Today I Learned

January 1st

Today I learned that the Ackermann function can be expressed using our generalized primitive recursion over $\NN.$ (continue reading...)

January 2nd

Today I learned some properties of equality. (continue reading...)

January 3rd

Today I learned the details of the Eckmann-Hamilton theorem in type theory. (continue reading...)

January 4th

Today I learned about type-theoretic homotopies between (dependent) functions. (continue reading...)

January 5th

Today I learned about equivalence of types. (continue reading...)

January 6th

Today I learned that equality types of Cartesian products are well-behaved in type theory. (continue reading...)

January 7th

Today I learned the proof of the Newton polygon theorem. (continue reading...)

January 8th

Today I learned the proof of the $abc$ conjecture for polynomials, and notably function fields. (continue reading...)

January 9th

Today I learned the proof of Fermat's last theorem for polynomials over a perfect field, as a corollary from yesterday's work; notably this holds for function fields. (continue reading...)

January 10th

Today I learned which elements of $\FF_q[[t]]^\times$ can be written as the sum of two squares for $q$ odd. (continue reading...)

January 11th

Today I learned about transporting functions. (continue reading...)

January 12th

Today I learned about the univalence axiom. (continue reading...)

January 13th

Today I learned the outline of how to use class field theory to classify primes of the form $x^2+ny^2,$ where $n\equiv1\pmod4$ and is squarefree. (continue reading...)

January 14th

Today I learned a somewhat reliable way to compute $L(1,\chi)$ for characters $\chi$ (that actually doesn't use the multiplicative structure of $\chi$). (continue reading...)

January 15th

Today I learned another interpretation of the fact that the class number measures the failure of unique prime factorization of elements, from this post . (continue reading...)

January 16th

Today I learned the Poisson summation formula, from Keith Conrad . (continue reading...)

January 17th

Today I learned a strengthening of Bauer's theorem, from Keith Conrad . (continue reading...)

January 18th

Today I learned an alternative explanation for the decimal expansion of fractions like $\frac1{9801},$ from here . (continue reading...)

January 19th

Today I learn an application of the (truncated) Poisson summation formula to compute the quadratic Gauss sum. (continue reading...)

January 20th

Today I learned an extension of the information-theoretic proof of the infinitude of primes, from here . (continue reading...)

January 21st

Today I learned an alternative proof of Wilson's theorem, from group actions. (continue reading...)

January 22nd

Today I learned some representation theory, from Artin 10.1.1: all representations of finite groups with dimension $1$ have image that is finite cyclic. (continue reading...)

January 23rd

Today I learned a proof of Maschke’s Theorem: every representation of a finite group $\rho:G\to\op{GL}(V)$ can be decomposed into a direct sum of irreducible representations. (continue reading...)

Today I learned the Ping-pong lemma: if a group $G$ generated by $a$ and $b$ acts on a set $X$ has subsets $X_1$ and $X_2$ which do not contain each other but $a^\bullet X_1\subseteq X_2$ and $b^\bullet X_2\subseteq X_1$ (for nonzero $\bullet$), then $G=\langle a,b\rangle$ is free. (continue reading...)