Roya Beheshti, Brian Lehmann, Eric Riedl, Sho Tanimoto
We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes pp we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic 00. We also investigate the principles of Geometric Manin’s Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over F2(t)mathbb F_2(t) or F3(t)mathbb {F}_{3}(t) such that the exceptional sets in Manin’s Conjecture are Zariski dense.
{"title":"Rational curves on del Pezzo surfaces in positive characteristic","authors":"Roya Beheshti, Brian Lehmann, Eric Riedl, Sho Tanimoto","doi":"10.1090/btran/138","DOIUrl":"https://doi.org/10.1090/btran/138","url":null,"abstract":"<p>We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We also investigate the principles of Geometric Manin’s Conjecture for weak del Pezzo surfaces. In the course of this investigation, we give examples of weak del Pezzo surfaces defined over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper F 2 left-parenthesis t right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb F_2(t)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> or <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper F 3 left-parenthesis t right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {F}_{3}(t)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> such that the exceptional sets in Manin’s Conjecture are Zariski dense.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131259596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.
{"title":"Varieties of general type with doubly exponential asymptotics","authors":"L. Esser, B. Totaro, Chengxi Wang","doi":"10.1090/btran/125","DOIUrl":"https://doi.org/10.1090/btran/125","url":null,"abstract":"We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133077070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a Riemannian manifold (Mm,g)(M^{m}, g) whose volume is the same as the standard sphere (Sm,ground)(S^{m}, g_{round}). If p>m2p!>!frac {m}{2} and ∫M{Rc
考虑到riemannefold (M M, g)， g)它的体积和标准球形差不多。如果p > m 2 p > >公元 frac{}{2}和 ∫ M { R c − ( m − 1 ) g } − p d v的int {M} !我们向他指出，从mm g (m)到g)，普通的Ricci flow的主动性质将永远存在，并将其转化为标准圈子。p - p选择是最佳的。
{"title":"Ricci curvature integrals, local functionals, and the Ricci flow","authors":"Yuanqing Ma, Bing Wang","doi":"10.1090/btran/155","DOIUrl":"https://doi.org/10.1090/btran/155","url":null,"abstract":"<p>Consider a Riemannian manifold <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M Superscript m Baseline comma g right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M^{m}, g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose volume is the same as the standard sphere <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper S Superscript m Baseline comma g Subscript r o u n d Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>r</mml:mi> <mml:mi>o</mml:mi> <mml:mi>u</mml:mi> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(S^{m}, g_{round})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than StartFraction m Over 2 EndFraction\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mspace width=\"negativethinmathspace\" /> <mml:mo>></mml:mo> <mml:mspace width=\"negativethinmathspace\" /> <mml:mfrac> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">p!>!frac {m}{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"integral Underscript upper M Endscripts left-brace upper R c minus left-parenthesis m minus 1 right-parenthesis g right-brace Subscript minus Superscript p Baseline d v\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>M</mml:mi> </mml:mrow> </mml:msub> <mml:mspace width=\"negativethinmathspace\" /> <mml:msubsup> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> <mm","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129496820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose (M,γ)(M, gamma ) is a balanced sutured manifold and KK is a rationally null-homologous knot in MM. It is known that the rank of the sutured Floer homology of M∖N(K)Mbackslash N(K) is at least twice the rank of the sutured Floer homology of MM. This paper studies the properties of KK when the equality is achieved for instanton homology. As an application, we show that if L⊂S3Lsubset S^3
假设(M， γ) (M, gamma)是一个平衡的缝合流形，K K是M M中的一个合理的零同源结。已知M∈N(K) M backslash N(K)的缝合花同调的秩至少是M的缝合花同调的秩的两倍。本文研究了K K在满足实例同调的等式时的性质。作为一个应用，我们证明了如果L∧S 3 L subset S^3是一个固定的链路，K K是L L补上的一个结，那么L∪K L cup K的瞬时链路花同调达到最小秩当且仅当K K是S 3∈L S^3 backslash L中的解结。
{"title":"On Floer minimal knots in sutured manifolds","authors":"Zhenkun Li, Yi Xie, Boyu Zhang","doi":"10.1090/btran/105","DOIUrl":"https://doi.org/10.1090/btran/105","url":null,"abstract":"<p>Suppose <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M comma gamma right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(M, gamma )</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a balanced sutured manifold and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a rationally null-homologous knot in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. It is known that the rank of the sutured Floer homology of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M minus upper N left-parenthesis upper K right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mi class=\"MJX-variant\" mathvariant=\"normal\">∖<!-- ∖ --></mml:mi>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Mbackslash N(K)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is at least twice the rank of the sutured Floer homology of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. This paper studies the properties of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> when the equality is achieved for instanton homology. As an application, we show that if <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L subset-of upper S cubed\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Lsubset S^3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126337158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a filling primitive geodesic curve in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the curve's canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length of generic curves. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.
{"title":"Volume bound for the canonical lift complement of a random geodesic","authors":"Tommaso Cremaschi, Yannick Krifka, D'idac Mart'inez-Granado, Franco Vargas Pallete","doi":"10.1090/btran/152","DOIUrl":"https://doi.org/10.1090/btran/152","url":null,"abstract":"Given a filling primitive geodesic curve in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the curve's canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length of generic curves. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127541427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article deals with the Lipschitz regularity of the “approximate” minimizers for the Bolza type control functional of the form [ Jt(y,u)≔∫tTΛ(s,y(s),u(s))ds+g(y(T))J_t(y,u)≔int _t^TLambda (s,y(s), u(s)),ds+g(y(T)) ] among the pairs (y,u)(y,u) satisfying a prescribed initial condition y(t)=x
本文研究了形式为[ J t ( y , u ) ≔ ∫ t T Λ ( s , y ( s ) , u ( s ) ) d s + g ( y ( T ) ) J_t(y,u)≔int _t^TLambda (s,y(s), u(s)),ds+g(y(T)) ]的Bolza型控制泛函在满足给定初始条件y(t)=x y(t)=x的(y,u)对(y,u)中的“近似”极小值的Lipschitz正则性，其中状态y y是绝对连续的。控制u u是可求和的，动态控制u u的形式为y ' =b(y)u y ' =b(y)u。对于b≡1 b equiv 1，上面的问题变成了变分法的问题。假设拉格朗日方程Λ (s,y,u) Lambda (s,y,u)在离原点(u u的径向凸性)的每条半直线上在变量u u上是凸的，或者在控制变量上是偏可微的，并且在时间变量上满足局部Lipschitz正则性，称为条件(s)。它可以是扩展值，在y y或u u上不连续。在u上不凸。我们假设一个非常温和的增长条件，实际上是对杜波依斯-雷蒙-厄德曼方程的破坏对于控制的高值，如果拉格朗日是强制的以及在一些几乎是线性的情况下，它是满足的。主要结果表明，给定任意可容许对(y,u) (y,u)，对于J＿t J＿t存在更方便的可容许对(y¯，u¯)(overline y, overline u)，其中u¯overline u有界，y¯overline y为Lipschitz，其边界和排名相对于
{"title":"Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems","authors":"C. Mariconda","doi":"10.1090/btran/80","DOIUrl":"https://doi.org/10.1090/btran/80","url":null,"abstract":"<p>This article deals with the Lipschitz regularity of the “approximate” minimizers for the Bolza type control functional of the form <disp-formula content-type=\"math/mathml\">\u0000[\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper J Subscript t Baseline left-parenthesis y comma u right-parenthesis colon-equal integral Subscript t Superscript upper T Baseline normal upper Lamda left-parenthesis s comma y left-parenthesis s right-parenthesis comma u left-parenthesis s right-parenthesis right-parenthesis d s plus g left-parenthesis y left-parenthesis upper T right-parenthesis right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>J</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>≔</mml:mo>\u0000 <mml:msubsup>\u0000 <mml:mo>∫<!-- ∫ --></mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mi>T</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mspace width=\"thinmathspace\" />\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">J_t(y,u)≔int _t^TLambda (s,y(s), u(s)),ds+g(y(T))</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000]\u0000</disp-formula> among the pairs <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis y comma u right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(y,u)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> satisfying a prescribed initial condition <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"y left-parenthesis t right-parenthesis equals x\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>x</mml:m","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"139 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123251146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a pair of knots K1K_1 and K0K_0, we consider the set of four-tuples of integers (g,c0,c1,c2)(g, c_0,c_1, c_2) for which there is a cobordism from K1K_1 to K0K_0 of genus gg having cic_i critical points of each index
{"title":"Critical point counts in knot cobordisms: abelian and metacyclic invariants","authors":"C. Livingston","doi":"10.1090/btran/139","DOIUrl":"https://doi.org/10.1090/btran/139","url":null,"abstract":"<p>For a pair of knots <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">K_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 0\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">K_0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we consider the set of four-tuples of integers <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis g comma c 0 comma c 1 comma c 2 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(g, c_0,c_1, c_2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for which there is a cobordism from <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 1\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">K_1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 0\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">K_0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of genus <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\">\u0000 <mml:semantics>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">g</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> having <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"c Subscript i\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mi>i</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">c_i</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> critical points of each index <inline-formula content-type=\"math/mathml\">\u0000<mml:mat","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"37 36","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114047058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a tree TT and a positive integer nn, let BnTB_nT denote the nn-strand braid group on TT. We use discrete Morse theory techniques to show that the cohomology ring H∗(BnT)H^*(B_nT) is encoded by an explicit abstract simplicial complex KnTK_nT that measures
对于一棵树T和一个积极的妥协n n n，让我们不要把braid小组放在T T上。我们用discrete莫尔斯理论techniques展示那个《cohomology环 H∗ ( B n T ) H ^ * (B_nT)是encoded by an explicit抽象simplicial情结 K n T K_nT那措施n n -local interactions》essential vertices of T T。我们在许多案子那个节目,(例如当T T是一个二进制树的 ), H∗ ( B n T ) H ^ * (B_nT)是《拳台“淡入脸intended: K n T K_nT。
{"title":"Cohomology ring of tree braid groups and exterior face rings","authors":"Jes'us Gonz'alez, Teresa I. Hoekstra-Mendoza","doi":"10.1090/btran/131","DOIUrl":"https://doi.org/10.1090/btran/131","url":null,"abstract":"<p>For a tree <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and a positive integer <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Subscript n Baseline upper T\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">B_nT</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> denote the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-strand braid group on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T\">\u0000 <mml:semantics>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">T</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. We use discrete Morse theory techniques to show that the cohomology ring <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript asterisk Baseline left-parenthesis upper B Subscript n Baseline upper T right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>B</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H^*(B_nT)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is encoded by an explicit abstract simplicial complex <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript n Baseline upper T\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mi>T</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K_nT</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that measures <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:seman","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123008913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We enrich the setting of strongly stable ideals (SSI): We introduce shift modules, a module category encompassing SSIs. The recently introduced duality on SSIs is given an effective conceptual and computational setting. We study SSIs in infinite dimensional polynomial rings, where the duality is most natural. Finally a new type of resolution for SSIs is introduced. This is the projective resolution in the category of shift modules.
{"title":"Shift modules, strongly stable ideals, and their dualities","authors":"Gunnar Fløystad","doi":"10.1090/btran/137","DOIUrl":"https://doi.org/10.1090/btran/137","url":null,"abstract":"We enrich the setting of strongly stable ideals (SSI): We introduce shift modules, a module category encompassing SSIs. The recently introduced duality on SSIs is given an effective conceptual and computational setting. We study SSIs in infinite dimensional polynomial rings, where the duality is most natural. Finally a new type of resolution for SSIs is introduced. This is the projective resolution in the category of shift modules.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115016246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give several characterizations of when a complete first-order theory T T is monadically NIP, i.e. when expansions of T T by arbitrary unary predicates do not have the independence property. The central characterization is a condition on finite satisfiability of types. Other characterizations include decompositions of models, the behavior of indiscernibles, and a forbidden configuration. As an application, we prove non-structure results for hereditary classes of finite substructures of non-monadically NIP models that eliminate quantifiers.
{"title":"Characterizations of monadic NIP","authors":"S. Braunfeld, M. Laskowski","doi":"10.1090/btran/94","DOIUrl":"https://doi.org/10.1090/btran/94","url":null,"abstract":"We give several characterizations of when a complete first-order theory \u0000\u0000 \u0000 T\u0000 T\u0000 \u0000\u0000 is monadically NIP, i.e. when expansions of \u0000\u0000 \u0000 T\u0000 T\u0000 \u0000\u0000 by arbitrary unary predicates do not have the independence property. The central characterization is a condition on finite satisfiability of types. Other characterizations include decompositions of models, the behavior of indiscernibles, and a forbidden configuration. As an application, we prove non-structure results for hereditary classes of finite substructures of non-monadically NIP models that eliminate quantifiers.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125554880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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