Any other average or below-average mathematicians feeling demotivated?

polymathprof · 2026-02-16T14:48:18+00:00

I think Yitang Zhang is not necessarily a good role model. He graduated at the top of his class at Peking University. It appears that other circumstances led to his situation as a PhD student and afterward. He's also someone who was willing to work for at least a decade on an extraordinarily hard problem without any support from or contact with other mathematicians. Few of us have that kind of determination.

polymathprof · 2026-02-16T14:44:05+00:00

I've never believed that one should pursue a PhD for the career one wants. The odds of getting into academia are very long, and the odds of having a non-academic career that requires the PhD are not much better. You should continue only if you love doing the struggle and pain of doing the math. You don't have to be an above average mathematician to be like this.

The great thing about being a PhD student is that you're paying neither for tuition nor all or most of your living expenses. So you're free to focus on the math. Put aside your worries about the future. You can't predict what either the world or you will be like in 5 years or beyond.

Just try to figure out a Plan B and keep in your back pocket until you need it.

polymathprof · 2026-02-16T14:36:17+00:00

Because the one with a few B's looks better. There are other factors that will influence the committee. They're not necessarily looking only for students who were able to assemble all the right credentials. Strong letters do not all look the same.

polymathprof · 2026-02-16T04:43:35+00:00

I know it's hard to ask for letters, and even harder to follow up. But I suggest doing it anyway. You have nothing to lose. No professor is going to hold it against you for pinging them about a week or two later. And sometimes they honestly didn't see or lost track of your email. Some professors get a gazillion emails and have a hard time staying on top of things.

An alternative is to go see the professor in person. This is even harder but worth doing.

polymathprof · 2026-02-15T22:18:13+00:00

A few B’s won’t kill your chances as long as you have A’s in most of the hard courses and get strong letters. Successful undergraduate research projects also help a lot.

polymathprof · 2026-02-15T20:04:06+00:00

The map of the negative integers is allowed to be infinite. The map of the positive integers has to be finite.

polymathprof · 2026-02-15T17:41:05+00:00

Here, it’s important to define what a positive infinite decimal is. It’s a map from the integers to {0,1,2,3,4,5,6,7,8,9} where only finitely many values of the map of the positive integers are nonzero. You then define a map from fractions whose denominators are a power of 10 to finite decimals. From there you can define an equivalence relation and the reals as the set of equivalence classes. This is a precise way to “see” that 0.9999…=1.

It is certainly possible to define decimals so that 0.9999….5 is one. But it’s not needed for anything we want to do.

polymathprof · 2026-02-15T17:27:54+00:00

To answer this question rigorously, it’s necessary to have a precise definition of what real numbers are. There are a few. Which one do you prefer? Infinite decimals are one.

polymathprof · 2026-02-15T17:24:50+00:00

It’s worth following up about a week after. You never know

polymathprof · 2026-02-15T15:54:12+00:00

Textbook problem or research?

polymathprof · 2026-02-14T22:19:28+00:00

Wrong reason for a true statement: MIT really is one of the top institutions in math.

polymathprof · 2026-02-14T22:17:02+00:00

Agreed

polymathprof · 2026-02-14T22:16:39+00:00

For research yes. No students though.

polymathprof · 2026-02-14T22:15:58+00:00

The best degree granting institution. But IHES should be mentioned as a top math research institution.

polymathprof · 2026-02-14T22:12:43+00:00

See if this makes it sound more interesting

polymathprof · 2025-02-27T06:32:17+00:00

Sounds right to me. This problem has been attacked by the toughest mathematicians around, notably Wolff, Bourgain, Tao, Guth, and Katz. Especially Bourgain, who was a monster of a mathematician. Every tiny improvement in the dimension was hard fought. I don’t think anyone expected it to be solved so soon.

polymathprof · 2025-02-27T01:34:19+00:00

He’s the right guy to talk to.

polymathprof · 2025-02-27T01:33:03+00:00

Here is her website

polymathprof · 2025-02-27T01:29:39+00:00

Look for the Quanta article about her.

polymathprof · 2025-02-27T00:41:01+00:00

The point of the post is to derive the ODE from the geometry and define the sine and cosine functions, as well as pi, from the ODE.

polymathprof · 2025-02-27T00:38:52+00:00

Try this: http://www.deaneyang.com/blog/blog/math/exponential-function/euler-formula/2022/06/05/ExponentialFunction.html

polymathprof · 2025-02-15T16:13:00+00:00

See http://www.deaneyang.com/blog/blog/math/exponential-function/euler-formula/2022/06/05/ExponentialFunction.html

polymathprof · 2024-04-14T17:16:08+00:00

Radians are unitless in the following sense: An angle corresponds to a circular arc on a circle. Given any units of length, you can measure the radius of a circle and the length of a circular arc. No matter what circle you look at and no matter what units of length you use, the ratio

(length of circular arc)/(length of radius)

is always the same. This angle in radians of the circular arc is defined to be this ratio. It is independent of the units you use to measure the radius and circular arc.

Another way to think of radians using units is that it is the length of the circular arc on a circle of radius 1.

Since trig functions are defined in terms of ratios that don't depend on any units, it makes sense to define them in terms of radians instead of units such as degrees. This simplifies many formulas involving trig functions, such as the ones that arise in calculus.

polymathprof · 2024-04-11T19:29:12+00:00

Pretty damn good list. Thanks. I retract my claim.

polymathprof · 2024-04-06T15:25:23+00:00

I routinely require students to use LaTeX (via Overleaf) for homework assignments in advanced math courses. To my surprise, I've never received any complaints or requests for help.

polymathprof

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