https://arxiv.org/abs/2412.21057
The rectifiable rectangular peg problem
Tomohiro Asano, Yuichi Ike
We give an affirmative answer to the rectangular peg problem for a large class of continuous Jordan curves that contains all rectifiable curves and Stromquist's locally monotone curves. Our proof is based on microlocal sheaf theory and inspired by recent work of Greene and Lobb.
The rectifiable rectangular peg problem
Tomohiro Asano, Yuichi Ike
We give an affirmative answer to the rectangular peg problem for a large class of continuous Jordan curves that contains all rectifiable curves and Stromquist's locally monotone curves. Our proof is based on microlocal sheaf theory and inspired by recent work of Greene and Lobb.
arXiv.org
The rectifiable rectangular peg problem
We give an affirmative answer to the rectangular peg problem for a large class of continuous Jordan curves that contains all rectifiable curves and Stromquist's locally monotone curves. Our proof...
Бена Тшишуку предложил сегодня челендж -- доказать теорему Делиня-Сулливана:
Каждое гиперболическое многообразие имеет конечное накрытие, которое является стабильно паралелизуемыми, то есть при умножении на тривиальное расслоение становится тривиальным.
Каждое гиперболическое многообразие имеет конечное накрытие, которое является стабильно паралелизуемыми, то есть при умножении на тривиальное расслоение становится тривиальным.
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Zenzeli
Бена Тшишуку предложил сегодня челендж -- доказать теорему Делиня-Сулливана: Каждое гиперболическое многообразие имеет конечное накрытие, которое является стабильно паралелизуемыми, то есть при умножении на тривиальное расслоение становится тривиальным.
MathOverflow
Well known theorems that have not been proved
I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be
Author(s): Yujiro Kawamata (川又 雄二郎); Chen Jiang (江辰) - Translator
Series: Cambridge Studies in Advanced Mathematics 214
Publisher: Cambridge University Press, Year: 2024
The finite generation theorem is a major achievement in modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic 0 is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar–Cascini–Hacon–McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori’s bend-and-break method, vanishing theorems, positivity theorems, and Siu’s analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum withself-contained explanations of terminology and theorems.
Series: Cambridge Studies in Advanced Mathematics 214
Publisher: Cambridge University Press, Year: 2024
The finite generation theorem is a major achievement in modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic 0 is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar–Cascini–Hacon–McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori’s bend-and-break method, vanishing theorems, positivity theorems, and Siu’s analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum withself-contained explanations of terminology and theorems.
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https://arxiv.org/abs/2407.11916
The volume intrinsic to a commutative graded algebra
Karim Alexander Adiprasito, Stavros Argyrios Papadakis, Vasiliki Petrotou
Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties. This paper illuminates a key aspect of these works, the behaviour of the canonical module under deformations in this moduli space. We demonstrate that even when there is no natural geometry around, we can give a viewpoint that behaves like it, effectively constructing geometry out of nothing, giving interpretation to intersection numbers without cycles. Moreover, we explore some properties of this normalization.
The volume intrinsic to a commutative graded algebra
Karim Alexander Adiprasito, Stavros Argyrios Papadakis, Vasiliki Petrotou
Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties. This paper illuminates a key aspect of these works, the behaviour of the canonical module under deformations in this moduli space. We demonstrate that even when there is no natural geometry around, we can give a viewpoint that behaves like it, effectively constructing geometry out of nothing, giving interpretation to intersection numbers without cycles. Moreover, we explore some properties of this normalization.
arXiv.org
The volume intrinsic to a commutative graded algebra
Recent works of the authors have demonstrated the usefulness of considering moduli spaces of Artinian reductions of a given ring when studying standard graded rings and their Lefschetz properties....
Forwarded from филозофъ, сиречь любомудръ
г҃./ д҃.
такожде и иныи тамо совокупляются преукрашени умомъ мужи и жены. иже несть от сѣдалища. токмо имутъ от сѣдалища мзду за оуменіе свое зело малую. нарицается мзда та: ставка. имя симъ: делатель от учения, еже новы письмены речется: научный работник. поелику раби суть.
такожде и иныи тамо совокупляются преукрашени умомъ мужи и жены. иже несть от сѣдалища. токмо имутъ от сѣдалища мзду за оуменіе свое зело малую. нарицается мзда та: ставка. имя симъ: делатель от учения, еже новы письмены речется: научный работник. поелику раби суть.
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https://arxiv.org/abs/2312.12294
Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
Andreas Bernig, Jan Kotrbatý, Thomas Wannerer
Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
The algebra of smooth translation-invariant valuations on convex bodies, introduced by this http URL in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a compact Kähler manifold: Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. Our main result establishes the hard Lefschetz theorem and the Hodge-Riemann relations in full generality. As a consequence, we obtain McMullen's quadratic inequalities, which are valid for strongly isomorphic polytopes and known to fail in general, for convex bodies with smooth and strictly positively curved boundary. Our proof is based on elliptic operator theory and on perturbation theory applied to unbounded operators on a natural Hilbert space completion of the space of smooth translation-invariant valuations.
Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
Andreas Bernig, Jan Kotrbatý, Thomas Wannerer
Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
The algebra of smooth translation-invariant valuations on convex bodies, introduced by this http URL in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a compact Kähler manifold: Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. Our main result establishes the hard Lefschetz theorem and the Hodge-Riemann relations in full generality. As a consequence, we obtain McMullen's quadratic inequalities, which are valid for strongly isomorphic polytopes and known to fail in general, for convex bodies with smooth and strictly positively curved boundary. Our proof is based on elliptic operator theory and on perturbation theory applied to unbounded operators on a natural Hilbert space completion of the space of smooth translation-invariant valuations.
arXiv.org
Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations
The algebra of smooth translation-invariant valuations on convex bodies, introduced by S.Alesker in the early 2000s, was in part proved and in part conjectured to satisfy properties formally...
https://arxiv.org/abs/2501.18150
Stability thresholds for big classes
Chenzi Jin, Yanir A. Rubinstein, Gang Tian
In 1987, the α-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to ℚ-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the δ-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles τ∈[0,1]. The special degenerate (collapsing) case τ=0 on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.
Stability thresholds for big classes
Chenzi Jin, Yanir A. Rubinstein, Gang Tian
In 1987, the α-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to ℚ-Fano varieties in terms of K-stability, and in 2017 Fujita related this circle of ideas to the δ-invariant of Fujita-Odaka. We introduce new invariants on the big cone and prove a generalization of the Tian-Odaka-Sano Theorem to all big classes on varieties with klt singularities, and moreover for all volume quantiles τ∈[0,1]. The special degenerate (collapsing) case τ=0 on ample classes recovers Odaka-Sano's theorem. This leads to many new twisted Kahler-Einstein metrics on big classes. Of independent interest, the proof involves a generalization to sub-barycenters of the classical Neumann-Hammer Theorem from convex geometry.
arXiv.org
Stability thresholds for big classes
In 1987, the $α$-invariant theorem gave a fundamental criterion for existence of Kahler-Einstein metrics on smooth Fano manifolds. In 2012, Odaka-Sano extended the framework to...
https://arxiv.org/abs/2501.18088
A characterization of uniruled compact Kähler manifolds
Wenhao Ou
We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Păun and of Druel to foliations on compact Kähler manifolds. As an application, we prove that a compact Kähler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.
A characterization of uniruled compact Kähler manifolds
Wenhao Ou
We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Păun and of Druel to foliations on compact Kähler manifolds. As an application, we prove that a compact Kähler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.
arXiv.org
A characterization of uniruled compact Kähler manifolds
We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Păun and of...
https://arxiv.org/abs/2501.18920
Not all sub-Riemannian minimizing geodesics are smooth
Yacine Chitour, Frédéric Jean, Roberto Monti, Ludovic Rifford, Ludovic Sacchelli, Mario Sigalotti, Alessandro Socionovo
A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a
negative answer to this question, exhibiting an example of a C2 but not C3 length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.
О
вроде как чуваки построили пример субримановой метрики
у которой минимизирующая геодезическая негладкая
big if true
Not all sub-Riemannian minimizing geodesics are smooth
Yacine Chitour, Frédéric Jean, Roberto Monti, Ludovic Rifford, Ludovic Sacchelli, Mario Sigalotti, Alessandro Socionovo
A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a
negative answer to this question, exhibiting an example of a C2 but not C3 length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.
О
вроде как чуваки построили пример субримановой метрики
у которой минимизирующая геодезическая негладкая
big if true
arXiv.org
Not all sub-Riemannian minimizing geodesics are smooth
A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a negative answer to this question, exhibiting an example of a $C^2$...
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Zenzeli
забавный факт что любая группа реализуется как группа автоморфизмов какого-то графа. для конечных груп это довольно просто если выбрать в группе G систему порождающих S, то группа автомофорфизов раскрашенного графа Кэли [рассматриваемого как ориентированный…
кстати хочется доказательство такой теормы:
любая конечная группа реализуется как группа автоморфизмов какой-то римановой поверхности. это хорошо известный факт и не сложный, также то же самое верно для гиперболических многообразий любой размерности (Белолипецкий-Луботцки)
но хочется вот типа сначала реализовать группу как группу автоморфизмов графа а потом по графу построить поверхность с такими же автоморфизмами.
я думаю по разговору с грэмом смитом что надо так делать --- сначала показать что граф можно сделать тривалентным (ну типа разрешить вершины с большой валентностью). потом склеить поверхность из соответсвующих штанов, где twist parameters координат Фенхеля-Нильсена будут кодировать направление, ну а длины ребер можем выбирать как хотим и выберем так чтобы паразитических автоморфизмов не было.
любая конечная группа реализуется как группа автоморфизмов какой-то римановой поверхности. это хорошо известный факт и не сложный, также то же самое верно для гиперболических многообразий любой размерности (Белолипецкий-Луботцки)
но хочется вот типа сначала реализовать группу как группу автоморфизмов графа а потом по графу построить поверхность с такими же автоморфизмами.
я думаю по разговору с грэмом смитом что надо так делать --- сначала показать что граф можно сделать тривалентным (ну типа разрешить вершины с большой валентностью). потом склеить поверхность из соответсвующих штанов, где twist parameters координат Фенхеля-Нильсена будут кодировать направление, ну а длины ребер можем выбирать как хотим и выберем так чтобы паразитических автоморфизмов не было.
any non-Abelian connected Lie group has a smooth fixed-point free action on Euclidean space
https://www.maths.ed.ac.uk/~v1ranick/papers/oliverphd.pdf
https://www.maths.ed.ac.uk/~v1ranick/papers/oliverphd.pdf