Forum of Mathematics Pi最新文献_第8页

Embedding codimension of the space of arcs 弧空间的嵌入余维

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-01-23 DOI: 10.1017/fmp.2021.19

C. Chiu, Tommaso de Fernex, Roi Docampo

引用次数: 7

Holomorphic anomaly equation for $({mathbb P}^2,E)$ and the Nekrasov-Shatashvili limit of local ${mathbb P}^2$ $（｛mathbb P｝^2，E）$的全纯异常方程和局部$｛math bb P｝^2的Nekrasov-Shatashvili极限$

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-01-15 DOI: 10.1017/fmp.2021.3

Pierrick Bousseau, H. Fan, Shuai Guo, Longting Wu

{"title":"Holomorphic anomaly equation for $({mathbb P}^2,E)$ and the Nekrasov-Shatashvili limit of local ${mathbb P}^2$","authors":"Pierrick Bousseau, H. Fan, Shuai Guo, Longting Wu","doi":"10.1017/fmp.2021.3","DOIUrl":"https://doi.org/10.1017/fmp.2021.3","url":null,"abstract":"Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $lambda _g$-insertion is related to Gromov-Witten theory of the total space of ${mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={mathbb P}^2$, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${mathbb P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${mathbb P}^2$ in the Nekrasov-Shatashvili limit.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2021.3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43779572","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}

引用次数: 7

Cordial elements and dimensions of affine Deligne–Lusztig varieties 仿射Deligne–Lusztig变种的Cordial元素和维数

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-01-10 DOI: 10.1017/fmp.2021.10

Xuhua He

引用次数: 17

On the derivation of the wave kinetic equation for NLS NLS波动动力学方程的推导

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2019-12-19 DOI: 10.1017/fmp.2021.6

Yu Deng, Z. Hani

{"title":"On the derivation of the wave kinetic equation for NLS","authors":"Yu Deng, Z. Hani","doi":"10.1017/fmp.2021.6","DOIUrl":"https://doi.org/10.1017/fmp.2021.6","url":null,"abstract":"Abstract A fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a large kinetic time scale \u0000$T_{mathrm {kin}} gg 1$\u0000 and in a limiting regime where the size L of the domain goes to infinity and the strength \u0000$alpha $\u0000 of the nonlinearity goes to \u0000$0$\u0000 (weak nonlinearity). For the cubic nonlinear Schrödinger equation, \u0000$T_{mathrm {kin}}=Oleft (alpha ^{-2}right )$\u0000 and \u0000$alpha $\u0000 is related to the conserved mass \u0000$lambda $\u0000 of the solution via \u0000$alpha =lambda ^2 L^{-d}$\u0000 . In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particular scaling law in which the \u0000$(alpha , L)$\u0000 limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to be two favourable scaling laws: when \u0000$alpha $\u0000 approaches \u0000$0$\u0000 like \u0000$L^{-varepsilon +}$\u0000 or like \u0000$L^{-1-frac {varepsilon }{2}+}$\u0000 (for arbitrary small \u0000$varepsilon $\u0000 ), we exhibit the wave kinetic equation up to time scales \u0000$O(T_{mathrm {kin}}L^{-varepsilon })$\u0000 , by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales \u0000$T_*ll T_{mathrm {kin}}$\u0000 and identify specific interactions that become very large for times beyond \u0000$T_*$\u0000 . In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond \u0000$T_*$\u0000 toward \u0000$T_{mathrm {kin}}$\u0000 for such scaling laws seems to require new methods and ideas.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2021.6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49053477","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}

引用次数: 41

On genus one mirror symmetry in higher dimensions and the BCOV conjectures 高维亏格单镜对称性与BCOV猜想

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2019-11-14 DOI: 10.1017/fmp.2022.13

Dennis Eriksson, Gerard Freixas i Montplet, Christophe Mourougane

引用次数: 8

Hensel minimality I Hensel极小性

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2019-09-30 DOI: 10.1017/fmp.2022.6

R. Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi

引用次数: 21

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields K3曲面约化的皮卡德秩在数域上的异常跳跃

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2019-09-16 DOI: 10.1017/fmp.2022.14

A. Shankar, A. Shankar, Yunqing Tang, Salim Tayou

引用次数: 13

Almost all orbits of the Collatz map attain almost bounded values Collatz映射的几乎所有轨道都达到几乎有界值

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2019-09-08 DOI: 10.1017/fmp.2022.8

T. Tao

{"title":"Almost all orbits of the Collatz map attain almost bounded values","authors":"T. Tao","doi":"10.1017/fmp.2022.8","DOIUrl":"https://doi.org/10.1017/fmp.2022.8","url":null,"abstract":"Abstract Define the Collatz map \u0000${operatorname {Col}} colon mathbb {N}+1 to mathbb {N}+1$\u0000 on the positive integers \u0000$mathbb {N}+1 = {1,2,3,dots }$\u0000 by setting \u0000${operatorname {Col}}(N)$\u0000 equal to \u0000$3N+1$\u0000 when N is odd and \u0000$N/2$\u0000 when N is even, and let \u0000${operatorname {Col}}_{min }(N) := inf _{n in mathbb {N}} {operatorname {Col}}^n(N)$\u0000 denote the minimal element of the Collatz orbit \u0000$N, {operatorname {Col}}(N), {operatorname {Col}}^2(N), dots $\u0000 . The infamous Collatz conjecture asserts that \u0000${operatorname {Col}}_{min }(N)=1$\u0000 for all \u0000$N in mathbb {N}+1$\u0000 . Previously, it was shown by Korec that for any \u0000$theta> frac {log 3}{log 4} approx 0.7924$\u0000 , one has \u0000${operatorname {Col}}_{min }(N) leq N^theta $\u0000 for almost all \u0000$N in mathbb {N}+1$\u0000 (in the sense of natural density). In this paper, we show that for any function \u0000$f colon mathbb {N}+1 to mathbb {R}$\u0000 with \u0000$lim _{N to infty } f(N)=+infty $\u0000 , one has \u0000${operatorname {Col}}_{min }(N) leq f(N)$\u0000 for almost all \u0000$N in mathbb {N}+1$\u0000 (in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a \u0000$3$\u0000 -adic cyclic group \u0000$mathbb {Z}/3^nmathbb {Z}$\u0000 at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2019-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44030374","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}

引用次数: 61

Bounds for sets with no polynomial progressions 无多项式级数集的界

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2019-09-01 DOI: 10.1017/fmp.2020.11

Sarah Peluse

引用次数: 17

Smoothing toroidal crossing spaces 平滑环形交叉空间

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2019-08-29 DOI: 10.1017/fmp.2021.8

Simon Felten, Matej Filip, Helge Ruddat

引用次数: 27