Schedule for: 25w5411 - Perspectives on Markov Numbers

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Meet and Greet at Professional Development Centre (PDC)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Alternate Title : Simple geodesics in arithmetic surfaces; Mapping class group dynamics and Diophantine approximation Abstract : We will introduce Markov numbers through the lens of hyperbolic geometry and mapping class group dynamics. After revisiting the work and expositions of Harvey Cohn, Andrew Haas and Caroline Series from a topological viewpoint, we will combine this picture with mapping class group dynamics or modular form arithmetic to propose new results and conjectures.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Cluster algebras are a class of combinatorial algebras introduced by Fomin and Zelevinsky in 2002. Rather than starting with a full list of generators, a cluster algebra begins with an initial seed, which contains a finite subset of generators as well as a combinatorial exchange relation. Then, we can find additional seeds and generators by mutating using this exchange relation. In this talk, we investigate a cluster algebra whose seeds specialize to Markov triples and discuss what has been learned by studying the additional cluster structure.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We describe all rational numbers with approximation constant at least 1/3 in terms of Markov fractions and their companions. In particular, a rational number has approximation constant exactly 1/3 if and only if it is the first left or right companion of a Markov fraction. The definitions and properties of Markov fractions and their companions are presented both algebraically and via hyperbolic geometry.

We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on $\mathbb P^2$ studied in 1980s by Dr\`ezet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov's result claiming that the ranks of the exceptional bundles on $\mathbb P^2$ are Markov numbers.

A conjecture of Baragar, Bourgain, Gamburd, and Sarnak states that the group generated by permutations and Vieta involutions should act transitively on the nonzero mod p points of the Markoff equation. While it is easy to see that this transitivity implies a strong approximation property for the equation (that its integral points surject onto its Z/p-points), we will explain how this also implies the connectedness of a certain moduli space of SL(2,p)-covers of elliptic curves. We will explain how this perspective helps to prove the transitivity/strong approximation conjecture, and how it also allows one to interpret Markoff numbers in terms of the monodromy of certain canonical covers/local systems on elliptic curves.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss a new proof of a special case of Will Chen’s theorem: that the number of vertices in a connected component of a Markoff mod p graph is divisible by p. We will also show how the proof can be generalized to provide information about arbitrary subgraphs of a Markoff mod p graph.

We use a combinatorial approach to Markov numbers as the number of perfect matchings of snake graphs. This approach was the main tool in a joint work with Lee, Li, and Rabideau to study the monotonicity of Markov numbers along lines of a fixed slope. As special cases, we obtained proofs for the three Aigner conjectures. If time permits, I will also discuss an extension of Markov's map, from Markov numbers to Lagrange numbers below 3, to a more general setting. On the combinatorial side this extension corresponds to replacing very special lattice paths, the Christoffel paths, by arbitrary lattice paths (below the diagonal).

We will discuss the geometry of groups generated by three $\pi$-rotations of the hyperbolic plane, their connection to Markov numbers and to mutations of quivers of rank 3. We also apply geometry of these groups to illustrate and interpret some known properties of Markov numbers. The talk is based on joint works with Pavel Tumarkin.

The Markov numbers can be generated by evaluating the initial cluster variables of the cluster algebra associated to a once punctured torus at 1. In joint work with Zachary Greenberg and Anna Wienhard, we use the non-commutative cluster algebra of Berenstein and Retakh associated to a punctured torus to define a noncommutative analog of the Markov equation. Given any ring with involution we can choose some valid collection of initial elements to evaluate the cluster variables of this algebra at to obtain non-commutative Markov numbers. In this talk I will give an overview of this construction along with many examples, including constructions of polynomial deformations of Markov numbers, Markov numbers over the Dual numbers, and group-ring Markov numbers.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Speakers TBD

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The number of Markoff triples \((a,b,c)\) with \(a\leq b\leq c \leq B\) is \[k\log(B)^2+O(\log(B)(\log\log B)^2,\] with \(k\) an explicitly computable constant (Zagier, 1982). The Markoff-Hurwitz equation \[w^2+x^2+y^2+z^2=4wxyz\] has an analogous tree of integer solutions, but the asymptotics are fractal: The number of integer solutions \((a,b,c,d)\) with \(0\leq a\leq b\leq c\leq d\leq B\) grows asymptotically like \[k(\log B)^\beta\], where \(\beta\) 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.

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.

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.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

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.

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.

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.

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.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.