing high school.

You are correct that the OP didn't directly say that the "training" of the "kids" was poor.

My view of such high school math competitions is that they waste the time and effort of the students and, otherwise, with good guidance, there would be significant results from the efforts 30 years later. In this sense, then the training was a "poor job".

But the OP did not say that directly -- that's my interpretation, heavily from what I saw used as efforts to have some high school students do better in math -- the efforts were a waste of time for all concerned and because the people directing the efforts didn't know enough math.

I didn't mention any activities at the college level. It's clear you have no idea what you're taking about.

Instead, you wrote:

> Harvard/MIT math contest (which is way more fun and difficult than the Temple School of Actuarial Sciences contest or Drexel's)

Sounds like college.

> It's clear you have no idea what you're taking about.

What's now fully clear is just that you are angry about something and want to attack me personally, not my writing or thinking but just me, personally. Not good.

To clarify: I think you don't know what kinds of problems they ask, what their contest format is like, what students learn and practice when preparing for them, how it affects students' interest in research mathematics and how it develops their abilities there. Why do I think that? Because instead of talking about the USAMO or ARML or the NSA's mail-in contest or anything specific about whether these contests actually harm or distract students' attention from what you think they should be doing, instead of rebutting the well-known "calculus trap" opinion or showing that you have any awareness of it, instead of showing knowledge of what students do to prepare for these contests, you've talked in abstractions, and there is no glue tying the words you are saying to objective reality.

Sorry not to be clear.

My view of efforts in grades 8-12 or so to have students do well in math competitions, e.g., the Math Olympiad, is that those efforts are not promising for helping those students do well with math or anything else later in life. Instead, mostly the efforts are just wasteful for all concerned.

The main reason for the waste is that the people, usually the high school math teachers, running the programs don't know enough math to give good direction to the students that might want to do especially well in math, well enough that 30 years later we would commonly see significant positive effects from that direction and the resulting efforts of the students.

So, the direction that is commonly given is to have the students study relatively useless material, say, recreational math, some puzzle problems, every tricky thing they can see in Pascal's triangle (the binomial coefficients, that is, the coefficients of (x + y)^n with one row for each value of positive integer n), and there are a lot of such tricky things.

But, right, the binomial coefficients have some nice properties, some of which are good to know, and an early volume of D. Knuth's The Art of Computer Programming has some good coverage. But, net, mostly to heck with the triangle and, if insist, then read the relevant sections of Knuth -- but even there Knuth has a good reason for presenting that material, and what's important is the applications Knuth makes, so, really, just read on in Knuth.

So, instead, students who do want to do better in math, and I did, should just proceed along the usual topics, texts, etc. for a ugrad math major. In particular, as a first goal, ASAP get a good college calculus text, get the basic prerequisites, and dig in.

By the way, the time I looked at the AP calculus materials I didn't like them and concluded that the people who wrote the materials didn't understand calculus very well. Instead, just get a good college calculus text.

For the rest I mentioned, about have to have a relatively applied ugrad math major to understand: So, would need to know some linear programming, network flows, optimization, integer programming, something about the question of P versus NP, and more.

For Lagrangian relaxation, that's a bit tricky and specialized and not commonly well known even among college math profs. I did give an outline in

https://news.ycombinator.com/item?id=8919311

but it's not really easy reading. But it's a cute topic with some a good intuitive views, a bit amazing in some ways, and at times shockingly powerful.

E.g., the guy who first did the many worlds interpretation of quantum mechanics, a Ph.D. physics student of J. Wheeler at Princeton, an expert in relativity theory, was H. Everett, and at one time he had a nice applied math shop near DC doing resource allocation for the US DoD using a first, simple version of Lagrangian relaxation. What I outlined in

https://news.ycombinator.com/item?id=8919311

is more general and powerful and makes some cute use of some convexity -- that there is convexity there is also surprising but true and powerful and, with the right setup, easy to prove.

When I view my posts at HN, my browser flows the lines ignoring my end of line indications except does honor blank lines.

I thought that such text flowing was arranged by some internal HN server logic.

I don't know who Dan Scales is but if you can't google someone nowadays then they might just work for the government ;-)