M@X — Math @ Xavier

Fermat’s Last Theorem is one of the most famous problems in mathematics, captivating mathematicians for over 350 years with its deceptively simple statement:
Theorem. (Fermat’s Last Theorem) The equation
\(x^n + y^n = z^n\)
has no solution in integers \(x,y,z\) for \(n\geq 3\) when \(xyz\neq 0\). After centuries of attempts, Andrew Wiles and Richard Taylor’s proof in 1995 marked a historic breakthrough. However, the story doesn’t end there. In the wake of Wiles and Taylor’s proof, mathematicians turned their attention to a broader class of equations known as Generalized Fermat Equations (GFE), which take the form: \begin{equation} X^p+Y^q=Z^r, \end{equation} where \(p,q,r\geq 2\). These equations lie at the intersection of number theory and algebraic geometry, requiring tools such as the “Modular Method”, Galois representation, level-lowering techniques, and the study of rational points on modular curves. Intriguingly, in 1997, an American banker and mathematician, Andrew Beal proposed a striking conjecture, now associated with a \$1 million prize: Conjecture (Beal). Let \(p, q, r\) be primes with \(\min\{p, q, r\}\geq 3\). Then the equation (1) has no solutions in coprime integer tuples \((X, Y, Z)\) with \( XYZ\neq 0\). Despite much effort, the conjecture remains unsolved except in a few special cases. In this expository talk, we will explore recent developments in Generalized Fermat Equations, focusing on joint work with Mike Bennett (University of British Columbia). Our findings offer new insights into this rich and evolving area of mathematics.

Let \(\sigma\) denote a permutation of \([n] = \{1, 2, \ldots, n\},\) and let \(O(\sigma)\) denote the order of \(\sigma.\) In 1967, Paul Erdős and Pál Turán proved the following lovely theorem, since referred to as the Erdős-Turán Law.

Theorem. If \(\sigma\) is a uniform random permutaton of \([n],\) then \[\mathbb{P}\left[\frac{\log O(\sigma) - \frac{1}{2}\log^{2}n}{\sqrt{\frac{1}{3}\log^{3}n}} \leq x\right] \to \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-t^2/2}\,dt\] as \(n \to \infty.\)
In other words, the order of a random permutation has an asymptotically lognormal distribution. However, Erdős and Turán's original proof is rather long and complicated, and so they invited mathematicians to find a simpler, more elegant proof from "The Book." Their challenge has inspired nearly six decades of ongoing research activity that lies in the intersection of analytic combinatorics and stochastic processes. In this expository talk, we will explore the many strengthenings and generalizations of the Erdős-Turán Law together with various applications and related problems.

TBA

TBA

TBA

TBA

TBA