What shall we learn today?

Propositions and logical reasoning

A proposition is a statement that is either true or false, never both. Connectives such as $\neg P$, $P \wedge Q$, $P \vee Q$, $P \to Q$ build compound statements, and truth tables let us verify equivalences such as $\neg(P \wedge Q) \equiv (\neg P) \vee (\neg Q)$ with certainty.

Predicate logic

Predicate logic extends propositional logic with quantifiers $\forall x\, P(x)$ ("for all $x$, $P(x)$") and $\exists x\, P(x)$ ("there exists $x$ such that $P(x)$"), letting us formalize statements about every element or some element of a domain, and reason about them with Tarski's satisfaction semantics.

Gödel's incompleteness theorems

Any formal system strong enough to describe arithmetic contains true statements it cannot prove, and cannot prove its own consistency — a fundamental limit discovered in 1931 that reshaped logic, computability, and the philosophy of mathematics.

abc conjecture

Predicts that for coprime $a+b=c$, $c$ is rarely much larger than the product of the distinct primes dividing $abc$ — proposed in 1985; a disputed 2012 proof claim was published in 2021 but is not accepted by most number theorists.

Angle trisection with compass and straightedge

Proved impossible in general by Pierre Wantzel in 1837: trisecting $60^\circ$ requires constructing $\cos(20^\circ)$, whose minimal polynomial $8x^3 - 6x - 1 = 0$ over $\mathbb{Q}$ has degree $3$, which is not a power of $2$.

Artin's conjecture on primitive roots

An integer $a$ is a primitive root modulo a prime $p$ if its powers generate the entire multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ of order $p-1$. In 1927, Emil Artin proposed to Helmut Hasse that any integer $a \ne -1$ that is not a square generates $(\mathbb{Z}/p\mathbb{Z})^{\times}$ for a positive proportion of all primes $p$, governed by an Euler product heuristic. Later computations by Lehmer revealed small entanglement corrections when the square-free part of $a$ is $1 \pmod{4}$, after which Heilbronn tightened the density formula. Christopher Hooley proved the corrected conjecture in 1967 assuming the Generalized Riemann Hypothesis, and D. R. Heath-Brown showed unconditionally in 1986 that at most two prime bases can fail — yet not a single specific base $a$ is unconditionally known to work.

Muhammad ibn Musa al-Khwarizmi

Persian mathematician and astronomer at the House of Wisdom in Baghdad whose treatise on restoring and balancing equations gave algebra its name and its first systematic treatment, and whose book on Indian numerals helped transmit the Hindu-Arabic decimal system to the western mathematical tradition.

Alan Turing

British mathematician and logician who defined the abstract computing machine that bears his name, laying the theoretical foundation of computer science, and played a central role in wartime efforts to break German Enigma ciphers.

Alexander Grothendieck

German-born mathematician in France who rebuilt algebraic geometry on the language of schemes, sheaves, topoi, and étale cohomology, and transformed homological algebra and category theory.