Martin’s Conjecture is a proposed classification of the definable functions on the Turing degrees. It is usually divided into two parts, the first of which classifies functions which are not above the identity and the second of which classifies functions which are above the identity. Slaman and Steel proved the second part of the conjecture for Borel functions which are order-preserving (i.e. which preserve Turing reducibility). We prove the first part of the conjecture for all order-preserving functions. We do this by introducing a class of functions on the Turing degrees which we call “measure-preserving” and proving that part 1 of Martin’s Conjecture holds for all measure-preserving functions and also that all nontrivial order-preserving functions are measure-preserving. Our result on measure-preserving functions has several other consequences for Martin’s Conjecture, including an equivalence between part 1 of the conjecture and a statement about the structure of the Rudin-Keisler order on ultrafilters on the Turing degrees.
{"title":"Part 1 of Martin’s Conjecture for order-preserving and measure-preserving functions","authors":"Patrick Lutz, Benjamin Siskind","doi":"10.1090/jams/1046","DOIUrl":"https://doi.org/10.1090/jams/1046","url":null,"abstract":"<p>Martin’s Conjecture is a proposed classification of the definable functions on the Turing degrees. It is usually divided into two parts, the first of which classifies functions which are <italic>not</italic> above the identity and the second of which classifies functions which are above the identity. Slaman and Steel proved the second part of the conjecture for Borel functions which are order-preserving (i.e. which preserve Turing reducibility). We prove the first part of the conjecture for all order-preserving functions. We do this by introducing a class of functions on the Turing degrees which we call “measure-preserving” and proving that part 1 of Martin’s Conjecture holds for all measure-preserving functions and also that all nontrivial order-preserving functions are measure-preserving. Our result on measure-preserving functions has several other consequences for Martin’s Conjecture, including an equivalence between part 1 of the conjecture and a statement about the structure of the Rudin-Keisler order on ultrafilters on the Turing degrees.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"186 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:annotation encoding="application/x-tex">infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category of non-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant motivic spectra, which turns out to be equivalent to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:annotation encoding="application/x-tex">infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:annotation encoding="application/x-tex">infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper P Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {P}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homotopy invariance and weighted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homotopy invariance, which we use in place of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper A Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ho
我们提出并证明了任意qcqs派生方案的代数K理论的康纳-弗洛伊德同构。为此，我们研究了非 A 1 mathbb {A}^1 -不变动机谱的稳定∞ infty -类，结果证明它等价于第一和第三作者之前引入的满足基本炸毁切除的基本动机谱的∞ infty -类。我们证明这个∞ infty -类满足 P 1 mathbb {P}^1 -同调不变性和加权 A 1 mathbb {A}^1 -同调不变性，我们用它来代替 A 1 mathbb {A}^1 -同调不变性，从而得到 A 1 mathbb {A}^1 -同调理论中几个关键结果的类似物。这些结果尤其允许我们定义一个普遍的定向动机 E ∞ mathbb {E}_infty -ring 谱 M G L mathrm {MGL} 。然后我们证明一个 qcqs 派生方案 X X 的代数 K 理论可以通过 Conner-Floyd 同构 [ M G L ∗∗ ( X ) ⊗ L Z [ β ± 1 ] ≃ K ∗∗ ( X ) 从它的 M G L mathrm {MGL} -同调中恢复出来、 mathrm {MGL}^{**}(X)otimes _{mathrm {L}{}mathbb {Z}[beta ^{pm 1}]simeq mathrm {K}{}^{**}(X)、 其中 L 是拉扎德环，K p , q ( X ) = K 2 q - p ( X ) mathrm {K}{}^{p,q}(X)=mathrm {K}{}_{2q-p}(X) 。最后，我们证明 M G L mathrm {MGL} 周期化版本的斯奈斯定理。
{"title":"Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory","authors":"Toni Annala, Marc Hoyois, Ryomei Iwasa","doi":"10.1090/jams/1045","DOIUrl":"https://doi.org/10.1090/jams/1045","url":null,"abstract":"<p>We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal infinity\"> <mml:semantics> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category of non-<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant motivic spectra, which turns out to be equivalent to the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal infinity\"> <mml:semantics> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal infinity\"> <mml:semantics> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-category satisfies <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P Superscript 1\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {P}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homotopy invariance and weighted <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-homotopy invariance, which we use in place of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ho","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"10 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Markus Land, Akhil Mathew, Lennart Meier, Georg Tamme
<p>We prove a purity property in telescopically localized algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory of ring spectra: For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">ngeq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-localization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> only depends on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis 0 right-parenthesis circled-plus midline-horizontal-ellipsis circled-plus upper T left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊕<!-- ⊕ --></mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>⊕<!-- ⊕ --></mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T(0)oplus dots oplus T(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-localization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This complements a classical result of Waldhausen in rational <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory. Combining our result with work o
我们证明了环谱的望远镜局部化代数 K K 理论的一个纯粹性：对于 n ≥ 1 ngeq 1，K ( R ) K(R) 的 T ( n ) T(n) 局部化只取决于 R R 的 T ( 0 ) ⊕ ⋯ ⊕ T ( n ) T(0)oplus dots oplus T(n) 局部化。这补充了瓦尔德豪森（Waldhausen）在有理 K K 理论中的一个经典结果。把我们的结果与克劳森-马修-瑙曼-诺尔的研究结合起来，我们会发现 L T ( n ) K ( R ) L_{T(n)}K(R) 事实上只取决于 T ( n - 1 ) ⊕ T ( n ) T(n-1)oplus T(n) -localization of R R，同样是 n ≥ 1 n geq 1。作为结果，我们推导出望远镜局部化 K K 理论的几个消失结果，以及 K ( R ) K(R) 和 T C ( τ ≥ 0 R ) TC(tau _{geq 0} R) 在 n ≥ 2 ngeq 2 时 T ( n ) T(n) 局部化之后的等价性。
{"title":"Purity in chromatically localized algebraic 𝐾-theory","authors":"Markus Land, Akhil Mathew, Lennart Meier, Georg Tamme","doi":"10.1090/jams/1043","DOIUrl":"https://doi.org/10.1090/jams/1043","url":null,"abstract":"<p>We prove a purity property in telescopically localized algebraic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory of ring spectra: For <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 1\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">ngeq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-localization of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K left-parenthesis upper R right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">K(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> only depends on the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T left-parenthesis 0 right-parenthesis circled-plus midline-horizontal-ellipsis circled-plus upper T left-parenthesis n right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>⊕<!-- ⊕ --></mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>⊕<!-- ⊕ --></mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T(0)oplus dots oplus T(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-localization of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=\"application/x-tex\">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This complements a classical result of Waldhausen in rational <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-theory. Combining our result with work o","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"246 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The singular set in the Stefan problem","authors":"Alessio Figalli, Xavier Ros-Oton, Joaquim Serra","doi":"10.1090/jams/1026","DOIUrl":"https://doi.org/10.1090/jams/1026","url":null,"abstract":"","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe
Let AA be drawn uniformly at random from the set of all n×nntimes n symmetric matrices with entries in {−1,1}{-1,1}. We show that [ P(det(A)=0)⩽e−cn,mathbb {P}( det (A) = 0 ) leqslant e^{-cn}, ] where c>0c>0 is an absolute constant, thereby resolving a long-standing conjecture.
让 A A 从所有 n × n times n 对称矩阵的集合中均匀随机抽取，这些矩阵的条目在 { - 1 , 1 } 中。 {-1,1} .我们证明 [ P ( det ( A ) = 0 )⩽ e - c n , mathbb {P}( det (A) = 0 )leqslant e^{-cn}, ] 其中 c > 0 c>0 是一个绝对常量，从而解决了一个长期存在的猜想。
{"title":"The singularity probability of a random symmetric matrix is exponentially small","authors":"Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe","doi":"10.1090/jams/1042","DOIUrl":"https://doi.org/10.1090/jams/1042","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be drawn uniformly at random from the set of all <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n times n\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">ntimes n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> symmetric matrices with entries in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet negative 1 comma 1 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{-1,1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that <disp-formula content-type=\"math/mathml\"> [ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P left-parenthesis det left-parenthesis upper A right-parenthesis equals 0 right-parenthesis less-than-or-slanted-equals e Superscript minus c n Baseline comma\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mo movablelimits=\"true\" form=\"prefix\">det</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {P}( det (A) = 0 ) leqslant e^{-cn},</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"c greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">c>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an absolute constant, thereby resolving a long-standing conjecture.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"30 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the n-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It’s conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Łojasiewicz gradient inequality.
无限自旋问题是一个古老的问题，涉及 n 体问题中完全碰撞轨道的旋转行为。人们早已知道，当一个解趋向于完全碰撞时，其归一化构型曲线必须收敛于归一化中心构型集。在平面 n-body 问题中，每个归一化构型都决定了一个旋转等效归一化构型圈，尤其是存在归一化中心构型圈。可以想象，通过无限自旋，全碰撞解可以收敛到这样一个圆，而不是圆上的某个点。在这里，我们证明了这是不可能的，至少在中心构型极限圆与其他中心构型圆隔离的情况下是如此。(人们认为所有中心构型都是孤立的，但一般情况下这一点并不清楚）。我们的证明依赖于中心流形定理与 Łojasiewicz 梯度不等式的结合。
{"title":"No infinite spin for planar total collision","authors":"Richard Moeckel, Richard Montgomery","doi":"10.1090/jams/1044","DOIUrl":"https://doi.org/10.1090/jams/1044","url":null,"abstract":"<p>The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the <italic>n</italic>-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It’s conceivable that by means of an <italic>infinite spin</italic>, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Łojasiewicz gradient inequality.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"38 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the minimum rank of a non-isotrivial local system of geometric origin on a suitably general nn-pointed curve of genus gg is at least 2g+12sqrt {g+1}. We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformations, which additionally answers questions of Biswas, Heu, and Hurtubise.
{"title":"Geometric local systems on very general curves and isomonodromy","authors":"Aaron Landesman, Daniel Litt","doi":"10.1090/jams/1038","DOIUrl":"https://doi.org/10.1090/jams/1038","url":null,"abstract":"We show that the minimum rank of a non-isotrivial local system of geometric origin on a suitably general <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-pointed curve of genus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=\"application/x-tex\">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is at least <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 StartRoot g plus 1 EndRoot\"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:msqrt> <mml:mi>g</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:msqrt> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">2sqrt {g+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We apply this result to resolve conjectures of Esnault-Kerz and Budur-Wang. The main input is an analysis of stability properties of flat vector bundles under isomonodromic deformations, which additionally answers questions of Biswas, Heu, and Hurtubise.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"179 S446","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful.
{"title":"Sectorial descent for wrapped Fukaya categories","authors":"Sheel Ganatra, John Pardon, Vivek Shende","doi":"10.1090/jams/1035","DOIUrl":"https://doi.org/10.1090/jams/1035","url":null,"abstract":"We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"16 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135218334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if Mmathcal M is any non-locally modular strongly minimal structure interpreted in an algebraically closed field KK of characteristic zero, then Mmathcal M itself interprets KK; in particular, any non-1-based structure interpreted in KK is mutually interpretable with KK. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.
{"title":"Restricted trichotomy in characteristic zero","authors":"Benjamin Castle","doi":"10.1090/jams/1037","DOIUrl":"https://doi.org/10.1090/jams/1037","url":null,"abstract":"We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper M\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is any non-locally modular strongly minimal structure interpreted in an algebraically closed field <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of characteristic zero, then <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper M\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> itself interprets <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; in particular, any non-1-based structure interpreted in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is mutually interpretable with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135689171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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