Geometric and Functional Analysis最新文献

{"title":"Persistent Homology of Function Spaces","authors":"Jonathan Block,Fedor Manin,Shmuel Weinberger","doi":"10.1007/s00039-026-00752-2","DOIUrl":"https://doi.org/10.1007/s00039-026-00752-2","url":null,"abstract":"","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"236 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149210377","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":"Real Heegaard Floer Homology","authors":"Gary Guth,Ciprian Manolescu","doi":"10.1007/s00039-026-00754-0","DOIUrl":"https://doi.org/10.1007/s00039-026-00754-0","url":null,"abstract":"","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"68 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149208879","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":"Measure Rigidity and Equidistribution for Fractal Carpets","authors":"Osama Khalil,Manuel Luethi,Barak Weiss","doi":"10.1007/s00039-026-00749-x","DOIUrl":"https://doi.org/10.1007/s00039-026-00749-x","url":null,"abstract":"Abstract Let $theta $ θ be a Bernoulli measure which is stationary for a random walk generated by finitely many contracting rational affine dilations of $mathbb{R}^{d}$ R d , and let $mathcal{K} = mathrm{supp}(theta )$ K = supp ( θ ) be the corresponding attractor. An example in dimension $d=1$ d = 1 is the Hausdorff measure on Cantor’s middle thirds set, and examples in higher dimensions include missing digits sets, Sierpiński carpets and Menger sponges. Let $nu $ ν denote the image of $theta $ θ under the map $mathcal{K} rightarrow mathrm{SL}_{d+1}(mathbb{R})/mathrm{SL}_{d+1}( mathbb{Z})$ K → SL d + 1 ( R ) / SL d + 1 ( Z ) which sends $mathbf{x}$ x to the lattice $Lambda _{mathbf{x}} = mathrm{span}_{mathbb{Z}} (mathbf{e}_{1}, ldots , mathbf{e}_{d}, mathbf{e}_{d+1}+ (mathbf{x},0))$ Λ x = span Z ( e 1 , … , e d , e d + 1 + ( x , 0 ) ) . We prove equidistribution of the pushforward measures $a_{n*}nu $ a n ∗ ν along any diverging sequence of diagonal matrices $(a_{n})subset mathrm{SL}_{d+1}(mathbb{R})$ ( a n ) ⊂ SL d + 1 ( R ) that expand the first $d$ d coordinates under a natural non-escape of mass condition. The latter condition is known to hold whenever $theta $ θ is absolutely friendly. We also show that weighted badly approximable vectors and Dirichlet-improvable vectors (for arbitrary norm) form a subset of $mathcal{K}$ K of $theta $ θ -measure zero. The key ingredient is a measure classification theorem for the stationary measures of an associated random walk on an $S$ S -arithmetic space, introduced by two of the authors in [KL23]. A new feature of this setting is that this random walk admits stationary measures which are not invariant.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"210 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148932797","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":"Eigenvalue Optimization in Higher Dimensions and $p$-Harmonic Maps","authors":"Denis Vinokurov","doi":"10.1007/s00039-026-00751-3","DOIUrl":"https://doi.org/10.1007/s00039-026-00751-3","url":null,"abstract":"","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"81 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148932798","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":"Bourgain’s Condition, Sticky Kakeya, and New Examples","authors":"Arian Nadjimzadah","doi":"10.1007/s00039-026-00750-4","DOIUrl":"https://doi.org/10.1007/s00039-026-00750-4","url":null,"abstract":"We prove that in all dimensions at least 3 and for any Hörmander-type phase function satisfying Bourgain’s condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture. This supports a conjecture of Guo–Wang–Zhang that an oscillatory integral operator satisfies the same <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L^{p}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> bounds as in the restriction conjecture exactly when its phase function satisfies Bourgain’s condition. Our result follows from a new geometric characterization of Bourgain’s condition in terms of straightening curved <jats:inline-formula> <jats:alternatives> <jats:tex-math>$delta $</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>δ</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -tubes in a <jats:inline-formula> <jats:alternatives> <jats:tex-math>$delta ^{1/2}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> -tube. We construct examples in all dimensions at least 3 which show this local straightening property does not persist in a larger tube and, in particular, these are the first phase functions satisfying Bourgain’s condition for which there is no diffeomorphism taking the corresponding families of curves to lines. This suggests that a general to sticky reduction in the spirit of Wang–Zahl needs substantial new ideas, and we take initial steps in this direction. We expect these examples to serve as a natural testing ground.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148883438","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":"Uniform Waist Inequalities in Codimension Two for Manifolds with Kazhdan Fundamental Group","authors":"Uri Bader, Roman Sauer","doi":"10.1007/s00039-026-00748-y","DOIUrl":"https://doi.org/10.1007/s00039-026-00748-y","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$M$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>M</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> be a closed Riemannian manifold with Kazhdan fundamental group. It is well known that the Buser-Cheeger inequality yields a uniform waist inequality in codimension 1 for the finite covers of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$M$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>M</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , which is basically another way of saying that the finite covers form an expander family. We show that the finite covers of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$M$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>M</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> also satisfy a uniform waist inequality in codimension 2.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"159 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148842929","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":"Non-Abelian Amplification and Bilinear Forms with Kloosterman Sums","authors":"Alexandru Pascadi","doi":"10.1007/s00039-026-00746-0","DOIUrl":"https://doi.org/10.1007/s00039-026-00746-0","url":null,"abstract":"We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli <jats:inline-formula> <jats:alternatives> <jats:tex-math>$c$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>c</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , using Fourier analysis on <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathrm{SL}_{2}(mathbb{Z}/cmathbb{Z})$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mi>c</mml:mi> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> and an amplification argument with non-abelian characters. For sums of length <jats:inline-formula> <jats:alternatives> <jats:tex-math>$sqrt{c}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msqrt> <mml:mi>c</mml:mi> </mml:msqrt> </mml:math> </jats:alternatives> </jats:inline-formula> , our method produces a non-trivial bound for all moduli except near-primes, saving <jats:inline-formula> <jats:alternatives> <jats:tex-math>$c^{-1/12}$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>c</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the Pólya–Vinogradov range for all moduli. We give applications to moments of twisted cuspidal <jats:inline-formula> <jats:alternatives> <jats:tex-math>$L$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>L</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> -functions, and to large sieve inequalities for exceptional cusp forms with composite levels.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"34 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768984","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":"A Paley–Wiener–Schwartz Theorem for Smooth Valuations on Convex Functions","authors":"Jonas Knoerr","doi":"10.1007/s00039-026-00747-z","DOIUrl":"https://doi.org/10.1007/s00039-026-00747-z","url":null,"abstract":"Continuous dually epi-translation invariant valuations on convex functions are characterized in terms of the Fourier–Laplace transform of the associated Goodey–Weil distributions. This description is used to obtain integral representations of the smooth vectors of the natural representation of the group of translations on the space of these valuations. As an application, a complete classification of all closed and affine invariant subspaces is established, yielding density results for valuations defined in terms of mixed Monge–Ampère operators.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768985","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":"Sharp Favard Length of Random Cantor Sets","authors":"Alan Chang, Pablo Shmerkin, Ville Suomala","doi":"10.1007/s00039-026-00743-3","DOIUrl":"https://doi.org/10.1007/s00039-026-00743-3","url":null,"abstract":"We show that for a large class of planar 1-dimensional random fractals &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$S$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , the Favard length &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$operatorname{Fav}(S(r))$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mo&gt;Fav&lt;/mml:mo&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of the neighborhood &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$S(r)$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is comparable to &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$log ^{-1}(1/r)$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:msup&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;−&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , matching a universal lower bound; up to now, this was only known in expectation for a few concrete models. In particular, we show that there exist 1-Ahlfors regular sets with the fastest possible Favard length decay. For a wide class of planar one-dimensional “grid random fractals”, including fractal percolation and its Ahlfors-regular variants, we further show that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$operatorname{Fav}(S(r))/log (1/r)$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mo&gt;Fav&lt;/mml:mo&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; converges almost surely, and we identify the limit explicitly. Furthermore, we prove that for some 1-dimensional Ahlfors-regular random fractals &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$S$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , the Favard length of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$S(r)$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;S&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; decays instead like &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$log log (1/r)/log (1/r)$&lt;/jats:tex-math&gt; &lt;mml:math","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"53 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148408882","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":"A New Approach to Strong Convergence II. The Classical Ensembles","authors":"Chi-Fang Chen, Jorge Garza-Vargas, Ramon van Handel","doi":"10.1007/s00039-026-00744-2","DOIUrl":"https://doi.org/10.1007/s00039-026-00744-2","url":null,"abstract":"The first paper in this series introduced a new approach to strong convergence of random matrices that is based primarily on soft arguments. This method was applied to achieve a refined qualitative and quantitative understanding of strong convergence of random permutation matrices and of more general representations of the symmetric group. In this paper, we introduce new ideas that make it possible to achieve stronger quantitative results and that facilitate the application of the method to new models.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"247 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148612813","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}