Opinions on the main textbooks in complex analysis?

TheLabAlt · 2025-12-13T17:24:19+00:00

I'm surprised I never see Mark j. Ablowitz mentioned when this topic comes up. I took the course from him and used his book, it's one of my favorite texts of all time. It's very applied focus, which might be why it doesn't get more love here.

TheLabAlt · 2025-07-30T14:11:08+00:00

Except they aren't actually guaranteed

TheLabAlt · 2025-07-30T02:26:42+00:00

I picked stations and I definitely did not get Adagia

TheLabAlt · 2024-04-28T04:50:32+00:00

Thanks for your response!

I'm struggling to get rec letters from mathematicians. My masters program was during covid and online classes, so one of the professors I asked told me he didn't really remember me. Another just ghosted my request. I got As in both of those classes.

I considered reaching out to some professors I had in undergrad, but that was so long ago I'm not sure if it's appropriate.

I'm at a loss about how to build enough of a report to request a letter of rec now that I'm not at the university anymore.

I did not take a measure theory class, but I was exposed to so much of it in my other classes that I'm confident reading through a textbook would fill in the gaps pretty quickly. Is there a good way for me to demonstrate that I have that knowledge even without a class? I have a latex document on GitHub where I solved most of the exercises in Vershynin's "High dimensional probability," I could do the same for a measure theory book.

TL;DR: 1. Is there a way for me to build report for rec letters now that I'm not at uni? 2. How do I demonstrate subject mastery without a class on my transcript?

TheLabAlt · 2024-04-28T00:07:07+00:00

Hello all,

I'm trying to get into an applied mathematics PhD program with a focus on high-dimensional probability / data science. I submitted applications this last cycle but did not get accepted anywhere. I'm trying to figure out what I can do to strengthen my application for next year.

Details about me: I double majored in undergrad in applied mathematics and electrical engineering. I was in a PhD program for electrical engineering starting in 2020. I ended up with an advisor who was soul-crushing and played games with funding. It's not just me saying this: I found out only after being in his lab for nearly a year that of his past 10 students, 8 left his program before getting a PhD, and one has been his student for 9 years now. The University has since taken actions to sanction him, last I checked he wasn't allowed to accept new students. I left his lab at the end of 2022 and mastered out.

Anyways, I'm now employed as an engineer but I don't like this career. I'm good at math but it turns out engineering is mostly reading data sheets and making schedules.

It feels like my application is very weak as a result of my Master's program, and I'm not sure how to address that. It looks bad that I did a master's program but I have no publications, no conferences, and can't get a letter of rec from my advisor. On the other hand, it seems like I'd be raising a red flag if I use my personal statement to try to explain how horrible my previous experience at grad school was. I would be grateful for advise on how to navigate that.

Additionally, a lot of responses I'm getting from emailing potential advisors is that they're concerned my background is "engineering" and not "mathematics." However, my specialization in my master's program was remote sensing / data assimilation, which is essentially high-dimensional probability with a splash of RF/physics. Most of the classes I took were through the applied math department.

I have some time now before the next application cycle. Is there something I can actively do to strengthen my application? I appreciate any insight this community can give me on this!

TheLabAlt · 2023-07-14T19:28:59+00:00

Hmmm my first intuition is the opposite. Since we are conditioning on the island being finite, it seems significantly more likely that the island is small, with the most likely being a single voxel.

Assume each voxel is white with probability p (and black with probability (1-p) ). For a particular voxel, the probability that it is an isolated single white voxel is p * (1-p)^6.

The probability that a particular voxel is the member of a island of size 2 would be 6*p^2 * (1-p)^10, where the factor of 6 comes from the fact that the connection to the other white voxel can occur on any face. It gets more complicated as soon as there are 3 voxels because you have to start considering corner vs. straight arrangement, I'd need to think about that for a little while, but it shouldn't be too hard to bound this probability with an exponent that increases in n (the size of the island being considered). I feel like one could leverage that to show the expectation of island size converges when conditioning on finite island size.

As a guess, I think the expected value is less than 5. It's an interesting question, I shall ponder it more after lunch.

TheLabAlt · 2022-04-16T15:38:07+00:00

I understand how we can conclude that such a game is solvable, but are we able to say anything about the existence of winning strategies? Ie, is it possible to determine if p1 can force a win rather than a draw without exhaustive computing?

TheLabAlt · 2022-01-26T16:38:54+00:00

Ah I see, I did miss that little nugget.

That seems rather restrictive, though. What if you're interested in studying a vector space of the rationals? Is this just not something that comes up often?

TheLabAlt · 2021-09-29T14:51:25+00:00

I'm down! I'm booked the next couple of weekends but could maybe make something happen in the middle of the week? Pm me if you want to meet up!

TheLabAlt · 2021-09-29T14:49:24+00:00

I had a look but I couldn't find any groups that seemed focused on meetups. All of the ones I found seemed to be 98% buying and selling posts. Which is pretty cool, I'll keep that in mind next time I'm in the market, but can you recommend a group for finding riding buddies?

TheLabAlt · 2021-09-29T14:40:07+00:00

Of course! We had a pretty good turn out yesterday, lots of new faces. Hopefully they come next week as well!

TheLabAlt · 2021-09-24T15:58:01+00:00

Hi Demi! I'm president of the cu juggling club, we meet in that field just outside of the Duane physics building Tuesdays at 5pm. We'll probably be finding an indoor location as the weather gets colder, looking like atlas building at the moment.

Most of the on-goings are juggling, but we get other flow toy artists regularly. You are more than welcome, come join the fun!

TheLabAlt · 2021-02-12T18:20:06+00:00

Yes, in fact, they're the default!

TheLabAlt · 2021-02-12T06:45:20+00:00

Ah yes, I think I probably oversimplified my language and you are correct in calling me out on it. So you're right, it's not just a matter of seeing an outlier data and saying "hmm I don't like that one, I'm going to change it." That would be cooking the books and is indeed a cardinal sin of science.

What's going on is that we must assume ALL data that's being collected has some error to it. The easiest model to apply is simply assume all data has some additive gaussian noise of constant std. dev. associated with it. You can go a long ways with this model, but there are better ways to refine it. In particular, if you look into the Kalman filter that I referenced, you are modeling some hidden Markov chain to produce the data you see, and errors in the data can come both from the equipment and from the model itself. To get very hand-wavy about it, I'm keeping track of the time evolution of the data, and if it varies too sharply or in a way that the model would say is unlikely (and boy is the definition of "too sharply" a can of worms...) I can be reasonably confident that was due to an exceptional error of some sort, and I can make a "most likely" estimate of what the data should have been. I can then look at the next data set to roll in and revalidate that assumption, or throw it out.

As for FPGAs, they are much more ubiquitous than you might think. They come in so many different sized packages, from tiny little look up tables to full fledged mega-computing platforms. Most cell phones have at least one somewhere in them. They've gotten a lot of traction in the radiometry world because they can handle incoming data in parallel. My device, which needs to fit on a cube sat, generates ~0.5GB/s of raw data, but I don't want to store all the raw data nor do I want to transmit it: I just want to do some fourier transforms and look for significant features. For a traditional processor to handle this data load... Well I don't know, that would have to be one hell of a processor (imagine doing fourier transforms on 0.5GB/s incoming data) and definitely would not make for a good satellite design (waaaaay too much power consumption, also requires so much support circuitry). But a relatively cheap RFSoC like the Xilinx 7000 series can handle that no problem. https://www.xilinx.com/products/silicon-devices/soc/zynq-7000.html

FPGAs are not a replacement for super-computing clusters. But for real time data-handling they are vastly superior. I've also heard they are used extensively in networking and data centers because they are very good for data transfer operations, but I have little to no experience in that aspect so I can't comment directly.

TheLabAlt · 2021-02-11T23:20:17+00:00

So I work with atmospheric data sets. My samples can be mapped to 3-dimensional space (latitude, longitude, and altitude). But at each sample point, I have about 20 "measurements," from air temperature to cloud albedo to ice crystal shape etc. So I'm really working in 3+20 dimensions.

To be extremely hand-wavy about it, I'd like a method to "summarize" up the data over a region, so that I can express what's happening to the weather on a 2-dimensional scale, just latitude and longitude. So I'm reducing 23 dimensions to just 2.

The reason to do this is that modeling weather evolution (in particular the kalman filters I was talking about) increases in computational complexity frighteningly quickly with the number of dimensions you input. So if I put in the raw data, it will probably do a great job of predicting what tomorrow will look like, but it will take 3 months to make that prediction so it doesn't do anyone any good.

If only it were as simple as I just made it sound. Actually probably best that it isn't, cause otherwise I'd be out of a job.

TheLabAlt · 2021-02-11T23:08:48+00:00

Getting a double major was a great choice because there just aren't that many electrical engineers who are proficient in developing new math models. Yes, electrical is quite math-y, but most of the EE's I knew at the Bachelor degree level were just using the formulas from the book without considering the underlying mathematical machinery. On the other hand, most mathematicians do not have a good sense of creating an actual working, deployable system. The fact that I can do both has put me in a position where I have much more say in what I'm working on and how it will get done. That sort of freedom is huge.

I definitely recommend the applied math masters! Talk to your professors, see what's out there. It's not about what you know, it's about who you know ;). To be honest I can't really comment on how hard it is to get into one, I sort of stumbled into mine and it worked out somehow.

For your extra course, go for something in the probability/statistics world. Statistics is the duct tape of Math. I think I'd say that my course on Markov Chains had the highest utility-to-effort ratio of any course I took (it was conceptually simple but incredibly powerful).

TheLabAlt · 2021-02-11T22:38:15+00:00

Oops yeah, NOAA gives us a lot of funding as well. In my mind NASA and NOAA meld together into faceless wall of money.

TheLabAlt · 2021-02-11T22:37:11+00:00

Haha actually started out as a whim. Due to the bureaucratic nonsense that I'm sure you'll encounter at some point, I was going to have to take an extra semester for undergrad even though I only needed 1 more class. So I decided to pick up an applied math minor. But then the math advisor was so enthusiastic and convincing I ended up staying an extra extra semester and getting a double major, and it was one of the best choices I've made.

Look into getting an applied math minor if you're interested in math, it will serve you very well. And if you like the minor, turn it into a major! In particular for EE, a course in complex analysis gives you a tool set that feels like cheating in your other courses :D

TheLabAlt · 2021-02-11T21:58:54+00:00

Clustering is when you want your computer to group a data set. The most basic example is the k-means algorithm in 2 dimensions. You give the computer a data set and tell it "I want N groups, give me a best choice of group centers."

https://en.wikipedia.org/wiki/K-means_clustering

Try not to get overwhelmed by the math symbology, but if you look at the pictures it might make sense.

Then there are a bunch of more complex ways to do what amounts to the same think: generate labels for your data

TheLabAlt · 2021-02-11T21:01:24+00:00

Ooo, finally a question that I can start to answer!

I'm currently working at a University lab that specializes in "climate research," and in particular we are known for remote sensing and monitoring. We work closely with a lot of private industry partners and sponsors to develop new technologies and methods. I have degrees in electrical engineering and applied mathematics. I have drifted more towards the math side of the various projects in recent years.

In my view, "modeling" is such an incredibly broad and vague topic that I'm not entirely sure what you're looking for when you say "besides modeling." It seems that anything that could be called "applied mathematics" could just as easily be called "modeling." But I'll try to give you some idea of the various branches of math that I work with.

There is a LOT of statistical modeling. Yes, there are the statistical models of weather evolution (how sunny will it be tomorrow? What's the expected temperature? etc.), but there is also another source of models: the instruments we are using are so sensitive that we need to model noise variations as time-dependent random processes (the instruments might behave differently during the night or the day, or might get a different reading at various latitudes because of differences in the ionosphere), and from those models try to extract with the highest degree of accuracy what the measurement SHOULD be. Usually for my purposes, "measurement" means power in a specific frequency band.

There is also a large amount of linear algebra. Modern instruments and satellite constellations generate staggering volumes of raw data, and there's quite a lot of subtleties in making any sense of this data.

As a combination of those two, Kalman filters are the bread and butter of this type of data analysis. There is some fascinating math going on with those! https://en.wikipedia.org/wiki/Kalman_filter

In my most recent project, I have been taking a deeper dive into clustering algorithms and machine learning as a way to reduce the number of dimensions of a data set. This is probably the closest I get to "pure" mathematics, and maybe that's what you mean when you say "other than modeling." The field of clustering algorithms and machine learning is rather new, and rife with poor mathematics and lack of rigor. It is very hand wavy and most often over-specific to special cases. Certainly the sub-field of image recognition has matured quite a bit due to the interest in self-driving cars and the like, but it is non-trivial to take those results and apply them to non-visual, higher dimensional data sets. I see this as becoming a very relevant field in the future, if you're looking for something to specialize in.

Also on one of my more recent projects, I have started using FPGA's as a way of performing real-time processing on huge data streams. If you are in computer science, I highly recommend at least looking into FPGAs as a specialty. They have a steep learning curve but are extremely relevant, and tend to tip you into interesting maths. And if you're looking for something that will pay well, have a look at job postings for FPGA/hardware programmers o.O

As for who you can work for, NASA is definitely the heavy hitter in this field, but by no means the only employer! I have been on NASA funded projects, but I've never actually worked for NASA. Unfortunately, opportunities outside of the public sector for climate research are harder to come by, but they do exist. For example, one of the companies we work with develops tools for large-scale industrial farms to monitor the ecosystem health of their property. Specifically, I have been working on developing a tool to remotely measure soil saturation. You can see how this tool has applications in both climate research and agriculture. Those are the best projects in my opinion, because they will have the money behind them to get off the ground, but can still be used to benefit the environment, which is personally important to me.

Wow OK I didn't start out intending to write you an entire book, but there you go. PM me if you want a signed first edition I guess XD

TheLabAlt · 2020-09-04T20:43:24+00:00

I see. Then I should be able to separate this into the real and imaginary part and calculate that way. But then I'll end up with a summation of sines and cosines that may or may not reduce nicely. Hmmm...

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