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Logarithmic Donaldson–Thomas theory Forum Math. Pi (IF 2.955) Pub Date : 2024-04-18 Davesh Maulik, Dhruv Ranganathan
Let X be a smooth and projective threefold with a simple normal crossings divisor D. We construct the Donaldson–Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on X relative to D. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties
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The Chromatic Fourier Transform Forum Math. Pi (IF 2.955) Pub Date : 2024-04-08 Tobias Barthel, Shachar Carmeli, Tomer M. Schlank, Lior Yanovski
We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology of $\pi $-finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$. We use this theory to generalize said duality in three different directions. First, we extend
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On the ergodic theory of the real Rel foliation Forum Math. Pi (IF 2.955) Pub Date : 2024-04-02 Jon Chaika, Barak Weiss
Let ${{\mathcal {H}}}$ be a stratum of translation surfaces with at least two singularities, let $m_{{{\mathcal {H}}}}$ denote the Masur-Veech measure on ${{\mathcal {H}}}$, and let $Z_0$ be a flow on $({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$ obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of
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Sharp well-posedness for the cubic NLS and mKdV in Forum Math. Pi (IF 2.955) Pub Date : 2024-04-02 Benjamin Harrop-Griffiths, Rowan Killip, Monica Vişan
We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s({{\mathbb {R}}})$ for any regularity $s>-\frac 12$. Well-posedness has long been known for $s\geq 0$, see [55], but not previously for any $s<0$. The scaling-critical value $s=-\frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48].
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Virasoro Constraints for Toric Bundles Forum Math. Pi (IF 2.955) Pub Date : 2024-02-05 Tom Coates, Alexander Givental, Hsian-Hua Tseng
We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of
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The least singular value of a random symmetric matrix Forum Math. Pi (IF 2.955) Pub Date : 2024-01-23 Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe
Let A be an $n \times n$ symmetric matrix with $(A_{i,j})_{i\leqslant j}$ independent and identically distributed according to a subgaussian distribution. We show that $$ \begin{align*}\mathbb{P}(\sigma_{\min}(A) \leqslant \varepsilon n^{-1/2} ) \leqslant C \varepsilon + e^{-cn},\end{align*} $$ where $\sigma _{\min }(A)$ denotes the least singular value of A and the constants $C,c>0 $ depend only on
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Degrees of maps and multiscale geometry Forum Math. Pi (IF 2.955) Pub Date : 2024-01-18 Aleksandr Berdnikov, Larry Guth, Fedor Manin
We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $X_k$ is the connected sum of k copies of $\mathbb CP^2$ for $k \ge 4$ , then we prove that the maximum degree of an L-Lipschitz self-map of $X_k$ is between $C_1 L^4 (\log L)^{-4}$ and $C_2 L^4 (\log L)^{-1/2}$ . More generally, we divide simply connected
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Smith theory and cyclic base change functoriality Forum Math. Pi (IF 2.955) Pub Date : 2024-01-15 Tony Feng
Lafforgue and Genestier-Lafforgue have constructed the global and (semisimplified) local Langlands correspondences for arbitrary reductive groups over function fields. We establish various properties of these correspondences regarding functoriality for cyclic base change: For $\mathbf {Z}/p\mathbf {Z}$ -extensions of global function fields, we prove the existence of base change for mod p automorphic
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The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime Forum Math. Pi (IF 2.955) Pub Date : 2023-12-22 Martin Ravn Christiansen, Christian Hainzl, Phan Thành Nam
We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of N fermions on a torus, interacting via a two-body repulsive potential proportional to $N^{-\frac {1}{3}}$ . In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic
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Global solutions for 1D cubic defocusing dispersive equations: Part I Forum Math. Pi (IF 2.955) Pub Date : 2023-12-04 Mihaela Ifrim, Daniel Tataru
This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are both small and localized. However, except
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On local Galois deformation rings Forum Math. Pi (IF 2.955) Pub Date : 2023-10-26 Gebhard Böckle, Ashwin Iyengar, Vytautas Paškūnas
We show that framed deformation rings of mod p representations of the absolute Galois group of a p-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application, we prove density results of loci with prescribed p-adic Hodge theoretic properties.
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Higher uniformity of arithmetic functions in short intervals I. All intervals Forum Math. Pi (IF 2.955) Pub Date : 2023-10-19 Kaisa Matomäki, Xuancheng Shao, Terence Tao, Joni Teräväinen
We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and
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The Local Langlands Conjecture for Forum Math. Pi (IF 2.955) Pub Date : 2023-10-19 Wee Teck Gan, Gordan Savin
We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.
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Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials Forum Math. Pi (IF 2.955) Pub Date : 2023-10-18 Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto
Iterating the skew RSK correspondence discovered by Sagan and Stanley in the late 1980s, we define deterministic dynamics on the space of pairs of skew Young tableaux $(P,Q)$ . We find that these skew RSK dynamics display conservation laws which, in the picture of Viennot’s shadow line construction, identify generalizations of Greene invariants. The introduction of a novel realization of $0$ -th Kashiwara
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On Thakur’s basis conjecture for multiple zeta values in positive characteristic Forum Math. Pi (IF 2.955) Pub Date : 2023-10-12 Chieh-Yu Chang, Yen-Tsung Chen, Yoshinori Mishiba
In this paper, we study multiple zeta values (abbreviated as MZV’s) over function fields in positive characteristic. Our main result is to prove Thakur’s basis conjecture, which plays the analogue of Hoffman’s basis conjecture for real MZV’s. As a consequence, we derive Todd’s dimension conjecture, which is the analogue of Zagier’s dimension conjecture for classical real MZV’s.
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Interpolation for Brill–Noether curves Forum Math. Pi (IF 2.955) Pub Date : 2023-10-11 Eric Larson, Isabel Vogt
In this paper, we determine the number of general points through which a Brill–Noether curve of fixed degree and genus in any projective space can be passed.
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Stellahedral geometry of matroids Forum Math. Pi (IF 2.955) Pub Date : 2023-10-09 Christopher Eur, June Huh, Matt Larson
We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob
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On a multi-parameter variant of the Bellow–Furstenberg problem Forum Math. Pi (IF 2.955) Pub Date : 2023-09-19 Jean Bourgain, Mariusz Mirek, Elias M. Stein, James Wright
We prove convergence in norm and pointwise almost everywhere on $L^p$ , $p\in (1,\infty )$ , for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment
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Hodge classes and the Jacquet–Langlands correspondence Forum Math. Pi (IF 2.955) Pub Date : 2023-09-06 Atsushi Ichino, Kartik Prasanna
We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\ell $ -adic cohomology is Galois invariant for all $\ell $ .
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Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture Forum Math. Pi (IF 2.955) Pub Date : 2023-08-24 Matthew Kwan, Ashwin Sah, Lisa Sauermann, Mehtaab Sawhney
An n-vertex graph is called C-Ramsey if it has no clique or independent set of size $C\log _2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of
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Exotic Monoidal Structures and Abstractly Automorphic Representations for Forum Math. Pi (IF 2.955) Pub Date : 2023-08-03 Gal Dor
We use the theta correspondence to study the equivalence between Godement–Jacquet and Jacquet–Langlands L-functions for ${\mathrm {GL}}(2)$ . We show that the resulting comparison is in fact an expression of an exotic symmetric monoidal structure on the category of ${\mathrm {GL}}(2)$ -modules. Moreover, this enables us to construct an abelian category of abstractly automorphic representations, whose
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Resolution of the Erdős–Sauer problem on regular subgraphs Forum Math. Pi (IF 2.955) Pub Date : 2023-07-24 Oliver Janzer, Benny Sudakov
In this paper, we completely resolve the well-known problem of Erdős and Sauer from 1975 which asks for the maximum number of edges an n-vertex graph can have without containing a k-regular subgraph, for some fixed integer $k\geq 3$ . We prove that any n-vertex graph with average degree at least $C_k\log \log n$ contains a k-regular subgraph. This matches the lower bound of Pyber, Rödl and Szemerédi
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A proof of the Erdős primitive set conjecture Forum Math. Pi (IF 2.955) Pub Date : 2023-06-14 Jared Duker Lichtman
A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum _{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative. As further applications of the method, we make progress towards
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New lower bounds for matrix multiplication and Forum Math. Pi (IF 2.955) Pub Date : 2023-05-29 Austin Conner, Alicia Harper, J.M. Landsberg
Let $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ denote the matrix multiplication tensor (and write $M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ ), and let $\operatorname {det}_3\in (\mathbb
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Factorisation de la cohomologie étale p-adique de la tour de Drinfeld Forum Math. Pi (IF 2.955) Pub Date : 2023-05-26 Pierre Colmez, Gabriel Dospinescu, Wiesława Nizioł
Résumé For a finite extension F of ${\mathbf Q}_p$ , Drinfeld defined a tower of coverings of (the Drinfeld half-plane). For $F = {\mathbf Q}_p$ , we describe a decomposition of the p-adic geometric étale cohomology of this tower analogous to Emerton’s decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finiteness theorem for the arithmetic étale cohomology
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The Asymptotic Statistics of Random Covering Surfaces Forum Math. Pi (IF 2.955) Pub Date : 2023-05-15 Michael Magee, Doron Puder
Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed
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Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics Forum Math. Pi (IF 2.955) Pub Date : 2023-05-08 Zhuchao Ji, Junyi Xie
The aims of this paper are to answer several conjectures and questions about the multiplier spectrum of rational maps and giving new proofs of several rigidity theorems in complex dynamics by combining tools from complex and non-Archimedean dynamics. A remarkable theorem due to McMullen asserts that, aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy
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Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals Forum Math. Pi (IF 2.955) Pub Date : 2023-05-02 Kenny De Commer, Sergey Neshveyev, Lars Tuset, Makoto Yamashita
We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators
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On the Torelli Lie algebra Forum Math. Pi (IF 2.955) Pub Date : 2023-04-14 Alexander Kupers, Oscar Randal-Williams
We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus g: Stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representations lying in the centre.
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Rigid continuation paths II. structured polynomial systems Forum Math. Pi (IF 2.955) Pub Date : 2023-04-14 Peter Bürgisser, Felipe Cucker, Pierre Lairez
This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random
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On the relative minimal model program for fourfolds in positive and mixed characteristic Forum Math. Pi (IF 2.955) Pub Date : 2023-03-24 Christopher Hacon, Jakub Witaszek
We show the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic $p>5$ : for contractions to ${\mathbb {Q}}$ -factorial fourfolds and in families over curves (‘semistable MMP’). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability
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The existence of the Kähler–Ricci soliton degeneration Forum Math. Pi (IF 2.955) Pub Date : 2023-03-10 Harold Blum, Yuchen Liu, Chenyang Xu, Ziquan Zhuang
We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .
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A Proof of the Extended Delta Conjecture Forum Math. Pi (IF 2.955) Pub Date : 2023-02-22 Jonah Blasiak, Mark Haiman, Jennifer Morse, Anna Pun, George H. Seelinger
We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is
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A Shuffle Theorem for Paths Under Any Line Forum Math. Pi (IF 2.955) Pub Date : 2023-02-22 Jonah Blasiak, Mark Haiman, Jennifer Morse, Anna Pun, George H. Seelinger
We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra
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Sharp smoothing properties of averages over curves Forum Math. Pi (IF 2.955) Pub Date : 2023-02-10 Hyerim Ko, Sanghyuk Lee, Sewook Oh
We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma $ in $\mathbb R^d$ , $d\ge 3$ . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity
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Stable anisotropic minimal hypersurfaces in Forum Math. Pi (IF 2.955) Pub Date : 2023-02-02 Otis Chodosh, Chao Li
We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf {R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular
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Prismatic Dieudonné Theory Forum Math. Pi (IF 2.955) Pub Date : 2023-01-06 Johannes Anschütz, Arthur-César Le Bras
We define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES129 (2019), 199–310), a category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to $\mathrm {DM}^{\mathrm {adm}}(R)$ . We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently
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Syntomic complexes and p-adic étale Tate twists Forum Math. Pi (IF 2.955) Pub Date : 2023-01-05 Bhargav Bhatt, Akhil Mathew
The primary goal of this paper is to identify syntomic complexes with the p-adic étale Tate twists of Geisser–Sato–Schneider on regular p-torsion-free schemes. Our methods apply naturally to a broader class of schemes that we call ‘F-smooth’. The F-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.
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Quantitative Heegaard Floer cohomology and the Calabi invariant Forum Math. Pi (IF 2.955) Pub Date : 2022-12-21 Daniel Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak, Sobhan Seyfaddini, Ivan Smith
We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any
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On some p-differential graded link homologies Forum Math. Pi (IF 2.955) Pub Date : 2022-12-19 You Qi, Joshua Sussan
We show that the triply graded Khovanov–Rozansky homology of knots and links over a field of positive odd characteristic p descends to an invariant in the homotopy category finite-dimensional p-complexes. A p-extended differential on the triply graded homology discovered by Cautis is compatible with the p-DG structure. As a consequence, we get a categorification of the Jones polynomial evaluated at
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Tits Alternative for -dimensional complexes Forum Math. Pi (IF 2.955) Pub Date : 2022-12-16 Damian Osajda, Piotr Przytycki
We prove the Tits Alternative for groups acting on $2$ -dimensional $\mathrm {CAT}(0)$ complexes with a bound on the order of the cell stabilisers.
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Local newforms for the general linear groups over a non-archimedean local field Forum Math. Pi (IF 2.955) Pub Date : 2022-12-13 Hiraku Atobe, Satoshi Kondo, Seidai Yasuda
In [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups of p-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of
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Planar random-cluster model: scaling relations Forum Math. Pi (IF 2.955) Pub Date : 2022-11-22 Hugo Duminil-Copin, Ioan Manolescu
This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in [1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $\beta $ , $\gamma $ , $\delta $ , $\eta $ , $\nu $ , $\zeta $ as well as $\alpha $ (when $\alpha \ge 0$ ). As a key input, we show the stability
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Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension Forum Math. Pi (IF 2.955) Pub Date : 2022-10-03 Mircea Mustaţă, Mihnea Popa
We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.
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Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields Forum Math. Pi (IF 2.955) Pub Date : 2022-09-26 Ananth N. Shankar, Arul Shankar, Yunqing Tang, Salim Tayou
Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline {K}}$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional
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Virasoro constraints for stable pairs on toric threefolds Forum Math. Pi (IF 2.955) Pub Date : 2022-09-06 Miguel Moreira, Alexei Oblomkov, Andrei Okounkov, Rahul Pandharipande
Using new explicit formulas for the stationary Gromov–Witten/Pandharipande–Thomas ( $\mathrm {GW}/{\mathrm {PT}}$ ) descendent correspondence for nonsingular projective toric threefolds, we show that the correspondence intertwines the Virasoro constraints in Gromov–Witten theory for stable maps with the Virasoro constraints for stable pairs proposed in [18]. Since the Virasoro constraints in Gromov–Witten
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On genus one mirror symmetry in higher dimensions and the BCOV conjectures Forum Math. Pi (IF 2.955) Pub Date : 2022-08-31 Dennis Eriksson, Gerard Freixas i Montplet, Christophe Mourougane
The mathematical physicists Bershadsky–Cecotti–Ooguri–Vafa (BCOV) proposed, in a seminal article from 1994, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck–Riemann–Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann–Roch theorem of Gillet–Soulé
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New lower bounds for van der Waerden numbers Forum Math. Pi (IF 2.955) Pub Date : 2022-07-13 Ben Green
We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that $w(3
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Quadratic Klein-Gordon equations with a potential in one dimension Forum Math. Pi (IF 2.955) Pub Date : 2022-07-11 Pierre Germain, Fabio Pusateri
This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time)
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The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials Forum Math. Pi (IF 2.955) Pub Date : 2022-07-01 Matteo Costantini, Martin Möller, Jonathan Zachhuber
For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification. The main new technical
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K-stability of Fano varieties via admissible flags Forum Math. Pi (IF 2.955) Pub Date : 2022-06-30 Hamid Abban, Ziquan Zhuang
We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst
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Undecidability of the Spectral Gap Forum Math. Pi (IF 2.955) Pub Date : 2022-06-10 Toby Cubitt, David Perez-Garcia, Michael M. Wolf
We construct families of translationally invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above
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On the Kottwitz conjecture for local shtuka spaces Forum Math. Pi (IF 2.955) Pub Date : 2022-05-26 David Hansen, Tasho Kaletha, Jared Weinstein
Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group
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Almost all orbits of the Collatz map attain almost bounded values Forum Math. Pi (IF 2.955) Pub Date : 2022-05-20 Terence Tao
Define the Collatz map ${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$ on the positive integers $\mathbb {N}+1 = \{1,2,3,\dots \}$ by setting ${\operatorname {Col}}(N)$ equal to $3N+1$ when N is odd and $N/2$ when N is even, and let ${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$ denote the minimal element of the Collatz orbit $N, {\operatorname
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Hensel minimality I Forum Math. Pi (IF 2.955) Pub Date : 2022-05-16 Raf Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi
We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel
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KP governs random growth off a 1-dimensional substrate Forum Math. Pi (IF 2.955) Pub Date : 2022-04-21 Jeremy Quastel, Daniel Remenik
The logarithmic derivative of the marginal distributions of randomly fluctuating interfaces in one dimension on a large scale evolve according to the Kadomtsev–Petviashvili (KP) equation. This is derived algebraically from a Fredholm determinant obtained in [MQR17] for the Kardar–Parisi–Zhang (KPZ) fixed point as the limit of the transition probabilities of TASEP, a special solvable model in the KPZ
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Grothendieck–Serre in the quasi-split unramified case Forum Math. Pi (IF 2.955) Pub Date : 2022-03-29 Kęstutis Česnavičius
The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete
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Combinatorial and harmonic-analytic methods for integer tilings Forum Math. Pi (IF 2.955) Pub Date : 2022-03-09 Izabella Łaba, Itay Londner
A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B. In
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On locally analytic vectors of the completed cohomology of modular curves Forum Math. Pi (IF 2.955) Pub Date : 2022-03-07 Lue Pan
We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak {gl}_2(\mathbb {Q}_p)$ . As applications, we prove a classicality result for overconvergent eigenforms of weight 1 and give a new proof of the Fontaine–Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent
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Point counting for foliations over number fields Forum Math. Pi (IF 2.955) Pub Date : 2022-03-04 Gal Binyamini
Let${\mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${\mathbb K}$. For an algebraic $V\subset {\mathbb M}$ over ${\mathbb K}$, write $\delta _{V}$ for the maximum of the degree and log-height of V. Write $\Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${\mathcal B}$ of a leaf ${\mathcal L}$. We prove effective