Breakthrough Prize Announces 2025 Laureates in Life Sciences, Fundamental Physics, and Mathematics: https://breakthroughprize.org/News/91

Dennis Gaitsgory wins the Breakthrough Prize in Mathematics for his central role in the proof of the geometric Langlands conjecture. The Langlands program is a broad research program spanning several fields of mathematics. It grew out of a series of conjectures proposing precise connections between seemingly disparate mathematical concepts. Such connections are powerful tools; for example, the proof of Fermat’s Last Theorem reduces to a particular instance of the Langlands conjecture. These Langlands program equivalences can be thought of as generalizations of the Fourier transform, a tool that relates waves to frequency spectrums and has widespread uses from seismology to sound engineering. In the case of the geometric Langlands conjecture, the proposed one-to-one correspondence is between two very different sets of objects, analogous to these spectrums and waves: on the spectrum side are abstract algebraic objects called representations of the fundamental group, which capture information about the kinds of loop that can wrap around certain complex surfaces; on the “wave” side are sheaves, which, loosely speaking, are rules assigning vector spaces to points on a surface. Gaitsgory has dedicated much of the last 30 years to the geometric Langlands conjecture. In 2013 he wrote an outline of the steps required for a proof, and after more than a decade of intensive research in 2024 he and his colleagues published the full proof, comprising over 800 pages spread over 5 papers. This is a monumental advance, expected to have deep implications in other areas of mathematics too, including number theory, algebraic geometry and mathematical physics.

New Horizons in Mathematics Prize: Ewain Gwynne, John Pardon, Sam Raskin
Maryam Mirzakhani New Frontiers Prize: Si Ying Lee, Rajula Srivastava, Ewin Tang

I'm currently working on my master's thesis. I took a course in C*-algebras, and later on operator k-theory, and chose the professor that taught those courses as my thesis advisor. The topic he gave me is related to quantitative operator k-theory and the coarse Baum Connes conjecture.

I know a master's thesis is supposed to be technical and unglamorous, but I can't help but feel that I skipped many steps between the basic course material and this more contemporary topic. Like I just now learned about these topics and now I had to jump into something complex instead of spending time gaining intuition beyond the main theorems and some examples.

Sometimes I get stuck on elementary results, and my advisor quickly explains why something is true or why the author of the paper did that. Most of the times those things seem like "common knowledge", except I feel I didn't have time to gain that common knowledge.

Is it normal to feel like this?

Recommend a book on differential equations that introduces the topic from a pure maths perspective without much applications.

Are there any books and/or introductory texts about doing mathematics constructively (for research purposes)? I think I'd like to do two things, for which I'd need guidance:

1. train my brain to not use law of excluded middle without noticing it
2. learn how to construct topoi (or some other kind of constructive model, if there are some), to prove consistency of a certain formula with the theory, similar to those where all real functions are continuous, all real functions are computable, set of all Dedekind cuts is countable, etc.

Is this something one might turn towards after getting a PhD in another area (modal logic), but with a postgraduate level of understanding category theory and topos theory?

I have a theory which I'd like to see if I could do constructively, which would include finding proofs of theorems, for which I need to be good at (1.), but also if the proof seems to be tricky, I'd need to be good at (2.), it seems.

Art For Mentats I: 2,584 Dots For Madam Kusama. Watercolor and fluorescent acrylic on paper 18x18".

I used Vogel's mathematical formula for spiral phyllotaxis and plotted this out by hand, dot-by-dot. I consecutively numbered each dot/node, and discovered some interesting stuff: The slightly larger pink dots are the Fibonacci dots, 1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584.

I did up to the 18th term in the sequence and it gave me 55:89 or 144:89 parastichy (the whorls of the spiral). Also note how the Fibonacci nodes trend towards zero degrees. Also, based on the table of data points I made, each of those Fibonacci nodes had an exact number of rotations around the central axis equal to Fibonacci numbers! Fascinating.

This question could well apply to physics or computer science as well. Say you’re working on a problem in your work as a researcher and come across a sub problem. This problem is rather vague and generic in nature, so maybe someone else in a completely unrelated field came across it as a sub problem but spun sliiiightly differently and solved it first. But you don’t really know what keywords to look for, because it’s not really critical to one specific area of study. It’s also not trivial enough to the point that you could spend two or so months scratching your head.

How much time and ink is spent mathematically « reinventing the wheel », i.e.

case 1. You solve the problem, but are unaware that this is already known in some other niche field and has been for 50 ish years

Case 2. You get stuck for some time but don’t get unstuck because even though you searched, you couldn’t find an existing solution because it may not have been worthy of its own paper even if it’s standard sleight of hand to some

Case 3. Oops your entire paper is basically the same thing as someone else just published less than two years ago but recent enough and in fields distant enough to yours that you have no way of keeping track of recent developments therein

Each of these cases represent some friction in the world of research. Imagine if maths researchers were a hive mind (for information retrieval only) so that the cogs of the machine were perfectly oiled. How much do we gain?

I understand it is subjective, that is why im curious to hear people's opinions.

I’ve always found it pretty obvious that a field is the “right” object to define a vector space over given the axioms of a vector space, and haven’t really thought about it past that.

Something I guess I’ve never made a connection with is the following. Say λ and α are in F, then by the axioms of a vector space

λ(v+w) = λv + λw

λ(αv) = αλ(v)

Which, when written like this, looks exactly like a linear transformation!

So I guess my question is, (V, +) forms an abelian group, so can you categorize a vector space completely as “a field acting on an abelian group linearly”? I’m familiar with group actions, but unsure if this is “a correct way of thinking” when thinking about vector spaces.

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

The construction of the vitali set and the subsequent proof of the existence of non-measurable sets under AC is mine. I just think it's fun and cute to play around with.

Okay, here goes. So I like Linear Algebra quite a bit (mostly because of the geometric interpretations, I still have not understood the ideas behind tensors), and also Group Theory (Mostly because every finite group can be interpreted as the symmetries of something). But I cannot get Rings, or Modules. I have learned about ideals, PIDs, UFDs, quotients, euclidean rings, and some specific topics in polynomial rings (Cardano and Vieta's formulas, symmetric functions, etc). I got a 9.3/10 in my latest algebra course, so it's not for lack of studying. But I still feel like I don't get it. What the fuck is a ring?? What is the intuitive idea that led to their definition? I asked an algebraic geometer at my faculty and he said the thing about every ring being the functions of some space, namely it's spectrum. I forgot the details of it. Furthermore, what the fuck is a module?? So far in class we have only classified finitely generated modules over a PID (To classify vector space endomorpisms and their Jordan normal form), which I guess are very loosely similar to a "vector space over Z". Also, since homomorphisms of abelian groups always have a ring structure, I guess you could conceptualize some modules as being abelian groups with multiplication by their function ring as evaluation (I think this also works for abelian-group-like structures, so vector spaces and their algebras, rings... Anything that can be restricted to an abelian group I would say). Basically, my problem is that in other areas of mathematics I always have an intution of the objects we are working with, doesn't matter if its a surface in 33 dimensions, you can always "feel" that there is something there BEHIND the symbols you write, and the formalism isn't the important part, its the ideas behind it. Essentially I don't care about how we write the ideas down, I care about what the symbols represent. I feel like in abstract algebra the symbols represent nothing. We make up some rules for some symbols because why the fuck not and then start moving them around and proving theorems about nothing.

Is this a product of my ignorance, I mean, there really are ideas besides the symbols, and I'm just not seeing it, or is there nothing behind it? Maybe algebra is literally that, moving symbols.

Aside: Also dont get why we define the dual space. The whole point of it was to get to inner products so we can define orthogonality and do geometry, so why not just define bilinear forms? Why make up a whole space, to then prove that in finite dimension its literally the same? Why have the transpose morphism go between dual spaces instead of just switching them around.

Edited to remove things that were wrong.

I took a course in Fourier analysis which covered trigonometric and Fourier series, parseval theorem, convolution and fourier transform of L1 and L2 functions, the coursework was so dry that it surprises me that people find it fascinating, I have a vague knowledge about the applications of Fourier transformation but still it doesn't "click" for me, how can I cure this ?

My favourite is aleph (ℵ) some might have seen it in Alan Becker's video. That big guy. What's your favourite symbol?

Everyday whenever I go out, I see such mathematical patterns everywhere around us and it’s so fascinating for me. As someone who loves maths, being able to see it everywhere especially in nature is something we take for granted, a small walk in the park and I see these. It’s almost as if there’s any god or whatever it is, its language is definitely mathematics. Truly inspiring

Are we supposed to finish any textbook as an undergraduate (or even master student), especially if one tries to do every exercise?

And some author suggests a more thorough style, i.e. thinking about how every condition is necessary in a theorem, constructing counterexamples etc. I doubt if you can finish even 1 book in 4 years, doing it this way.

In this comment for example,

Intersection theory is much more well behaved. For example, over C, Bezout's theorem says that a curve of degree d and another of degree e in the projective plane meet in d*e points. This doesn't hold over the affine plane as intersection points may occur at infinity. [This is in part due to the fact that degree d curves can be deformed to d lines in a way that preserves intersection, and lines intersect correctly in projective space, basically by construction.]

Maps from a space X to a projective space have a nice description that is intrinsic in X. They are given by sections of some line bundle on X

They have a nice cellular decomposition in terms of smaller protective spaces and so are a proto-typical example of such things like toric varieties and CW complexes.

So projective spaces have

• nice intersection properties,
• deformation properties,
• deep ties with line bundles,
• nice recursive/cellular properties,
• nice duality properties.

You see them in blowups, rational equivalence, etc. Projective geometry is also a lot more "symmetric" than affine; for instance instead of rotations around 1 point and translations, we just have rotations around 1 point. Or instead of projections from 1 point (like stereographic projection), and projection along a direction (e.g. perpendicular to a hyperplane), we just have projection from 1 point.

So why does this silly innocuous little idea of "adding points for each direction of line in affine space" simultaneously produce miracle after miracle after miracle? Is there some unifying framework in which we see all these properties arise hand in hand, instead of all over the place in an ad-hoc and unpredictable manner?

See also

https://math.stackexchange.com/questions/1641100/why-is-a-projective-variety-the-best-kind

https://www.cis.upenn.edu/~jean/gma-v2-chap5.pdf discussing how e.g. circle is not a parameterized algebraic curve (it is a parameterized rational curve), but parameterized rational curve in general are "central"/"projective" projections of parameterized algebraic curves in 1 higher dimension. "Clearing denominators"

https://www.reddit.com/r/math/comments/y1ljfe/why_are_complex_varieties_and_manifolds_often/

https://arxiv.org/html/2410.07207v1

https://math.stackexchange.com/questions/1179312/why-are-projective-spaces-and-varieties-preferable

https://www.ams.org/bookstore/pspdf/cbms-134-prev.pdf (some nice history on elimination theory, resultants, which inevitably --- but still mysteriously, to me --- brings one to homogeneous polynomials)

https://mathoverflow.net/questions/26755/what-if-anything-makes-homogeneous-polynomials-so-great in particular, geometric niceties ("proto Bezout"?) "in projective space, varieties of complementary dimensions always intersect". Algebraic niceties include "putting a grading on an algebra usually organizes the algebra into a collection of finite-dimensional vector spaces, each indexed by a natural number. This opens the door to induction arguments" and "clearing denominators"

https://math.stanford.edu/~vakil/0708-216/216class32.pdf, https://math.stanford.edu/~vakil/725/class21.pdf

https://people.math.wisc.edu/~jwrobbin/951dir/divisors.pdf

questions/queries like "why complex projective space best compactification", "complex projective manifold", "complex geometry projective duality"

Mine is probably either the Twin Prime Conjecture or the Odd Perfect Number problem, so simple to state, yet so difficult to prove :D

Pretty much the title, I guess. I usually don't remember a lot more than a sort of broad theme of a course and a few key results here and there, after a couple of semesters of doing the course. Maybe a bit more of the finer details if I repeatedly use ideas from the course in other courses that I'd take currently. I definitely would not remember any big proof unless the idea of the proof itself is key to the result, and that's being generous.

I understand that its not possible to fully remember everything you'd learn, especially if you're not constantly in touch with the topics, but how would you 'optimize' how much you remember out of a course/self studying a book? Does writing some sort of short notes help? What methods have you tried that helps you in remembering things well? How do you prioritize learning the math that you'd use regularly vs learning things out of your own interest, that you may not particularly visit again in a different course/research work?