And that's it for today's #MathTopicOfTheDay ! I'm still in the process of recovery from covid, so I feel like my head's all over the place and it's hard to do justice to this amazing part of math. Hopefully the articles I linked to make up for it!

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Today's #MathTopicOfTheDay is Faltings' theorem, formerly known as the Mordell conjecture! It says that a nonsingular algebraic curve over the rationals (or some other number field) of genus 2 or greater only has a finite number of rational points:

https://en.wikipedia.org/wiki/Faltings%27s_theorem

#Math #Mathematics

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I'm still recovering from covid so I won't be writing a thread, but today's #MathTopicOfTheDay is de Rham's theorem, which relates singular cohomology (one of the first cohomology theories one learns in an algebraic topology course) and de Rham cohomology (which is a more "analytic" kind of cohomology, using differential forms):

https://en.wikipedia.org/wiki/De_Rham_cohomology#De_Rham's_theorem

#Math #Mathematics

Before I end I want to highlight a very recent breakthrough made possible by the developments on these topics, again by Prof. Ana Caraiani this time with Prof. James Newton - the analogue of Prof. Andrew Wiles' proof of the Shimura-Taniyama-Weil modularity conjecture, for elliptic curves over quadratic imaginary fields! Here is a talk by Prof. Caraiani on this work:

https://www.youtube.com/watch?v=MM6sebSdtCc

And that's it for today's #MathTopicOfTheDay (I think this is my longest thread yet in this series)!

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This is an example of a modular curve (often we also compactify it first to make it a projective variety). Things called "modular forms" (the topic of yesterday's #MathTopicOfTheDay) "live" on this space (more technically, they are sections of certain sheaves on such a variety).

SL_2(Z) is not the only arithmetic group we can use to form an arithmetic manifold out of the upper half-plane. We can also use for example "congruence subgroups" of SL_2(Z).

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Today's #MathTopicOfTheDay is arithmetic manifolds! Once again there appears to be no Wikipedia page for this except in the special case of arithmetic hyperbolic 3-manifolds, so I will link instead to a very cool survey by Prof. Jared Weinstein (arithmetic manifolds are discussed in section 5, in particular 5.3):
https://math.bu.edu/people/jsweinst/CEB/BAMS.pdf

#Math #Mathematics

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I think I know what I'm going to post about tomorrow but if anyone has any suggestions for #MathTopicOfTheDay for the coming days let me know!

I've posted six days of #MathTopicOfTheDay !

So far I've posted about:

Day 1: SO(3)
Day 2: Cyclotomic fields
Day 3: Grassmannians
Day 4: Fontaine-Wintenberger theorem
Day 5: Sierpiński space
Day 6: Ramanujan's conjectures

#Math #Mathematics

And that's it for today's #MathTopicOfTheDay! I'm ending this thread with a blog post by @antoinechambertloir from two years ago also on this topic:

https://freedommathdance.blogspot.com/2020/12/celebrating-ramanujans-birthday.html

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Today's #MathTopicOfTheDay is the Ramanujan conjecture! It is named after Srinivasa Ramanujan, who was born on this day in 1887. It concerns the Ramanujan tau function, which are the Fourier coefficients of the discriminant modular form (a cusp form of level 1 and weight 12).

https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture

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#Math #Mathematics #Ramanujan

There are also other applications of the Sierpiński space (Wikipedia says it also shows up in the theory of computation, but I don't know much about that area so I'll leave the reader to check out that article).

And that's it for today's #MathTopicOfTheDay ! Here's a nice category theory blog post on the amazing blog math3ma which involves the Sierpinski space:

https://www.math3ma.com/blog/the-sierpinski-space-and-its-special-property

Today's #MathTopicOfTheDay is the Sierpiński space! It is my favorite topological space. It is named after Wacław Sierpiński, who also studied fractals like the Sierpiński triangle and the Sierpiński carpet (a 2D version of the Cantor set). However, unlike infinite things like fractals, the Sierpiński space is a very *finite* thing. It's underlying set only has two elements!

https://en.m.wikipedia.org/wiki/Sierpi%C5%84ski_space

That's about it for today's #MathTopicOfTheDay ! For learning more about the Fontaine-Wintenberger theorem and perfectoid fields, I really like this blog post by Prof. Alex Youcis:

https://ayoucis.wordpress.com/2017/02/20/the-fontaine-winterberger-theorem-going-full-tilt/

For more on perfectoid spaces, I learned from the Berkeley Notes on p-adic Geometry (now a book) by Profs. Peter Scholze and Jared Weinstein:

https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf

Alright, the #MathTopicOfTheDay is the Fontaine-Wintenberger theorem, which inspired the theory of perfectoid spaces! I usually post a Wikipedia article on here but since there seems to be none for the Fontaine-Wintenberger theorem, I'll link to Prof. Bhargav Bhatt's article "What is... a Perfectoid Space?" instead:

https://www.ams.org/notices/201409/rnoti-p1082.pdf

Related to today's #MathTopicOfTheDay , but more advanced and specialized, is the "affine Grassmannian". I don't even know why it's called "affine" (it's not an affine scheme, it's not even a scheme!) although I do kinda know why it's called a Grassmannian (the relationship is not straightforward!). It's very closely related to things I'm interested in and I hope to know more about it someday.

Since vector spaces and linear algebra is ubiquitous in math, one can imagine that the Grassmannian finds application in many areas of math as well!

Anyway that's it for today's #MathTopicOfTheDay and I'll just add a couple things. First, sorry for omitting discussion about the field of scalars, second, the Grassmannian itself has a generalization, the flag variety. Maybe I'll discuss it on some other day, for now I'll leave a link to the Wikipedia page:

https://en.m.wikipedia.org/wiki/Generalized_flag_variety

Today's #MathTopicOfTheDay is Grassmannians! A Grassmannian (named after the 19th-century mathematician Hermann Grassmann) is the moduli space of subspaces (of a fixed dimension) of some vector space:

https://en.m.wikipedia.org/wiki/Grassmannian

And that's it for today's #MathTopicOfTheDay! There's so much fascinating stuff regarding cyclotomic fields that I can't cover them all in this short thread. Hopefully the links make up for it, and next time we'll cover other interesting topics possibly from other parts of math!

Today's #MathTopicOfTheDay is cyclotomic fields! They are fields (sets for which you can add, subtract, multiply, and divide, except by zero) obtained by "adjoining" the nth roots of 1 (for some n) to the rational numbers:

https://en.wikipedia.org/wiki/Cyclotomic_field

The nth roots of 1 (or as we often call them, the nth roots of unity) form n evenly spaced points of a circle of radius 1 in the complex plane, and this is where the name comes from - "cyclotomic" means "dividing a circle".

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Anyway that's it for now, hopefully I can continue this #MathTopicOfTheDay series with other topics from other parts of math as well. A nice reference for SO(3) is in section 9.4 of the book Algebra by Michael Artin (second edition).