?

Shortcut

PrevPrev Article

NextNext Article

Larger Font Smaller Font Up Down Go comment Print
?

Shortcut

PrevPrev Article

NextNext Article

Larger Font Smaller Font Up Down Go comment Print
Extra Form
일정시작 2020-11-25
배경색상 #FF5733

Date: November 25, 2020 

Time: AM10~12
Speaker: Kyungyong Lee (University of Nebraska-Linocln)

Title : On the ordering of the Markov numbers

 

Abstract:

The Markov numbers are the positive integers that appear in the solutions of the equation x2+y2+z2=3xyz. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics. It is known that the Markov numbers can be labeled by the lattice points (q,p) in the first quadrant and below the diagonal whose coordinates are coprime. In this paper, we consider the following question. Given two lattice points, can we say which of the associated Markov numbers is larger? A complete answer to this question would solve the uniqueness conjecture formulated by Frobenius in 1913. Using tools from cluster algebras, we give a partial answer in terms of the slope of the line segment that connects the two lattice points. We prove that the Markov number with the greater x-coordinate is larger than the other if the slope is at least −87 and that it is smaller than the other if the slope is at most −54. As a special case, namely when the slope is equal to 0 or 1, we obtain a proof of two conjectures from Aigner's book "Markov's theorem and 100 years of the uniqueness conjecture". This is joint work with Li Li, Michelle Rabideau, and Ralf Schiffler.

 

 

 

 

 

 

 


List of Articles
No. Subject Author Date Views
Notice QSMS Mailing List Registration qsms 2021.09.09 2472
22 [QSMS Monthly Seminar] Closed orbits and Beatty sequences qsms 2021.06.24 1605
21 [QSMS Monthly Seminar] Symmetric functions and super duality qsms 2021.05.28 1616
20 [정수론 집중강연 2021.08.03, 08.05] Geometric approaches to the local Langlands correspondence I-IV qsms 2021.07.29 1680
19 [Seminar 2021.04.23] Quiver Hecke algebras and unipotent quantum coordinate ring qsms 2021.04.09 1751
18 [Seminar 2021.04.02] CM congruence and the derivatives of p-adic L-functions for imaginary quadratic fields qsms 2021.04.19 1821
17 [Seminar 2021.05.28] Ramanujan continued fractions of order sixteen qsms 2021.05.21 1924
16 [Seminar 2020.12.03~05]Visual Studio Code를 사용하여 편리한 LaTeX 환경 구축하기 qsms 2020.12.02 1975
15 [Seminar 2021.06.15, 06.22] Cluster algebras with a view toward the Jacobian conjecture qsms 2021.05.28 2018
» [Seminar 2020.11.25] On the ordering of the Markov numbers (AM 10:00~12:00) qsms 2020.11.18 2024
13 [Seminar 2021.04.29] A realization of type A finite $W$-(super)algebra in terms of (super) Yangian qsms 2021.04.09 2480
12 [초청강연 2020.11.20] Zeta함수와 L함수에 대한 역사적 고찰 qsms 2020.11.18 3615
11 [2021 KMS Spring Meeting] Related talks qsms 2021.04.30 4772
10 [Seminar 2021.04.16] Hecke system of harmonic Maass functions and applications to modular curves of higher genera qsms 2021.04.19 5584
9 [QSMS Monthly Seminar] Introduction to cluster algebras (and monoidal categorification) qsms 2020.11.18 6383
8 [QSMS Monthly Seminar] Canonical basis qsms 2020.11.18 6750
7 [Seminar 2021.04.02] Quantum affine analog of Kazhdan-Lusztig positivity for non-simply laced types via simply laced types qsms 2021.03.30 8057
6 [Seminar 2021.03.03] Zeta함수와 L함수에 대한 역사적 고찰X qsms 2021.03.02 8961
5 [Seminar 05.14, 05.21, 05.28, 06.04] Introduction to derived categories qsms 2021.04.16 16506
4 [QSMS Monthly Seminar] Peterson conjecture via Lagrangian correspondences and wonderful compactifications qsms 2020.11.18 20557
3 [SNU Number Theory Seminiar] 2021.10.08 qsms 2021.08.23 21520
Board Pagination Prev 1 2 3 4 5 6 7 8 Next
/ 8