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過去の記録
過去の記録 ～06/07｜本日 06/08 | 今後の予定 06/09～
2023年02月06日(月)
応用解析セミナー
16:00-18:10 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催（通常と開催曜日が異なりますのでご注意下さい）
Marek Fila 氏 (Comenius University) 16:00-17:00
Solutions with moving singularities for nonlinear diffusion equations (ENGLISH)
Fast diffusion equation: uniqueness of solutions with a moving singularity (ENGLISH)
https://forms.gle/nKa4XATuuGPwZWbUA
対面・オンラインハイブリッド開催（通常と開催曜日が異なりますのでご注意下さい）
Marek Fila 氏 (Comenius University) 16:00-17:00
Solutions with moving singularities for nonlinear diffusion equations (ENGLISH)
[ 講演概要 ]
We give a survey of results on solutions with singularities moving along a prescribed curve for equations of fast diffusion or porous medium type. These results were obtained in collaboration with J.R. King, P. Mackova, J. Takahashi and E. Yanagida.
Petra Mackova 氏 (Comenius University) 17:10-18:10We give a survey of results on solutions with singularities moving along a prescribed curve for equations of fast diffusion or porous medium type. These results were obtained in collaboration with J.R. King, P. Mackova, J. Takahashi and E. Yanagida.
Fast diffusion equation: uniqueness of solutions with a moving singularity (ENGLISH)
[ 講演概要 ]
This talk focuses on open questions in the area of the uniqueness of distributional solutions of the fast diffusion equation with a given source term. The existence of different sets of such solutions is known from previous research, and the natural next issue is to examine their uniqueness. Assuming that the source term is a measure, the existence of different classes of solutions is known, however, their uniqueness is an open problem. The existence of a class of asymptotically radially symmetric solutions with a singularity that moves along a prescribed curve was proved by M. Fila, J. Takahashi, and E. Yanagida. More recently, it has been established by M. Fila, P. M., J. Takahashi, and E. Yanagida that these solutions solve the corresponding problem with a moving Dirac source term. In this talk, we discuss the uniqueness of these solutions. This is a joint work with M. Fila.
[ 参考URL ]This talk focuses on open questions in the area of the uniqueness of distributional solutions of the fast diffusion equation with a given source term. The existence of different sets of such solutions is known from previous research, and the natural next issue is to examine their uniqueness. Assuming that the source term is a measure, the existence of different classes of solutions is known, however, their uniqueness is an open problem. The existence of a class of asymptotically radially symmetric solutions with a singularity that moves along a prescribed curve was proved by M. Fila, J. Takahashi, and E. Yanagida. More recently, it has been established by M. Fila, P. M., J. Takahashi, and E. Yanagida that these solutions solve the corresponding problem with a moving Dirac source term. In this talk, we discuss the uniqueness of these solutions. This is a joint work with M. Fila.
https://forms.gle/nKa4XATuuGPwZWbUA
代数幾何学セミナー
13:00-14:30 数理科学研究科棟(駒場) 123号室
レクチャーシリーズ第２回目
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties. (English)
レクチャーシリーズ第２回目
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties. (English)
[ 講演概要 ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
2023年01月31日(火)
代数幾何学セミナー
14:30-16:00 数理科学研究科棟(駒場) ハイブリッド開催/002号室
いつもと時間が異なります.
Shiji Lyu 氏 (プリンストン大学)
Some properties of splinters via ultrapower (English)
いつもと時間が異なります.
Shiji Lyu 氏 (プリンストン大学)
Some properties of splinters via ultrapower (English)
[ 講演概要 ]
A Noetherian (reduced) ring is called a splinter if it is a direct summand of every finite ring extension of it. This notion is related to various interesting notions of singularities, but far less properties are known about splinters.
In this talk, we will discuss the question of "regular ascent"; in the simplest (but already essential) form, we ask, for a Noetherian splinter R, is the polynomial ring R[X] always a splinter. We will see how ultrapower, a construction mainly belonging to model theory, is involved.
A Noetherian (reduced) ring is called a splinter if it is a direct summand of every finite ring extension of it. This notion is related to various interesting notions of singularities, but far less properties are known about splinters.
In this talk, we will discuss the question of "regular ascent"; in the simplest (but already essential) form, we ask, for a Noetherian splinter R, is the polynomial ring R[X] always a splinter. We will see how ultrapower, a construction mainly belonging to model theory, is involved.
2023年01月27日(金)
代数幾何学セミナー
13:00-14:30 数理科学研究科棟(駒場) 056号室
全4回:1/27 (金) 13:00―14:30, 数理科学研究科056室 2/6 (月) 13:00―14:30, 数理科学研究科123室, 2/17 (金) 10:00―11:30, 数理科学研究科123室, 2/20 (月) 10:00ー11:30, 数理科学研究科056室
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties (English)
全4回:1/27 (金) 13:00―14:30, 数理科学研究科056室 2/6 (月) 13:00―14:30, 数理科学研究科123室, 2/17 (金) 10:00―11:30, 数理科学研究科123室, 2/20 (月) 10:00ー11:30, 数理科学研究科056室
Chenyang Xu 氏 (プリンストン大学)
K-stability of Fano varieties (English)
[ 講演概要 ]
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
The notion of K-stability of Fano varieties was first introduced to characterize the existence of Kahler-Einstein metric. Recently, a purely algebro-geometric theory has been developed and it has yielded many striking results, such as the solution of the Yau-Tian-Donaldson Conjecture for all Fano varieties, as well as the construction of a projective moduli scheme, called K-moduli, parametrizing K-polystable Fano varieties.
In this lecture series, I will survey the recent progress. The first two lectures will be devoted to explain the evolution of algebraic geometer’s understanding of various aspects of the notion of K-stability. The Lecture 3 and 4 will be devoted to discuss the construction of the K-moduli space.
博士論文発表会
9:15-10:30 数理科学研究科棟(駒場) 118号室
松本 圭峰 氏 (東京大学大学院数理科学研究科)
Integral Derived Invariants and Motives
(整数導来不変量とモチーフ)
松本 圭峰 氏 (東京大学大学院数理科学研究科)
Integral Derived Invariants and Motives
(整数導来不変量とモチーフ)
博士論文発表会
9:15-10:30 数理科学研究科棟(駒場) 122号室
矢部 貴大 氏 (東京大学大学院数理科学研究科)
On classification of 2-generated axial algebras of Jordan and Majorana type
(Jordan型及びMajorana型の二元生成軸代数の分類について)
矢部 貴大 氏 (東京大学大学院数理科学研究科)
On classification of 2-generated axial algebras of Jordan and Majorana type
(Jordan型及びMajorana型の二元生成軸代数の分類について)
博士論文発表会
9:15-10:30 数理科学研究科棟(駒場) 126号室
鶴橋 知典 氏 (東京大学大学院数理科学研究科)
On microscopic interpretation for convex integration and self- similar structure of vortices in turbulence
(凸積分法に関する微視的表現と乱流渦の自己相似構造について)
鶴橋 知典 氏 (東京大学大学院数理科学研究科)
On microscopic interpretation for convex integration and self- similar structure of vortices in turbulence
(凸積分法に関する微視的表現と乱流渦の自己相似構造について)
博士論文発表会
11:00-12:15 数理科学研究科棟(駒場) 118号室
佐藤 謙 氏 (東京大学大学院数理科学研究科)
A group action on higher Chow cycles on a family of Kummer surfaces
(あるクンマー曲面族の上の高次チャウサイクルへの群作用について)
佐藤 謙 氏 (東京大学大学院数理科学研究科)
A group action on higher Chow cycles on a family of Kummer surfaces
(あるクンマー曲面族の上の高次チャウサイクルへの群作用について)
博士論文発表会
11:00-12:15 数理科学研究科棟(駒場) 122号室
原子 秀一 氏 (東京大学大学院数理科学研究科)
Manifolds Graded by an Arbitrary Abelian Group
(任意のアーベル群で次数付けられた多様体)
原子 秀一 氏 (東京大学大学院数理科学研究科)
Manifolds Graded by an Arbitrary Abelian Group
(任意のアーベル群で次数付けられた多様体)
博士論文発表会
11:00-12:15 数理科学研究科棟(駒場) 126号室
佐藤 翔一 氏 (東京大学大学院数理科学研究科)
Various problems for properties of solutions to fractional partial differential equations
(非整数階偏微分方程式の解の性質に関する諸問題)
佐藤 翔一 氏 (東京大学大学院数理科学研究科)
Various problems for properties of solutions to fractional partial differential equations
(非整数階偏微分方程式の解の性質に関する諸問題)
博士論文発表会
13:00-14:15 数理科学研究科棟(駒場) 118号室
査 承晗 氏 (東京大学大学院数理科学研究科)
Integral Structures in the Local Algebra of a Singularity
(特異点の局所代数の整構造について)
査 承晗 氏 (東京大学大学院数理科学研究科)
Integral Structures in the Local Algebra of a Singularity
(特異点の局所代数の整構造について)
博士論文発表会
13:00-14:15 数理科学研究科棟(駒場) 122号室
里見 貴志 氏 (東京大学大学院数理科学研究科)
Refinement of Young’s convolution inequality on locally compact groups and generalizations of related inequalities
(局所コンパクト群上のYoung の畳み込み不等式の精密化と関連の不等式の拡張)
里見 貴志 氏 (東京大学大学院数理科学研究科)
Refinement of Young’s convolution inequality on locally compact groups and generalizations of related inequalities
(局所コンパクト群上のYoung の畳み込み不等式の精密化と関連の不等式の拡張)
2023年01月26日(木)
博士論文発表会
9:15-10:30 数理科学研究科棟(駒場) 122号室号室
奥田 伸樹 氏 (東京大学大学院数理科学研究科)
Fourier-Mukai transforms for non-commutative complex tori
(非可換複素トーラスのフーリエ・向井変換)
奥田 伸樹 氏 (東京大学大学院数理科学研究科)
Fourier-Mukai transforms for non-commutative complex tori
(非可換複素トーラスのフーリエ・向井変換)
博士論文発表会
11:00-12:15 数理科学研究科棟(駒場) 122号室
浅香 猛 氏 (東京大学大学院数理科学研究科)
Earthquake theorem and cluster algebras
(地震定理とクラスター代数)
浅香 猛 氏 (東京大学大学院数理科学研究科)
Earthquake theorem and cluster algebras
(地震定理とクラスター代数)
博士論文発表会
11:00-12:15 数理科学研究科棟(駒場) 126号室
林 晃平 氏 (東京大学大学院数理科学研究科)
On universality of the Kardar-Parisi-Zhang equation in high temperature regime
(高温相におけるKardar-Parisi-Zhang 方程式の普遍性について)
林 晃平 氏 (東京大学大学院数理科学研究科)
On universality of the Kardar-Parisi-Zhang equation in high temperature regime
(高温相におけるKardar-Parisi-Zhang 方程式の普遍性について)
博士論文発表会
13:00-14:15 数理科学研究科棟(駒場) 118号室
王 龍 氏 (東京大学大学院数理科学研究科)
Studies on the Cone Conjecture, Automorphisms, and Arithmetic Degrees
(錐予想, 自己同型と算術次数の研究)
王 龍 氏 (東京大学大学院数理科学研究科)
Studies on the Cone Conjecture, Automorphisms, and Arithmetic Degrees
(錐予想, 自己同型と算術次数の研究)
博士論文発表会
13:00-14:15 数理科学研究科棟(駒場) 122号室
キム ミンギュ 氏 (東京大学大学院数理科学研究科)
Finite path integral model and toric code based on homological algebra
(ホモロジー代数に基づく有限経路積分モデルとトーリックコード)
キム ミンギュ 氏 (東京大学大学院数理科学研究科)
Finite path integral model and toric code based on homological algebra
(ホモロジー代数に基づく有限経路積分モデルとトーリックコード)
博士論文発表会
14:45-16:00 数理科学研究科棟(駒場) 118号室
鶴崎 修功 氏 (東京大学大学院数理科学研究科)
Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by sl2 subalgebras which are generalizations of principal sl2 subalgebras
(主sl2部分代数の一般化であるsl2部分代数によるrank2対称双曲型Kac-Moody Lie 代数の既約分解)
鶴崎 修功 氏 (東京大学大学院数理科学研究科)
Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by sl2 subalgebras which are generalizations of principal sl2 subalgebras
(主sl2部分代数の一般化であるsl2部分代数によるrank2対称双曲型Kac-Moody Lie 代数の既約分解)
2023年01月20日(金)
談話会・数理科学講演会
15:30-16:30 ハイブリッド開催
数理科学研究科所属以外の方は、オンラインでのご参加（参考URLから参加登録）をお願いいたします。
Mikhail Bershtein 氏 (HSE大学, Skoltech)
Kyiv formula and its applications (ENGLISH)
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUrduioqjouG9wBfhl35VPxN_K92oa1wB4P
数理科学研究科所属以外の方は、オンラインでのご参加（参考URLから参加登録）をお願いいたします。
Mikhail Bershtein 氏 (HSE大学, Skoltech)
Kyiv formula and its applications (ENGLISH)
[ 講演概要 ]
The Kyiv formula gives the generic tau function of Painleve' equation (and more generally isomonodromy deformation equations) in terms of conformal blocks or Nekrasov partition function. I will explain the statement, examples and different approaches to the proof. If time permits, I will discuss some applications of this formula.
[ 参考URL ]The Kyiv formula gives the generic tau function of Painleve' equation (and more generally isomonodromy deformation equations) in terms of conformal blocks or Nekrasov partition function. I will explain the statement, examples and different approaches to the proof. If time permits, I will discuss some applications of this formula.
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUrduioqjouG9wBfhl35VPxN_K92oa1wB4P
東京名古屋代数セミナー
10:30-12:00 オンライン開催
オンライン開催の詳細は講演参考URLをご覧ください。
狩野 隼輔 氏 (東北大学)
Tropical cluster transformations and train track splittings (Japanese)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
オンライン開催の詳細は講演参考URLをご覧ください。
狩野 隼輔 氏 (東北大学)
Tropical cluster transformations and train track splittings (Japanese)
[ 講演概要 ]
Fock-Goncharovは箙に対し、クラスター代数と呼ばれる組み合わせ構造を持つような概形であるクラスター多様体を定義した。
この概形は良い正値性を持つことから、半体値集合を考えることができる。
箙が点付き曲面の三角形分割から得られるとき、トロピカル半体値集合は曲面の測度付き葉層構造の空間の適切な拡張と同一視される。
クラスター多様体のトロピカル半体値集合はクラスター構造から定まるPL構造を持つが、一方で曲面の測度付き葉層構造の空間にはトレイントラックと呼ばれるグラフを用いたPL構造が定まることが知られている。
本講演では、Goncharov-Shenのクラスター多様体上のLandau-Ginzburgポテンシャル関数のトロピカル化を通してトレイントラックを翻訳し、2つのPL構造が同値であることを確認する。
またこれの応用として、一般の擬Anosov写像類が符号安定性と呼ばれる性質を持つことを説明する。
ミーティングID: 820 6834 6105
パスコード: 039914
[ 参考URL ]Fock-Goncharovは箙に対し、クラスター代数と呼ばれる組み合わせ構造を持つような概形であるクラスター多様体を定義した。
この概形は良い正値性を持つことから、半体値集合を考えることができる。
箙が点付き曲面の三角形分割から得られるとき、トロピカル半体値集合は曲面の測度付き葉層構造の空間の適切な拡張と同一視される。
クラスター多様体のトロピカル半体値集合はクラスター構造から定まるPL構造を持つが、一方で曲面の測度付き葉層構造の空間にはトレイントラックと呼ばれるグラフを用いたPL構造が定まることが知られている。
本講演では、Goncharov-Shenのクラスター多様体上のLandau-Ginzburgポテンシャル関数のトロピカル化を通してトレイントラックを翻訳し、2つのPL構造が同値であることを確認する。
またこれの応用として、一般の擬Anosov写像類が符号安定性と呼ばれる性質を持つことを説明する。
ミーティングID: 820 6834 6105
パスコード: 039914
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023年01月19日(木)
情報数学セミナー
16:50-18:35 数理科学研究科棟(駒場) 123号室
成定 真太郎 氏 (KDDI総合研究所)
符号暗号とその求解アルゴリズム (Japanese)
成定 真太郎 氏 (KDDI総合研究所)
符号暗号とその求解アルゴリズム (Japanese)
[ 講演概要 ]
耐量子暗号の候補である符号暗号について紹介するとともに、符号暗号の解読アルゴリズムであるInformation Set Decoding (ISD)について解説する。また、ISDの量子アルゴリズムについて紹介する。
耐量子暗号の候補である符号暗号について紹介するとともに、符号暗号の解読アルゴリズムであるInformation Set Decoding (ISD)について解説する。また、ISDの量子アルゴリズムについて紹介する。
2023年01月18日(水)
代数学コロキウム
17:00-18:00 ハイブリッド開催
Kestutis Cesnavicius 氏 (Paris-Saclay University)
The affine Grassmannian as a presheaf quotient (English)
Kestutis Cesnavicius 氏 (Paris-Saclay University)
The affine Grassmannian as a presheaf quotient (English)
[ 講演概要 ]
The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.
The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.
2023年01月17日(火)
トポロジー火曜セミナー
17:00-18:00 オンライン開催
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
Chenghan Zha 氏 (東京大学大学院数理科学研究科)
Integral structures in the local algebra of a singularity (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
Chenghan Zha 氏 (東京大学大学院数理科学研究科)
Integral structures in the local algebra of a singularity (ENGLISH)
[ 講演概要 ]
We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor lattice can be identified with an appropriate relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity. Furthermore, we figure out the image of the Milnor lattice of the singularity of an invertible polynomial of chain type using the basis of middle homology constructed by Otani-Takahashi. We calculated the Seifert form of the basis as well.
[ 参考URL ]We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor lattice can be identified with an appropriate relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity. Furthermore, we figure out the image of the Milnor lattice of the singularity of an invertible polynomial of chain type using the basis of middle homology constructed by Otani-Takahashi. We calculated the Seifert form of the basis as well.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2023年01月16日(月)
複素解析幾何セミナー
10:30-12:00 数理科学研究科棟(駒場) 128号室
対面・オンラインのハイブリッド形式で行います。オンライン参加される場合は参考URLからご登録ください。
小池貴之 氏 (大阪公立大学)
Holomorphic foliation associated with a semi-positive class of numerical dimension one (Japanese)
https://forms.gle/hYT2hVhDE3q1wDSh6
対面・オンラインのハイブリッド形式で行います。オンライン参加される場合は参考URLからご登録ください。
小池貴之 氏 (大阪公立大学)
Holomorphic foliation associated with a semi-positive class of numerical dimension one (Japanese)
[ 講演概要 ]
Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bidegree $(1,1)$ on $X$.
When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero.
As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.
[ 参考URL ]Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bidegree $(1,1)$ on $X$.
When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero.
As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.
https://forms.gle/hYT2hVhDE3q1wDSh6
2023年01月13日(金)
離散数理モデリングセミナー
13:15-14:45 数理科学研究科棟(駒場) 126号室
Andy Hone 氏 (University of Kent)
An infinite sequence of Heron triangles with two rational medians (English)
Andy Hone 氏 (University of Kent)
An infinite sequence of Heron triangles with two rational medians (English)
[ 講演概要 ]
Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, despite an assertion by Schubert that even two rational medians are impossible, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z x Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.
Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, despite an assertion by Schubert that even two rational medians are impossible, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z x Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.
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