2020.11.18 21:54

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

조회 수 18499 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 35802
152 [QSMS Seminar 2021.07.02] Lefschetz fibrations on cotangent bundles 2021.06.29 58711
151 [초청강연 2020.11.27] Zeta함수와 L함수에 대한 역사적 고찰II 2020.11.23 45592
150 [SNU Number Theory Seminiar] 2021.10.08 2021.08.23 36794
149 [QSMS Monthly Seminar] Peterson conjecture via Lagrangian correspondences and wonderful compactifications 2020.11.18 35077
148 [Seminar 05.14, 05.21, 05.28, 06.04] Introduction to derived categories 2021.04.16 31771
147 [Seminar 2021.03.03] Zeta함수와 L함수에 대한 역사적 고찰X 2021.03.02 24042
146 [Seminar 2021.04.02] Quantum affine analog of Kazhdan-Lusztig positivity for non-simply laced types via simply laced types 2021.03.30 23662
145 [Seminar 2021.04.16] Hecke system of harmonic Maass functions and applications to modular curves of higher genera 2021.04.19 21942
144 [QSMS Monthly Seminar] Canonical basis 2020.11.18 21107
143 [2021 KMS Spring Meeting] Related talks 2021.04.30 21037
142 [QSMS Monthly Seminar] Introduction to cluster algebras (and monoidal categorification) 2020.11.18 20536
141 [SNU Number Theory Seminar 31 Mar] Height gap theorem and almost law 2023.03.02 20492
140 [QSMS Monthly Seminar 2023-03-24] Rational points on abelian varieties 2023.03.24 20240
139 [SNU Number Theory Seminar 2022-05-13] A singular function containing all Lagrange numbers less than three 2022.04.12 20067
138 [QSMS Representation theory seminar - 2023-10-17] A Data-Scientific Approach to Representation Theory: Case Study of Kronecker Coefficients 2023.09.27 20054
137 [2022 Global KMS International Conference 10/19 ~ 10/21] Related talks 2023.01.05 20040
136 [SNU Number Theory Seminar 2022-04-15] A note on non-ordinary primes for some genus-zero arithmetic groups 2022.04.12 19935
135 [QSMS-BK21 Symplectic Seminar 10 Mar] Mirror Symmetry for Log Calabi-Yau Surfaces 2023.03.09 19776
134 [SNU Number Theory Seminar 14 Apr] Homological interpretation of Hecke \$L\$-values over real quadratic fields 2023.03.31 19767
133 [SNU Number Theory Seminar 6 ~ 7 Feb] Prismatic F-crystals and applications to p-adic Galois representations 2023.02.13 19707
Board Pagination Prev 1 2 3 4 5 6 7 8 Next
/ 8