{"title":"Constructions of Turán systems that are tight up to a multiplicative constant","authors":"Oleg Pikhurko","doi":"10.1016/j.aim.2025.110148","DOIUrl":"10.1016/j.aim.2025.110148","url":null,"abstract":"<div><div>For positive integers <span><math><mi>n</mi><mo>⩾</mo><mi>s</mi><mo>></mo><mi>r</mi></math></span>, the <em>Turán function</em> <span><math><mi>T</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> is the smallest size of an <em>r</em>-graph with <em>n</em> vertices such that every set of <em>s</em> vertices contains at least one edge. Also, define the <em>Turán density</em> <span><math><mi>t</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span> as the limit of <span><math><mi>T</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>. The question of estimating these parameters received a lot of attention after it was first raised by Turán in 1941. A trivial lower bound is <span><math><mi>t</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>⩾</mo><mn>1</mn><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mrow><mi>s</mi><mo>−</mo><mi>r</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. In the 1990s, de Caen conjectured that <span><math><mi>r</mi><mo>⋅</mo><mi>t</mi><mo>(</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>)</mo><mo>→</mo><mo>∞</mo></math></span> as <span><math><mi>r</mi><mo>→</mo><mo>∞</mo></math></span> and offered 500 Canadian dollars for resolving this question.</div><div>We disprove this conjecture by showing more strongly that for every integer <span><math><mi>R</mi><mo>⩾</mo><mn>1</mn></math></span> there is <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> (in fact, <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> can be taken to grow as <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mspace></mspace><mi>R</mi><mi>ln</mi><mo></mo><mi>R</mi></math></span>) such that <span><math><mi>t</mi><mo>(</mo><mi>r</mi><mo>+</mo><mi>R</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>⩽</mo><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>r</mi><mo>+</mo><mi>R</mi></mrow></mtd></mtr><mtr><mtd><mi>R</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> as <span><math><mi>r</mi><mo>→</mo><mo>∞</mo></math></span>, that is, the trivial lower bound is tight for every <em>R</em> up to a multiplicative constant <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110148"},"PeriodicalIF":1.5,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The affine Springer fiber – sheaf correspondence","authors":"Eugene Gorsky , Oscar Kivinen , Alexei Oblomkov","doi":"10.1016/j.aim.2025.110143","DOIUrl":"10.1016/j.aim.2025.110143","url":null,"abstract":"<div><div>Given a semisimple element in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting variety of the Langlands dual group. The construction uses affine Springer theory and can be thought of as an incarnation of 3d mirror symmetry. For the group <span><math><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the corresponding partial resolution is <span><math><msup><mrow><mi>Hilb</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>×</mo></mrow></msup><mo>×</mo><mi>C</mi><mo>)</mo></math></span>. We also consider a quantization of this construction for homogeneous elements.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110143"},"PeriodicalIF":1.5,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series","authors":"Johannes Droschl","doi":"10.1016/j.aim.2025.110145","DOIUrl":"10.1016/j.aim.2025.110145","url":null,"abstract":"<div><div>In this paper we prove a conjecture of Kudla and Rallis, see <span><span>[12, Conjecture V.3.2]</span></span>. Let <em>χ</em> be a unitary character, <span><math><mi>s</mi><mo>∈</mo><mi>C</mi></math></span> and <em>W</em> a symplectic vector space over a non-archimedean field with symmetry group <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span>. Denote by <span><math><mi>I</mi><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> the degenerate principal series representation of <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>⊕</mo><mi>W</mi><mo>)</mo></math></span>. Pulling back <span><math><mi>I</mi><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> along the natural embedding <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>×</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>↪</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>⊕</mo><mi>W</mi><mo>)</mo></math></span> gives a representation <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>W</mi><mo>,</mo><mi>W</mi></mrow></msub><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> of <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>×</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span>. Let <em>π</em> be an irreducible smooth complex representation of <span><math><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></math></span>. We then prove<span><span><span><math><msub><mrow><mi>dim</mi></mrow><mrow><mi>C</mi></mrow></msub><mo></mo><msub><mrow><mi>Hom</mi></mrow><mrow><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo><mo>×</mo><mi>G</mi><mo>(</mo><mi>W</mi><mo>)</mo></mrow></msub><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>W</mi><mo>,</mo><mi>W</mi></mrow></msub><mo>(</mo><mi>χ</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>,</mo><mi>π</mi><mo>⊗</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>∨</mo></mrow></msup><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo></math></span></span></span> We also give analogous statements for <em>W</em> orthogonal or unitary. This gives in particular a new proof of the conservation relation of the local theta correspondence for symplectic-orthogonal and unitary dual pairs.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110145"},"PeriodicalIF":1.5,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stability of fixed points in Poisson geometry and higher Lie theory","authors":"Karandeep J. Singh","doi":"10.1016/j.aim.2025.110132","DOIUrl":"10.1016/j.aim.2025.110132","url":null,"abstract":"<div><div>We provide a uniform approach to obtain sufficient criteria for a (higher order) fixed point of a given bracket structure on a manifold to be stable under perturbations. Examples of bracket structures include Lie algebroids, Lie <em>n</em>-algebroids, singular foliations, Lie bialgebroids, Courant algebroids and Dirac structures in split Courant algebroids admitting a Dirac complement. We in particular recover stability results of Crainic-Fernandes for zero-dimensional leaves, as well as the stability results of higher order singularities of Dufour-Wade.</div><div>These stability problems can all be shown to be specific instances of the following problem: given a differential graded Lie algebra <span><math><mi>g</mi></math></span>, a differential graded Lie subalgebra <span><math><mi>h</mi></math></span> of finite codimension in <span><math><mi>g</mi></math></span> and a Maurer-Cartan element <span><math><mi>Q</mi><mo>∈</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, when are Maurer-Cartan elements near <em>Q</em> in <span><math><mi>g</mi></math></span> gauge equivalent to elements of <span><math><msup><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>?</div><div>We show that the vanishing of a finite-dimensional cohomology group associated to <span><math><mi>g</mi><mo>,</mo><mi>h</mi></math></span> and <em>Q</em> implies a positive answer to the question above, and therefore implies stability of fixed points of the geometric structures described above.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110132"},"PeriodicalIF":1.5,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277975","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":"Prescribed mean curvature min-max theory in some non-compact manifolds","authors":"Liam Mazurowski","doi":"10.1016/j.aim.2025.110133","DOIUrl":"10.1016/j.aim.2025.110133","url":null,"abstract":"<div><div>This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension <span><math><mn>3</mn><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mn>7</mn></math></span> and consider a smooth function <span><math><mi>h</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><mi>R</mi></math></span> which is asymptotic to a positive constant near infinity. We show that, under certain additional assumptions on <em>h</em>, there exists a closed hypersurface Σ in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> with mean curvature prescribed by <em>h</em>. Second, let <span><math><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> be an asymptotically flat 3-manifold with no boundary and fix a constant <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span>. We show that, under an additional assumption on <em>M</em>, it is possible to find a closed surface Σ of constant mean curvature <em>c</em> in <em>M</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110133"},"PeriodicalIF":1.5,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143277976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Curvature operators on Kähler manifolds","authors":"Barry Minemyer","doi":"10.1016/j.aim.2025.110142","DOIUrl":"10.1016/j.aim.2025.110142","url":null,"abstract":"<div><div>We prove that there exist Kähler manifolds that are not homotopy equivalent to a quotient of complex hyperbolic space but which admit a Riemannian metric with nonpositive curvature operator. This shows that Kähler manifolds do not satisfy the same type of rigidity with respect to the curvature operator as quaternionic hyperbolic and Cayley hyperbolic manifolds and are thus more similar to real hyperbolic manifolds in this setting. Along the way we also calculate explicit values for the eigenvalues of the curvature operator with respect to the standard complex hyperbolic metric.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110142"},"PeriodicalIF":1.5,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143274451","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":"Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy","authors":"A.J. Homburg , J.S.W. Lamb , D.V. Turaev","doi":"10.1016/j.aim.2025.110131","DOIUrl":"10.1016/j.aim.2025.110131","url":null,"abstract":"<div><div>We consider reversible vector fields in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span> such that the set of fixed points of the involutory reversing symmetry is <em>n</em>-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110131"},"PeriodicalIF":1.5,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A connected sum formula for embedded contact homology","authors":"Luya Wang","doi":"10.1016/j.aim.2025.110130","DOIUrl":"10.1016/j.aim.2025.110130","url":null,"abstract":"<div><div>Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a connected sum formula for embedded contact homology.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110130"},"PeriodicalIF":1.5,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165801","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":"Unital operads, monoids, monads, and bar constructions","authors":"J.P. May, Ruoqi Zhang, Foling Zou","doi":"10.1016/j.aim.2024.110065","DOIUrl":"10.1016/j.aim.2024.110065","url":null,"abstract":"<div><div>We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital <figure><img></figure>-sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of symmetric sequences. The monads associated to unital operads are the ones of interest in iterated loop space theory and factorization homology, among many other applications. Our new description of unital operads allows an illuminating comparison between the two-sided monadic bar constructions used in such applications and “classical” monoidal two-sided bar constructions. It also allows a more conceptual understanding of the scanning map central to non-abelian Poincaré duality in factorization homology.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110065"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164287","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}
Jean-Baptiste Campesato , Goulwen Fichou , Adam Parusiński
{"title":"Motivic, logarithmic, and topological Milnor fibrations","authors":"Jean-Baptiste Campesato , Goulwen Fichou , Adam Parusiński","doi":"10.1016/j.aim.2024.110075","DOIUrl":"10.1016/j.aim.2024.110075","url":null,"abstract":"<div><div>We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration. We give two equivalent constructions: the first one extending the classical Kato–Nakayama log-space, and the second one, more geometric, based on the real oriented multigraph construction, a version of the real oriented deformation to the normal cone. As an application, we recover A'Campo's model of the topological Milnor fibration, by quotienting the motivic Milnor fibration with suitable powers of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>, and show that it determines the classical motivic Milnor fibre.</div><div>We also give precise formulae expressing how the introduced objects change under blowings-up. As an application, we show that the motivic Milnor fibre is well-defined as an element of a suitable Grothendieck ring without requiring that the Lefschetz motive be invertible.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110075"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164291","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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