2020.11.18 21:54

[Seminar 2020.11.25] On the ordering of the Markov numbers (AM 10:00~12:00)

조회 수 1534 추천 수 0 댓글 0
?

단축키

Prev이전 문서

Next다음 문서

크게 작게 위로 아래로 댓글로 가기 인쇄
?

단축키

Prev이전 문서

Next다음 문서

크게 작게 위로 아래로 댓글로 가기 인쇄
일정시작 2020-11-25 #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.

 제목+내용제목내용댓글이름닉네임아이디태그
List of Articles
번호 제목 글쓴이 날짜 조회 수
공지 QSMS Mailing List Registration 2021.09.09 1182
107 weekplan_672_2020 2020.11.18 0
106 [Series of lectures 2/21, 2/23, 2/28] An introduction to geometric representation theory and 3d mirror symmetry 2023.02.02 36
105 [2022 Global KMS International Conference 10/19 ~ 10/21] Related talks 2023.01.05 43
104 [QSMS Geometry Seminar 20221230] Morphism spaces between coisotropic A-branes 2022.12.27 63
103 [QSMS-BK21 Symplectic Geometry Seminar 2022-11-04] Introduction to Floer Homotopy: Kuranishi Structures 2022.10.26 65
102 [SNU Number Theory Seminar 2022-11-04] Approximations with mod 2 congruence conditions 2022.10.27 66
101 [QSMS Monthly Seminar 2022-12-02] Extended crystal structures of Hernandez-Leclerc categories 2022.12.02 71
100 [SNU Number Theory Seminar 2022-11-11] Mixed moments and the joint distribution of random groups 2022.10.24 75
99 [QSMS Monthly Seminar 2022-09-30] Monotone Lagrangians via toric degenerations: algebraic approach 2022.09.30 79
98 [QSMS Seminar 2022-12-2] N=2 supersymmetric W algebras from Hamiltonian reduction 2022.11.25 80
97 [QSMS Monthly Seminar 2022-10-28] Oscillator representations of quantum affine algebras   2022.10.30 81
96 [SNU Number Theory Seminar 2022-12-02] Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one 2022.11.28 82
95 [QSMS 집중강연 Representation theory 2023-02-13] Mini-course on quantum symmetric pairs II 2023.01.05 82
94 [QSMS-BK21 Symplectic Geometry Seminar 2022-12-09] Introduction to Floer Homotopy: Formal Groups and Complex Cobordism 2022.11.28 89
93 [QSMS-BK21 Symplectic Geometry Seminar 2022-09-30] Introduction to Floer Homotopy: Bordism and Flow Categories 2022.09.26 94
92 [SNU Number Theory Seminar 2022-09-16] Height-moduli and lower order terms 2022.09.06 103
91 [QSMS Seminar 2022-10-25] Rational liftings and gRSK 2022.10.08 123
90 [QSMS 집중강연 Representation theory 2023-02-07] Mini-course on quantum symmetric pairs I 2023.01.03 147
89 [QSMS Seminar 2022-09-27] Categories of Whittaker Modules over Lie superalgebras and Categorification of Fock Spaces 2022.09.20 150
88 [QSMS Seminar 2022-10-11] (q,t)-Cartan matrix and representation theories related to the specializations of the matrix 2022.09.18 183
Board Pagination Prev 1 2 3 4 5 6 Next
/ 6