Pub Date : 2026-04-01Epub Date: 2026-01-23DOI: 10.1016/j.aim.2026.110799
Di Wu, Xi Zhang
Given a flat vector bundle over a compact Riemannian manifold, the Corlette-Donaldson theorem indicates that it admits harmonic metrics if and only if it is semi-simple. We extend this equivalence to arbitrary vector bundles without any additional hypotheses, it can be viewed as a Riemannian Donaldson-Uhlenbeck-Yau correspondence. Furthermore, we prove an equivalence of categories in Sasakian geometry, relating projective flat vector bundles to Higgs bundles. Along the way, a transparent proof is also provided for the Reeb invariance of harmonic metrics in Sasakian geometry that had required Sasakian curvature theory and spinorial trick before, the Reeb invariance plays a crucial role in defining stability of basic Higgs bundles and establishing Sasakian Corlette-Simpson correspondence.
{"title":"Harmonic metrics and semi-simpleness","authors":"Di Wu, Xi Zhang","doi":"10.1016/j.aim.2026.110799","DOIUrl":"10.1016/j.aim.2026.110799","url":null,"abstract":"<div><div>Given a flat vector bundle over a compact Riemannian manifold, the Corlette-Donaldson theorem indicates that it admits harmonic metrics if and only if it is semi-simple. We extend this equivalence to arbitrary vector bundles without any additional hypotheses, it can be viewed as a Riemannian Donaldson-Uhlenbeck-Yau correspondence. Furthermore, we prove an equivalence of categories in Sasakian geometry, relating projective flat vector bundles to Higgs bundles. Along the way, a transparent proof is also provided for the Reeb invariance of harmonic metrics in Sasakian geometry that had required Sasakian curvature theory and spinorial trick before, the Reeb invariance plays a crucial role in defining stability of basic Higgs bundles and establishing Sasakian Corlette-Simpson correspondence.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110799"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025721","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}
Pub Date : 2026-04-01Epub Date: 2026-01-29DOI: 10.1016/j.aim.2026.110823
Andrii Ilienko , Ilya Molchanov , Tommaso Visonà
We obtain a complete characterization of planar monotone σ-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or σ-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.
{"title":"Integer-valued valuations","authors":"Andrii Ilienko , Ilya Molchanov , Tommaso Visonà","doi":"10.1016/j.aim.2026.110823","DOIUrl":"10.1016/j.aim.2026.110823","url":null,"abstract":"<div><div>We obtain a complete characterization of planar monotone <em>σ</em>-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or <em>σ</em>-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110823"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080759","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}
Pub Date : 2026-04-01Epub Date: 2026-01-28DOI: 10.1016/j.aim.2026.110805
Matt Tyler
A partition is a t-core partition if t is not one of its hook lengths. Let be the number of t-core partitions of N. In 1999, Stanton conjectured if . This was proved for t fixed and N sufficiently large by Anderson, and for small values of t by Kim and Rouse. In this paper, we prove Stanton's conjecture in general.
Our approach is to find a saddle point asymptotic formula for , valid in all ranges of t and N. This includes the known asymptotic formulas for as special cases, and shows that the behavior of depends on how compares in size to N. For example, our formula implies that if , then for suitable constants A and B defined in terms of κ.
{"title":"Asymptotics for t-core partitions and Stanton's conjecture","authors":"Matt Tyler","doi":"10.1016/j.aim.2026.110805","DOIUrl":"10.1016/j.aim.2026.110805","url":null,"abstract":"<div><div>A partition is a <em>t-core partition</em> if <em>t</em> is not one of its hook lengths. Let <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> be the number of <em>t</em>-core partitions of <em>N</em>. In 1999, Stanton conjectured <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> if <span><math><mn>4</mn><mo>≤</mo><mi>t</mi><mo>≠</mo><mi>N</mi><mo>−</mo><mn>1</mn></math></span>. This was proved for <em>t</em> fixed and <em>N</em> sufficiently large by Anderson, and for small values of <em>t</em> by Kim and Rouse. In this paper, we prove Stanton's conjecture in general.</div><div>Our approach is to find a saddle point asymptotic formula for <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span>, valid in all ranges of <em>t</em> and <em>N</em>. This includes the known asymptotic formulas for <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> as special cases, and shows that the behavior of <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo></math></span> depends on how <span><math><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> compares in size to <em>N</em>. For example, our formula implies that if <span><math><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>κ</mi><mi>N</mi><mo>+</mo><mi>o</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, then <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>exp</mi><mo></mo><mrow><mo>(</mo><mn>2</mn><mi>π</mi><msqrt><mrow><mi>A</mi><mi>N</mi></mrow></msqrt><mo>)</mo></mrow></mrow><mrow><mi>B</mi><mi>N</mi></mrow></mfrac><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow></math></span> for suitable constants <em>A</em> and <em>B</em> defined in terms of <em>κ</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110805"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080757","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}
Pub Date : 2026-04-01Epub Date: 2026-02-03DOI: 10.1016/j.aim.2026.110826
Emma Brakkee , Chiara Camere , Annalisa Grossi , Laura Pertusi , Giulia Saccà , Sasha Viktorova
Generalizing work of Markushevich–Tikhomirov and Arbarello–Saccà–Ferretti, we use relative Prym varieties to construct Lagrangian fibered symplectic varieties in infinitely many dimensions. We then give criteria for when the construction yields primitive symplectic varieties, respectively, irreducible symplectic varieties. The starting point of the construction is a K3 surface endowed with an anti-symplectic involution and an effective linear system on the quotient surface. We give sufficient conditions on the linear system to ensure that the relative Prym varieties satisfy the criteria above. As a consequence, we produce infinite series of irreducible symplectic varieties.
{"title":"Irreducible symplectic varieties via relative Prym varieties","authors":"Emma Brakkee , Chiara Camere , Annalisa Grossi , Laura Pertusi , Giulia Saccà , Sasha Viktorova","doi":"10.1016/j.aim.2026.110826","DOIUrl":"10.1016/j.aim.2026.110826","url":null,"abstract":"<div><div>Generalizing work of Markushevich–Tikhomirov and Arbarello–Saccà–Ferretti, we use relative Prym varieties to construct Lagrangian fibered symplectic varieties in infinitely many dimensions. We then give criteria for when the construction yields primitive symplectic varieties, respectively, irreducible symplectic varieties. The starting point of the construction is a <em>K</em>3 surface endowed with an anti-symplectic involution and an effective linear system on the quotient surface. We give sufficient conditions on the linear system to ensure that the relative Prym varieties satisfy the criteria above. As a consequence, we produce infinite series of irreducible symplectic varieties.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110826"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174564","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}
Pub Date : 2026-04-01Epub Date: 2026-01-22DOI: 10.1016/j.aim.2026.110804
Naichung Conan Leung , Qin Li , Ziming Nikolas Ma
In this paper, we explore the quantization of Kähler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form for a certain smooth (non-formal) function . If is real-valued then corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian G-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.
{"title":"Symmetry in deformation quantization and geometric quantization","authors":"Naichung Conan Leung , Qin Li , Ziming Nikolas Ma","doi":"10.1016/j.aim.2026.110804","DOIUrl":"10.1016/j.aim.2026.110804","url":null,"abstract":"<div><div>In this paper, we explore the quantization of Kähler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form <span><math><mi>f</mi><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mfrac><mrow><mi>ħ</mi></mrow><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo>(</mo><mi>Δ</mi><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>c</mi><mo>)</mo></math></span> for a certain smooth (non-formal) function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. If <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is real-valued then <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian <em>G</em>-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110804"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025724","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}
Pub Date : 2026-04-01Epub Date: 2026-02-02DOI: 10.1016/j.aim.2026.110822
Feng Dai , Yinqin Li , Dachun Yang , Wen Yuan , Yirui Zhao
<div><div>In this article, we introduce a family of local lifted Hardy–Littlewood maximal operators <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> on the upper half-plane and prove that, for any given <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>∈</mo><mi>R</mi></math></span>, the estimate, with the implicit positive constant independent of <em>f</em>,<span><span><span><math><mrow><munder><mi>sup</mi><mrow><mi>θ</mi><mo>,</mo><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></munder><mo></mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msup><munder><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><munderover><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></munderover></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mi>λ</mi><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mi>γ</mi></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></msup></mrow></munder><msup><mrow><mi>t</mi></mrow><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><mi>d</mi><mi>t</mi><mspace></mspace><mi>d</mi><mi>x</mi><mo>≲</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><mo>|</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span></span></span> holds for all <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> if and only if either <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>≠</mo><mn>0</mn></math></span> or <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>γ</mi><mo>∉</mo><mo>[</mo><mo>−</mo><mi>n</mi><mo>,</mo><mn>0</mn><mo>]</mo></math></span>. Moreover, we use <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> to refine the Lusin–Lipschitz inequality and the pointwise domination for the generalized approximation to the identity, which connect various differential operators with related one-parameter families of linear operators. As applications, we obtain several new weak-type representations for the norms of these differential operators, including endpoint estimates and extensions to spaces of homogeneous type, which gives an affirmative answer
{"title":"Sharp weak-type estimate for local lifted Hardy–Littlewood maximal operators with applications to generators of linear operator families and Hardy(–Sobolev) spaces","authors":"Feng Dai , Yinqin Li , Dachun Yang , Wen Yuan , Yirui Zhao","doi":"10.1016/j.aim.2026.110822","DOIUrl":"10.1016/j.aim.2026.110822","url":null,"abstract":"<div><div>In this article, we introduce a family of local lifted Hardy–Littlewood maximal operators <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> on the upper half-plane and prove that, for any given <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>∈</mo><mi>R</mi></math></span>, the estimate, with the implicit positive constant independent of <em>f</em>,<span><span><span><math><mrow><munder><mi>sup</mi><mrow><mi>θ</mi><mo>,</mo><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></munder><mo></mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msup><munder><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><munderover><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></munderover></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mi>λ</mi><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mi>γ</mi></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></msup></mrow></munder><msup><mrow><mi>t</mi></mrow><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><mi>d</mi><mi>t</mi><mspace></mspace><mi>d</mi><mi>x</mi><mo>≲</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><mo>|</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span></span></span> holds for all <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> if and only if either <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>≠</mo><mn>0</mn></math></span> or <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>γ</mi><mo>∉</mo><mo>[</mo><mo>−</mo><mi>n</mi><mo>,</mo><mn>0</mn><mo>]</mo></math></span>. Moreover, we use <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> to refine the Lusin–Lipschitz inequality and the pointwise domination for the generalized approximation to the identity, which connect various differential operators with related one-parameter families of linear operators. As applications, we obtain several new weak-type representations for the norms of these differential operators, including endpoint estimates and extensions to spaces of homogeneous type, which gives an affirmative answer","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110822"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174565","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 characterize the limiting root distribution μ of a sequence of polynomials with nonnegative roots and degree d, in terms of their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients of to Voiculescu's S-transform of μ.
In the framework of finite free probability, we interpret these ratios of coefficients as a new notion of finite S-transform, which converges to in the large d limit. It also satisfies several analogous properties to those of the S-transform in free probability, including multiplicativity and monotonicity.
The proof of the main theorem is based on various ideas and new results relating finite free probability and free probability. In particular, we provide a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation.
This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of to ⊠ and prove a finite approximation of the Tucci–Haagerup–Möller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.
{"title":"S-transform in finite free probability","authors":"Octavio Arizmendi , Katsunori Fujie , Daniel Perales , Yuki Ueda","doi":"10.1016/j.aim.2026.110803","DOIUrl":"10.1016/j.aim.2026.110803","url":null,"abstract":"<div><div>We characterize the limiting root distribution <em>μ</em> of a sequence of polynomials <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> with nonnegative roots and degree <em>d</em>, in terms of their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> to Voiculescu's <em>S</em>-transform <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> of <em>μ</em>.</div><div>In the framework of finite free probability, we interpret these ratios of coefficients as a new notion of finite <em>S</em>-transform, which converges to <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> in the large <em>d</em> limit. It also satisfies several analogous properties to those of the <em>S</em>-transform in free probability, including multiplicativity and monotonicity.</div><div>The proof of the main theorem is based on various ideas and new results relating finite free probability and free probability. In particular, we provide a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation.</div><div>This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of <span><math><msub><mrow><mo>⊠</mo></mrow><mrow><mi>d</mi></mrow></msub></math></span> to ⊠ and prove a finite approximation of the Tucci–Haagerup–Möller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110803"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080764","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}
Pub Date : 2026-04-01Epub Date: 2026-01-28DOI: 10.1016/j.aim.2026.110809
Xiaofa Chen
We introduce the notion of exact dg category, which provides a differential graded enhancement of Nakaoka–Palu's notion of extriangulated category. We give a definition in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We introduce the notion of stable dg category, and prove that the -category of an exact dg category is triangulated if and only if is stable. We illustrate our theory with several examples including the homotopy category of two-term complexes and Amiot's fundamental domain for generalized cluster categories.
{"title":"Exact dg categories I: Foundations","authors":"Xiaofa Chen","doi":"10.1016/j.aim.2026.110809","DOIUrl":"10.1016/j.aim.2026.110809","url":null,"abstract":"<div><div>We introduce the notion of exact dg category, which provides a differential graded enhancement of Nakaoka–Palu's notion of extriangulated category. We give a definition in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We introduce the notion of stable dg category, and prove that the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>-category of an exact dg category <span><math><mi>A</mi></math></span> is triangulated if and only if <span><math><mi>A</mi></math></span> is stable. We illustrate our theory with several examples including the homotopy category of two-term complexes and Amiot's fundamental domain for generalized cluster categories.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110809"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080760","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}
Inspired by work of the first and second authors, this paper studies the Gromov width of the disk cotangent bundle of spheroids and Zoll spheres of revolution. This is achieved with the use of techniques from integrable systems and embedded contact homology capacities.
{"title":"Gromov width of the disk cotangent bundle of spheres of revolution","authors":"Brayan Ferreira , Vinicius G.B. Ramos , Alejandro Vicente","doi":"10.1016/j.aim.2025.110761","DOIUrl":"10.1016/j.aim.2025.110761","url":null,"abstract":"<div><div>Inspired by work of the first and second authors, this paper studies the Gromov width of the disk cotangent bundle of spheroids and Zoll spheres of revolution. This is achieved with the use of techniques from integrable systems and embedded contact homology capacities.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"487 ","pages":"Article 110761"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928076","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}
Pub Date : 2026-03-01Epub Date: 2026-01-14DOI: 10.1016/j.aim.2025.110774
Avy Soffer , Xiaoxu Wu
For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in an adapted way the Free Channel Wave Operator, and further tools from the recent works [21], [22], [35]. This work generalizes the results of the first part of [21], [22] to arbitrary dimension, and non-radial data.
{"title":"On the large time asymptotics of Schrödinger type equations with general data","authors":"Avy Soffer , Xiaoxu Wu","doi":"10.1016/j.aim.2025.110774","DOIUrl":"10.1016/j.aim.2025.110774","url":null,"abstract":"<div><div>For the Schrödinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in an adapted way the Free Channel Wave Operator, and further tools from the recent works <span><span>[21]</span></span>, <span><span>[22]</span></span>, <span><span>[35]</span></span>. This work generalizes the results of the first part of <span><span>[21]</span></span>, <span><span>[22]</span></span> to arbitrary dimension, and non-radial data.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110774"},"PeriodicalIF":1.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981067","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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