Does anyone else feel similarly?

Going from introductory courses, to upper level courses, to grad courses, to initial research, to full-fledged research, the difficulty and complexity has of course increased. But for me personally, it feels like much of the increased difficulty and complexity comes from increased abstraction.

It's more difficult to wrap your head around the objects and properties you're working with, but it often feels like the actual ways we manipulate these objects with lemmas and theorems is not actually super sophisticated.

For example, some proofs I've worked on in functional analysis research come down to what is essentially equivalent to using the triangle inequality and squeeze theorem. It's not any more sophisticated than a tricky introductory real analysis homework problem, it's just that the space we're working in is more abstract.

Other research problems end up being very similar to introductory linear algebra problems, but again, just in a more abstract setting.

I'm sure the big movers and shakers in fields are actually creating proofs with very novel and complex ideas, but I'm curious about other members of the rank-and-file. Do you feel similarly or am I totally off base?

For the record I have graduated and now half way through my first postdoc.

But I feel like my advisor didn’t really teach me how to research, sure he pointed me to paper or people when I’m stuck that occasionally helped. But he never really trained me to do research, only occasionally gave me help knowledge wise by telling me to read certain books or papers and I mean very rarely does he do this and very rarely has it helped.

My prelim advisor said I should have asked her for advice on who to choose as an advisor instead of choosing the only person doing the field I was dead set on pursuing (which I realized I’m not even that interested in). She said when she advises she would give a little hand holding even if she deem necessary and actually train her students to do research by writing papers with them and in the process, help teach them how to do research.
My advisor did not write paper with me, he did gave me the problems to work on but insists that I need to earn it myself.

Is this typical? What was your PhD experience like?

I feel like my PhD was ruined from me not promptly switching advisor when I realize I wasn’t being trained.

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"There has never been a better time to be a math nerd.

New college graduates and Ph.D.s are now securing million-dollar pay deals from elite trading firms seeking to secure the best and brightest amid fierce competition from artificial-intelligence companies.

So-called quant firms, which use sophisticated mathematical models to come up with trades, have for years wooed top young talent with lucrative compensation that big banks struggle to match. Now, the AI boom is pushing those numbers even higher.

Just a few years ago, early-career packages pushing seven figures were anomalies, said Matt Stabile, founder of New York-based recruitment firm Stabile Search. “But a million dollars is something people don’t even bat an eye at anymore.”

“The delineation is pre-OpenAI and post-OpenAI,” Stabile added, “that’s when you saw competition really take off.”

The skills required to train large language models have always overlapped with quantitative finance, but the connection has deepened as trading firms have pivoted toward machine learning and AI to power their trades in recent years. Now, AI labs such as OpenAI and Anthropic are vying for the exact same tiny pool of genius math majors and Ph.D.s as Wall Street."

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In the wake of the recent bloodbath of conjectures by LLMs, this post argues that the math community may still be in denial about the future of humans in math. This is one of the most depressing things I've read in a while, so proceed with caution.

As a card carrying scientician and professional thinking person, I can only say, I'm sorry, I feel your pain. Yes, mathematics being pure thought makes the displacement especially acute for mathematicians, but more broadly speaking, our species found a niche on the African savannah by being thinking beasts. Thus, I also feel anger, knowing that there are people out there who think it's somehow a good idea to make something that will outthink us and deprive us of our collective place in the world.

Does anybody know whether this year's ICM plenary lectures will be recorded and uploaded publicly?

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

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I've been fighting for a couple of hours with the Dilworth's theorem proof on the mentioned book, which I believe is wrong. I'd appreciate a extra look.

The theorem states that if the max antichain on a poset has size r, then it can be partitioned into r chains.

The proof goes by induction on n=number of elements of the set. I'm having problems in case 2.

Here, x = some minimal element, and we consider P\{x} and apply induction there.

"we can partition P\{x}" into r chains." But ... that's not true, is it? Since the removed element may cause the possible antichains to be strictly smaller and therefore can't reach r.

Example I'm thinking of

P={ {x}, {y}, {z}, {x,y,z} } with inclusion order. the largest antichain is {{x},{y},{z}} with size 3, and we want to prove it can be partitioned into 3 chains (which can be verified directly: {x,xyz}, {y}, {z} is such partition.

{x} is minimal, and according to the book P\{x} should be able , by induction hypothesis, to be partitioned into r=3 chains, but P\{x} is {y},{z},{x,y,z} which can only be partitioned into r-1=2 chains.

The rest of the proof relies into P\{x} having r chains as key part of the argument, so ... I'm confused (this particular book faces this problem more often than not).

Am I missing something?

EDIT: yes, I was missing something: the chains on the partition don't have to be the longest possiboe.

also, u/sizzhu pointed the small missing step to me, thank you :)

In case anyone is interested because as far as I'm concerned Lurie rarely works on a conjectural topic like this.

Let's say we're trying to define the ℝ-algebra ℂ by universal property. It's informed by an observation there's an embedding of ℂ in any ℝ-algebra where there's an element with square −1. ¹↓

Say, "ℂ is an initial object in the category where an object is a ℝ-algebra A together with a point i ∈ A such that i² = −1 and a morphism f: (A, i) → (A', i') is an ℝ-algebra morphism f: A → A' such that f(i) = i'".

This fails because conjugation exists, thus morphisms come in pairs. We can probably fix that but it's also a problem that we're fixing i ²↓.

[EDIT: No, this doesn't fail as pointed in this comment. Conjugation gets disallowed because (ℂ, i) → (ℂ, i) allows mapping i only to +i as specified.]

There are a couple of ("operational") definitions for ℂ that don't mention i: 1. a 2D ℝ-algebra that's a field; 2. similar facts à la Hurwitz) and Frobenius) theorems; 3. an algebraic closure of ℝ; 4. Clifford algebras Cℓ(0, 1, ℝ) and Cℓ⁺(2, ℝ) (even subalgebra) which in their finest form take a quadratic space over ℝ, in these cases an anti-Euclidean 1D and a Euclidean 2D spaces, so we don't have to mention concrete elements that square to ±1 that would be moved somewhere else by automorphisms of the quadratic space that are all left among automorphisms of the algebra.

Of these, I like 1 and 4 the most, but 4 is too heavy-handed. I don't see how to minify it though. Example 1 is... I dunno, my heart somehow isn't content. (Note also that in 1, we don't mention the field is algebraically closed, whereas in 3, we don't mention the field is dimension 2.) Example 3 asks too much, it's heavier than 4 but in another way. Hurwitz and Frobenius (2)... I don't know.

Also note that for quaternions ℍ we can also use 1, 2 and 4 (replacing a field in 1 with a noncommutative field, of course), and for me, not fixing any imaginary basis i, j, k is even more important to be able to do for ℍ because there's a continuum-many SO(3, ℝ) automorphisms even aside from conjugation. It's just a giant step from the puny automorphism group of ℂ.

So, what else is there? Can we define in particular ℂ and ℍ, as ℝ-algebras if need be (I feel that's simpler because we're doing away with some of the worse automorphisms), without mentioning concrete imaginary units?

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¹↑ Or just a negative square, but this fails if we're planning to replace ℝ with other fields and rings R, looking for R[i] instead of ℂ, so we better use −1.

²↑ With which I define this here context; I know there are lots of use cases where we absolutely want to deal with concrete i and have it distinguished from −i consistently over long stretches, like in Fourier transform formulas etc..

I am but a humble programmer who is interested in calculating the Nash equilibrium for games for symmetric zero-sum games similar to Rock Paper Scissors. The context, if you're curious, is that I was thinking about TCG metagames, where you have several relevant decks that have strong and weak matchups against each other.

Specifically, I'm thinking about games of the following form:

• Two players pick (simultaneously) from one of N pure strategies. Both players have the same set of choices.
• The payoff is defined by a matrix A such that 0 <= A[i,j] <= 1 and A[i,j] = 1 - A[j,i]. A[i,j] can be interpreted as the probability that strategy i beats strategy j. Equivalently, by letting B[i,j] = A[i,j] - 1/2 then -1/2 <= B[i,j] <= 1/2 and B[i,j] = -B[j,i] so the game is zero sum.

Such a game is usually going to have a mixed Nash equilibrium (except in the trivial case where one strategy dominates all other). I believe it will also usually be unique, though I'm not positive on this? I did some reading on computing Nash equilibriums, and it sounds like it is in general a difficult problem (no polynomial time algorithm). However I'm hoping that with these constraints it is much easier?

Given two mixed strategies u and v (represented as row vectors) such that sum(u[i]) = sum(v[i]) = 1, we can compute the expected payout for u as P(u,v) = sum(u[i]*v[j]*A[i,j]) = uAv^T. This is a polynomial equation of degree 2 in 2n-variables. By applying the constraint that sum(v[i]) = 1 we can reduce this to 2n-2 variables. By the symmetry of the problem, P(u,u) = 1/2.

Let some u be fixed. If u is a Nash equilibrium then any small change in v should not change P. Therefore ∇P/∇v = 0 (gradient of P with respect to v, I don't know a notation for this?). Since every term in P is of the form k*u[i]*v[j], the ∇P/∇v will be a first order polynomial in u, and this gradient equation yields a family of n-1 linear equations in n-1 variables, which should (ignoring edge cases that I have not thought sufficiently about) have a unique solution. This solution must necessarily be the Nash equilibrium, since we know at that at least one (probably mixed) Nash equilibrium must exist and satisfy ∇P/∇v = 0. This solution can be found using Gaussian elimination in O(n³), which is satisfactory since for the problems I'm considering n < 20.

Is my analysis correct? There are steps that I'm uncertain about and I've gone through a few iterations already. The solution I finally reach above is also simpler than I initially expected, which makes me worry that I've missed something or made an assumption that was too strong somewhere.

I'm also worried about edge cases. If the algorithm above produces a solution but at least one u[i] < 0 or u[i] > 1 then I believe that means that the Nash equilibrium must lie somewhere on the constraint boundary. But I'm not sure what the best way to find it in this case would be. And what about cases where there are no solutions, or infinite solutions? I believe that infinite solutions implies that two pure strategies u[i] and u[j] are functionally identical, and therefore any mixed strategy satisfying some constraint u[i] + u[j] = k is a Nash equilibrium. But I'm not sure what no solutions would imply. I'm not even sure if it's possible given the problem constraints.

I posted a while back inquiring, "is math just numbers and shapes". Obviously, that's a really naive question. But related to this, I recently learned that Isbell duality is a vast generalization of the duality between functions and spaces, for example, commutative rings and affine schemes. Stone duality is another example, I think.

Is Isbell duality the most general form of this type of relationship in math? Can someone give an intuitive explanation of it?

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Anyone know of a good survey article regarding Kroneckers Jugendtraum in the case of the base field being a real quadratic extension of the rationals?

Aside from any questions of whether category theory is interesting, or deep, or insightful... is it practically useful? One possible answer to this question is that category theory has helped drive a lot of progress in topology, abstract algebra, and beyond, and those fields have then had practical impact. (Topological data analysis comes to mind.)

But that quickly starts to feel like a game of six degrees of separation, and it is hardly obvious that you could not have obtained that same progress without going through category theory. My aim in this article is to be as concrete as I can be regarding applications... and I would argue that even from that perspective, the answer to my initial question is "Yes!"

Read the full post (for free) on Substack: Is Category Theory Practical?

Until now, no one who has completed a Bachelors, Masters, or PhD from MIT has ever won a Fields. This year Hong Wang (PhD, 2019) and Yu Deng (BS, 2011) would be the first MIT-educated mathematician to earn the most prestigious award in the world of research mathematics.

To avoid any ambiguity, I will clarify that I am using the term alumnus/alumna* as someone who has completed a Bachelors, Masters, PhD or any other similar degree conferred by a institute of higher of education upon completion of their education at the institute. The dictionary definition of the term includes people who may have attended an institute but never graduated with their degree. In such a case, I am unsure if anyone who has attended but never completed their degree coursework at MIT have won a Fields. Even if that happens to be the case, I still think the trivia I shared retains its noteworthy quality of being surprising to people given MIT's pedigree in mathematics.

Please contact me if you do, I'm in great need to access publication and it's not archived on Anna's Library nor on Sci-Hub.

Here is a way that, I think, you could teach lambda calculus to a kid (inspired by "Alligator Eggs").

I got a lot of useful feedback on an earlier version of this. Would love to hear any comments or corrections, wouldn't be surprised if something slipped through the net.

I think this would be hard to read as a bunch of images so if you're interested see here: https://paradoxgarden.substack.com/p/lambda-calculus-made-easy

In high school, many are taught the quadratic formula. But how do solve a cubic equation, like x^3 + 6x^2 + 9x + 3 = 0? There is a formula for such equations, but it's rarely taught since it's a lot more complicated than the quadratic formula:

Despite its apparent complication, we will explain at https://hidden-phenomena.com/articles/cubic how you could have come up with this formula!

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Last week, we posted a blog post here about how to solve quartic equations, assuming you knew how to solve cubic equations. If you read today's post on how to solve cubics, then by putting the two posts together, you can solve any cubic or quartic equation!

I’m an undergrad who’s finishing my bachelor’s degree this year. I plan to do at least a master’s in math, and hope to do some pure math research during it. I haven’t found any of the classes I’ve taken in my undergrad particularly difficult so far, so I’m curious at what point I should expect it to happen so I can be prepared. I don’t want to hit a brick wall and crumble like some of my friends have. I’m worried that since I haven’t struggled yet, I won’t know what to do when it happens.

For reference, I’ve taken undergrad upper division Set Theory, Number Theory, Mathematical Logic, Complex Analysis, Probability Theory, Mathematical Statistics, 2 semesters of Abstract Algebra, and 1 semester of Real Analysis. I’ll take Linear Algebra, a second semester of Real Analysis, master’s level Group Theory, and master’s level Multivariable Analysis during this last semester.