Thursday, January 30 |
07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
09:00 - 09:30 |
Arthur Baragar: Orbits of rational points on K3 surfaces ↓ The number of Markoff triples (a,b,c) with a≤b≤c≤B is klog(B)2+O(log(B)(loglogB)2, with k an explicitly computable constant (Zagier, 1982). The Markoff-Hurwitz equation w2+x2+y2+z2=4wxyz has an analogous tree of integer solutions, but the asymptotics are fractal: The number of integer solutions (a,b,c,d) with 0≤a≤b≤c≤d≤B grows asymptotically like k(logB)β, where β is a constant in the interval (2.430,2.477). In this talk, I will explore a similar situation with orbits of curves on certain K3 surfaces, and in particular draw attention to a vexing question about orbits of rational points. (TCPL 201) |
10:00 - 10:30 |
Dani Kaufman: Non-commutative Markov Numbers ↓ Fricke's trace identity is a cubic relation between the traces of any pair of 2-by-2 matrices of determinant 1, their product, and their commutator. It allows the Markoff numbers to be interpreted as traces of matrices, opening rich connections with many topics in mathematics. This talk will discuss a proof of Fricke's identity using the spin representation into a four-variable orthogonal group. (TCPL 201) |
10:45 - 11:15 |
Coffee Break (TCPL Foyer) |
11:15 - 11:45 |
Ian Agol: A new proof of the Markov theorem ↓ We give a new variation on a proof of Markov’s theorem that the worst approximable numbers correspond to solutions to the Markov equation. This is inspired by Series’ proof using immersed curves on the modular torus. (Online) |
12:00 - 13:30 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
13:45 - 14:15 |
Aaron Calderon: Twist tori equidistribute in moduli space ↓ Every hyperbolic surface can be described by the lengths and twist of the curves of a pants decomposition. Fixing lengths and taking arbitrary twists creates an immersed torus inside the moduli space of curves, which turns out to be related to the unipotent-like ``earthquake flow.’’ Mirzakhani conjectured twist tori equidistribute as lengths are taken to infinity: in this talk, I will discuss joint work with James Farre in which we prove this conjecture along ``most’’ sequences. The key tool is a bridge that allows for the transfer of theorems between flat and hyperbolic geometry. (Online) |
14:30 - 15:00 |
Esther Banaian: Orbifold Markov Numbers ↓ It is known that Markov numbers can be viewed as specializations of cluster variables in the cluster algebra from a once-punctured torus. This connection has inspired formulas for Markov numbers involving continued fractions and these formulas in turn can be used to better understand Markov numbers. We consider similar formulas for solutions to several variants of the Markov equation coming from triangulated orbifolds. This is based on joint work with Archan Sen. (TCPL 201) |
15:15 - 15:45 |
Coffee Break (TCPL Foyer) |
15:45 - 16:15 |
Alfonso Sorrentino: Markov numbers, Fock's function, and Mather’s β function. ↓ In 1997, Fock introduced a fascinating function intrinsically linked to Markov numbers. In this talk, I will describe a joint work with A. Veselov, where we investigate the interplay between Fock's function, Federer–Gromov's stable norm in Riemannian geometry, and Mather's β-function in Hamiltonian dynamics. (TCPL 201) |
16:30 - 17:00 |
Elisa Bellah: Markoff Triples and Linear Recurrence Sequences ↓ In 2024, Chen built upon results of Bourgain, Gamburd, and Sarnak to prove that Strong Approximation holds for the Markoff surface for all but finitely many primes p. That is, the modulo p solutions to the Markoff equation are covered by the integer solutions if the prime p is large enough. In this talk, we discuss how certain associated order two linear recurrence sequences can help us study the size of integer lifts of Markoff mod p points. We also discuss how studying the periods of these sequences can help us guarantee paths between special families of Markoff mod p points for primes p where p+1 has large enough 2-adic valuation. (Online) |
17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |