6 ms·

I'm a third year maths student at the moment and having a bit of an existential career choice crisis. I don't know if academia is for me, because I don't think

by hgibbs 12y ago

I'm a third year maths student at the moment and having a bit of an existential career choice crisis. I don't know if academia is for me, because I don't think it is worth it unless you are top tier, and having done an internship at a big bank neither does finance, mainly due to the stress. What kind of resources or subject areas do you think are valuable to study while I will have time? How lucrative is the industry in its current state and how good current employment opportunities are?

eru 12y ago
You can find a refuge at Google. You'll need to pick up some programming skills, but we have lots of math-y problems once you're here.
- hgibbs 12y ago
  I've basically done half a cs major already. My part time job is some stats programming work for a mining consultancy company and I was a strategist intern at Goldman over the summer so that is very good to hear!
  - graycat 12y ago
    You did well at GS! 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.
graycat 12y ago
For academics, for one possibility, look into B-schools: At the research universities, the B-school research requirement seems to be to publish nearly anything. Some of the journals popular in B-school publishing have not very high standards for quality. 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.
- hgibbs 12y ago
  Cheers, I really appreciate that you took the time to write this all out. I am glad that there seem to be so many options.
  - graycat 12y ago
    Here's an example of applied math in industry: 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.

by hgibbs 12y ago

eru 12y ago

You can find a refuge at Google. You'll need to pick up some programming skills, but we have lots of math-y problems once you're here.

hgibbs 12y ago
I've basically done half a cs major already. My part time job is some stats programming work for a mining consultancy company and I was a strategist intern at Goldman over the summer so that is very good to hear!
- graycat 12y ago
  You did well at GS! 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.

graycat 12y ago

For academics, for one possibility, look into B-schools: At the research universities, the B-school research requirement seems to be to publish nearly anything. Some of the journals popular in B-school publishing have not very high standards for quality. 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.

hgibbs 12y ago
Cheers, I really appreciate that you took the time to write this all out. I am glad that there seem to be so many options.
- graycat 12y ago
  Here's an example of applied math in industry: 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.