過去の記録

過去の記録 ~02/15本日 02/16 | 今後の予定 02/17~

代数幾何学セミナー

13:30-15:00   数理科学研究科棟(駒場) 122号室
今週は月曜日にセミナーを行います。13:30-15:00と15:30-17:00の2講演あります。This week's seminar will be held on Monday and consist of two lectures: 13:30-15:00 and 15:30-17:00.
Will Donovan 氏 (IPMU)
Perverse sheaves of categories and birational geometry (English)
[ 講演概要 ]
Kapranov and Schechtman have initiated a program to study perverse sheaves of categories, or perverse schobers. It is expected that examples arise from birational geometry, in particular from webs of flops. I explain progress towards constructing these objects for Grothendieck resolutions (work of the above authors with Bondal), and for 3-folds (joint work of myself and Wemyss).

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
井上瑛二 氏 (東京大学)
Kähler-Ricci soliton, K-stability and moduli space of Fano
manifolds (JAPANESE)
[ 講演概要 ]
Kähler-Ricci soliton is a kind of canonical metrics on Fano manifolds and is a natural generalization of Kähler-Einstein metric in view of Kähler-Ricci flow.

In this talk, I will explain the following good geometric features of Fano manifolds admitting Kähler-Ricci solitons:
1. Volume minimization, reductivity and uniqueness results established by Tian&Zhu.
2. Relation to algebraic (modified) K-stability estabilished by Berman&Witt-Niström and Datar&Székelyhidi.
3. Moment map picture for Kähler-Ricci soliton (‘real side’)
4. Moduli stack (‘virtual side’) and moduli space of them

A result in 1 is indispensable for the formulation in 3 and 4, and explains why we should consider solitons, beyond Einstein metrics. I also show an essential idea in the construction of the moduli space of Fano manifolds admitting Kähler-Ricci solitons and give some remarks on technical key point.

2018年05月16日(水)

FMSPレクチャーズ

14:45-15:45   数理科学研究科棟(駒場) 122号室
M.M. Lavrentʼev, Jr. 氏 (Novosibirsk State University)
Some strongly degenerate parabolic equations (joint with Prof. A. Tani) (ENGLISH)
[ 講演概要 ]
We consider some nonlinear 1D parabolic equations with the positive leading coefficient which is not away from zero. "Hyperbolic phenomena" (gradient blowing up phenomena) were reported in literature for such models. We describe special cases of regular solvability for degenerate equations under study.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_MMLavrentev.pdf

2018年05月15日(火)

トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
岡 睦雄 氏 (東京理科大学)
超曲面混合特異点理論とある予想 (JAPANESE)
[ 講演概要 ]
Consider a real algebraic variety of real codimension 2 defined by $V:=\{g(\mathbf x,\mathbf y)=h(\mathbf x,\mathbf y)=0\}$ in $\mathbb C^n=\mathbb R^n\times \mathbb R^n$. Put $\mathbf z=\mathbf x+i\mathbf y$ and consider complex valued real analytic function $f=g+ih$. Replace the variables $x_1,y_1\dots, x_n,y_n$ using the equality $x_j=(z_j+\bar z_j)/2,\, y_j=(z_j-\bar z_j)/2i$. Then $f$ can be understood to be an analytic functions of $z_j,\bar z_j$. We call $f$ a mixed function. In this way, $V=\{f(\mathbf z,\bar{\mathbf z})=0\}$ and we can use the techniques of complex analytic functions and the singularity theory developed there. In this talk, we explain basic properties of the singularity of mixed hyper surface $V(f)$ and give several open questions.

2018年05月14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
山田澄生 氏 (学習院大学)
Harmonic map and the Einstein equation in five dimension (JAPANESE)
[ 講演概要 ]
We present a new method in constructing 5-dimensional stationary solutions to the vacuum Einstein equation. In 1917, H. Weyl expressed the Schwarzschild black hole solution using a cylindical coordinate system, and consequently realized that the metric is completely determined by a harmonic function. Since then, the relation between harmonic maps and the Einstein equation has been explored mostly by physicists, which they call the sigma model of the Einstein equation. In this talk, after explaining the historical background, we demonstrate that in 5D, the Einstein spacetimes can have a wide range of black hole horizons in their topological types. In particular we establish an existence theorem of harmonic maps, which subsequently leads to constructions of 5D spacetimes with black hole horizons of positive Yamabe types, namely $S^3$, $S^2 \times S^1$, and the lens space $L(p,q)$. This is a joint work with Marcus Khuri and Gilbert Weinstein.

東京確率論セミナー

16:00-17:30   数理科学研究科棟(駒場) 126号室
村山 拓也 氏 (京都大学大学院理学研究科)
Chordal Komatu-Loewner equation for a family of continuously growing hulls (JAPANESE)
[ 講演概要 ]
Loewner方程式は,複素平面上の単連結な領域における単葉函数族の極値問題に古くから用いられてきた.近年では,統計力学模型のスケーリング極限を記述する確率的Loewner発展(SLE)の構成に応用され,函数論,確率論,数理物理と様々な分野にまたがって注目を受けている.本講演ではこの方程式を多重連結領域へと拡張したKomatu-Loewner方程式について紹介する.特に,先行研究の結果を「連続な」増大殻(hull)へ一般化することで,これまで上手く扱えなかった多重連結領域上の問題に対し新たなアプローチが得られる様子を概説する.

2018年05月11日(金)

FMSPレクチャーズ

15:00-17:00   数理科学研究科棟(駒場) 123号室
全5回講演の(5)
Sug Woo Shin 氏 (University of California, Berkeley)
Introduction to the Langlands-Rapoport conjecture (ENGLISH)
[ 講演概要 ]
In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.
(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.
(ii) Statement of the conjecture: After setting up the language of
Galois gerbs, we state the Langlands-Rapoport conjecture.
(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.
(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of
Hecke-Frobenius correspondences.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SugWooShin.pdf

講演会

13:00-14:00   数理科学研究科棟(駒場) 123号室
Alex Youcis 氏 (University of California, Berkeley)
The Langlands-Kottwitz method for deformation spaces of Hodge type
[ 講演概要 ]
Cohomology of global Shimura varieties is an object of universal importance in the Langlands program. Given a Shimura datum (G,X) and a (sufficiently nice) representation ¥xi of G, one obtains an l-adic sheaf F_{¥xi,l} on Sh(G,X) with a G(A_f)-structure. Thus, in the standard way, the cohomology group H^*(Sh(G,X),F_¥xi) has an admissible action of Gal(¥overline{E}/E) ¥times G(A_f), where E=E(G,X) is the reflex field of (G,X). Extending work of Kottwitz, Scholze, and others we discuss a method for computing the traces of this action, more specifically of an element ¥tau ¥times g where ¥tau ¥in W_{E_¥mathfrak{p}} for some prime ¥mathfrak{p} of E dividing p and g ¥in G(A_f^p) ¥times G(Z_p), in terms of a weighted point count on the Shimura variety's special fiber, as well as the traces of various local Shimura varieties over E_¥mathfrak{p}, at least in the case when (G,X) is a abelian-type Shimura datum unramified at p.

談話会・数理科学講演会

15:30-16:30   数理科学研究科棟(駒場) 056号室
入江 慶 氏 (東京大学大学院数理科学研究科)
周期Reeb軌道および極小超曲面に対する生成的(generic)稠密定理

(日本語)
[ 講演概要 ]
次のふたつの結果について説明する:(1)3次元閉多様体上の$C^\infty$位相についてgenericなReeb力学系において,周期軌道が稠密に存在する(講演者).(2)次元が3以上7以下の閉多様体上の$C^\infty$位相についてgenericなRiemann計量において,極小閉超曲面が稠密に存在する(Marques-Neves-講演者).

(1)の証明にはHutchings等によるEmbedded Contact Homologyの理論,(2)の証明にはMarques-Neves等によるAlmgren-Pitts理論の最近の進展を用いる.これらは技術的には相当異なる理論であるが,どちらも無限次元空間上のMorse理論(あるいはmin-max理論)といえるもので,結果として定義されるmin-max値はいくつかのよく似た性質を満たす.特に,これらのmin-max値の漸近挙動から多様体の体積が復元されるという性質(Laplacianの固有値に対するWeylの法則の類似)が,いずれの証明においても重要な役割を果たす.

2018年05月10日(木)

FMSPレクチャーズ

15:00-17:00   数理科学研究科棟(駒場) 123号室
全5回講演の(4)
Sug Woo Shin 氏 (University of California, Berkeley)
Introduction to the Langlands-Rapoport conjecture (ENGLISH)
[ 講演概要 ]
In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.
(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.
(ii) Statement of the conjecture: After setting up the language of
Galois gerbs, we state the Langlands-Rapoport conjecture.
(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.
(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of
Hecke-Frobenius correspondences.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SugWooShin.pdf

講演会

11:00-12:00   数理科学研究科棟(駒場) 123号室
Alexander Bertoloni Meli 氏 (University of California, Berkeley)
The Cohomology of Rapoport-Zink Spaces of EL-Type

[ 講演概要 ]
I will discuss Rapoport-Zink spaces of EL-type and how to explicitly compute a certain variant of their cohomology in terms of the local Langlands correspondence for general linear groups. I will then show how this computation can be used to resolve certain cases of a conjecture of Harris.

2018年05月09日(水)

代数学コロキウム

17:30-18:30   数理科学研究科棟(駒場) 056号室
Sug Woo Shin 氏 (University of California, Berkeley)
Endoscopy and cohomology of U(n-1,1) (ENGLISH)
[ 講演概要 ]
We apply the endoscopic classification of automorphic representations for inner forms of unitary groups to bound the growth of cohomology in congruence towers of locally symmetric spaces associated with U(n-1,1). Our bound is sharper than the bound predicted by Sarnak-Xue for general locally symmetric spaces. This is joint work with Simon Marshall.

(本講演は「東京北京パリ数論幾何セミナー」として, インターネットによる東大数理, Morningside Center of Mathematics と IHES の双方向同時中継で行います.今回は東京からの中継です.)

FMSPレクチャーズ

15:00-17:00   数理科学研究科棟(駒場) 123号室
全5回講演の(3)
Sug Woo Shin 氏 (University of California, Berkeley)
Introduction to the Langlands-Rapoport conjecture (ENGLISH)
[ 講演概要 ]
In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.
(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.
(ii) Statement of the conjecture: After setting up the language of
Galois gerbs, we state the Langlands-Rapoport conjecture.
(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.
(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of
Hecke-Frobenius correspondences.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SugWooShin.pdf

作用素環セミナー

16:45-18:15   数理科学研究科棟(駒場) 126号室
磯野優介 氏 (京大数理研)
Factoriality, Connes' invariants and fullness of amalgamated free products (English)

2018年05月08日(火)

FMSPレクチャーズ

15:00-17:00   数理科学研究科棟(駒場) 123号室
全5回講演の(2)
Sug Woo Shin 氏 (University of California, Berkeley)
Introduction to the Langlands-Rapoport conjecture (ENGLISH)
[ 講演概要 ]
In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.
(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.
(ii) Statement of the conjecture: After setting up the language of
Galois gerbs, we state the Langlands-Rapoport conjecture.
(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.
(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of
Hecke-Frobenius correspondences.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SugWooShin.pdf

トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Dan Cristofaro-Gardiner 氏 (University of California, Santa Cruz)
Beyond the Weinstein conjecture (ENGLISH)
[ 講演概要 ]
The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室
鈴木拓 氏 (宇都宮大)
Higher order families of lines and Fano manifolds covered by linear
spaces
(Japanese (writing in English))
[ 講演概要 ]
In this talk, for an embedded Fano manifold $X$, we introduce higher
order families of lines and a new invariant $S_X$. They are line
versions of higher order minimal families of rational curves and the
invariant $N_X$ which were introduced in my previous talk on 4th
November 2016. In addition, $S_X$ is related to the dimension of
covering linear spaces. Our goal is to classify Fano manifolds $X$ which
have large $S_X$.

講演会

13:00-14:00   数理科学研究科棟(駒場) 122号室
Sander Mack-Crane 氏 (University of California, Berkeley)
Langlands-Rapoport for the Modular Curve

[ 講演概要 ]
We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.

数値解析セミナー

16:50-18:20   数理科学研究科棟(駒場) 002号室
齊藤宣一 氏 (東京大学大学院数理科学研究科)
数値解析の諸相 (Japanese)
[ 講演概要 ]
本講演の前半では,さまざまな反例を検討することで,数値計算手法の収束性の研究の重要性を再確認したい.とくに,偏微分方程式の数値解析においては,解の正則性,特異性,様々な摂動に対する安定性に正面から取り組むことが必須になる.具体的には,正則性の欠如や領域の近似が原因で,近似解の収束が任意に遅くなったり,意図しない問題を正しく解いてしまうことがありうることを紹介する.後半は,数学を専門とする学生・院生に対する数値解析の教育(やユーザーへの啓蒙)について,講演者自身の反省を述べたい.

多くの皆様のご協力のおかげで,本セミナーは100回目を迎えることができました. 第100回目は一つの総括の意味を込めて,主催者の一人による初歩的あるいは総括的講演を,あまり形式張らずに行いたいと思います.参加者の方々と議論ができれば幸いです.

統計数学セミナー

15:00-16:10   数理科学研究科棟(駒場) 052号室
千葉 航平 氏 (東京大学)
LAN property for stochastic differential equations driven by fractional Brownian motion of Hurst parameter 1/4 < H < 1/2
[ 講演概要 ]
We consider the problem of estimating the drift parameter of solution to the stochastic differential equation driven by a fractional Brownian motion with Hurst parameter less than 1/2 under complete observation. We derive a formula for the likelihood ratio and prove local asymptotic normality when 1/4 < H < 1/2. Our result shows that the convergence rate is $T^{-1/2}$ for the parameters satisfying a certain equation and $T^{-(1-H)}$ for the others.
In this talk, we outline the proof of local asymptotic normality and explain how the different rates of convergence occur and where we use the assumption H > 1/4. We also mention some remaining problems and future directions. This talk is based on arXiv:1804.04108.

2018年05月07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
林本厚志 氏 (長野高専)
Proper holomorphic mappings and generalized pseudoellipsoids (JAPANESE)
[ 講演概要 ]
We study the classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions.

Huang proved some classification theorems of proper holomorphic mappings between balls of different dimensions, which are called gap theorems. Our present theorems are their weakly pseudoconvex versions.

In the theorem, classified mapping is so-called a variables splitting mapping and each component is derived from a homogeneous proper polynomial mapping between balls.

The essential methods are the ''good'' decompositions of CR vector bundle and reduction the mapping under consideration to the mapping of balls. By this reduction, we can apply Huang's gap theorem.

東京確率論セミナー

16:00-17:30   数理科学研究科棟(駒場) 126号室
沙川貴大 氏 (東京大学工学部)
孤立量子多体系における熱力学第二法則
(JAPANESE)
[ 講演概要 ]
可逆な量子力学から不可逆な熱力学が如何にして創発するかは、19世紀以来の物理学の難問の一つである。本講演では、多体系の量子力学に基づいて熱力学を理解する研究の背景と、最近の我々の結果について紹介する。我々は、熱浴の初期状態がエネルギー固有状態で時間発展がユニタリの場合について、熱力学第二法則および「ゆらぎの定理」と呼ばれる関係式を厳密に証明した[1]。その際の重要な概念は、固有状態熱化仮説とLieb-Robinson限界である。本講演では、固有状態熱化仮説に関する我々の数値的な研究の結果[2]も合わせて紹介する。
[1] Eiki Iyoda, Kazuya Kaneko, and Takahiro Sagawa, Phys. Rev. Lett. 119, 100601 (2017).
[2] Toru Yoshizawa, Eiki Iyoda, Takahiro Sagawa, arXiv:1712.07289, accepted by Phys. Rev. Lett. (2018).
[ 講演参考URL ]
http://www.taksagawa.com

FMSPレクチャーズ

15:00-17:00   数理科学研究科棟(駒場) 123号室
全5回講演の(1)
Sug Woo Shin 氏 (University of California, Berkeley)
Introduction to the Langlands-Rapoport conjecture (ENGLISH)
[ 講演概要 ]
In 1970s Langlands envisioned a program to compute the Hasse-Weil zeta functions of Shimura varieties as an alternating product of automorphic L-functions, which in particular implies the meromorphic continuation and functional equation for the zeta functions. In 1987, Langlands and Rapoport formulated a precise and far-reaching conjecture describing the set of points of Shimura varieties modulo p as an essential step towards the goal. The program has been largely carried out by Langlands, Kottwitz, and others for PEL-type Shimura varieties with striking applications to the local and global Langlands correspondences (which in turn led to further applications). We have started to understand the more general Hodge-type and abelian-type cases only recently, thanks to Kisin's work on the Langlands-Rapoport conjecture in the good reduction case. The lecture aims to give a gentle introduction to his seminal paper. After a brief introduction, the lecture is divided into four parts.
(i) Shimura varieties: We introduce Shimura varieties of Hodge type and abelian type and their integral models.
(ii) Statement of the conjecture: After setting up the language of
Galois gerbs, we state the Langlands-Rapoport conjecture.
(iii) Sketch of Kisin's proof: We sketch Kisin's proof of the conjecture for Shimura varieties of Hodge type.
(iv) Counting fixed points: Following forthcoming work of Kisin, Y. Zhu, and the speaker, we explain how to apply the Langlands-Rapoport conjecture to count fixed-points of
Hecke-Frobenius correspondences.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SugWooShin.pdf

2018年05月02日(水)

作用素環セミナー

16:45-18:15   数理科学研究科棟(駒場) 126号室
Gabor Szabo 氏 (Copenhagen Univ.)
Classification of Rokhlin flows (English)

2018年04月24日(火)

代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室
陳韋中 氏 (東大数理)
BIRATIONAL BOUNDEDNESS OF RATIONALLY CONNECTED CALABI–YAU 3-FOLDS
(English)
[ 講演概要 ]
Firstly, we show that rationally connected Calabi–Yau 3- folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected 3-folds of ε-CY type form a birationally bounded family for ε > 0. Then we focus on ε-lc log Calabi–Yau pairs (X, B) such that coefficients of B are bounded from below away from zero. We show that such pairs are log bounded modulo flops. As a consequence, we show that rationally connected klt Calabi–Yau 3-folds with mld bounding away from 1 are bounded modulo flops.

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