arXiv subjects

Samuel Johnston

Publications and source records attributed to Samuel Johnston.

4 recordsLinked to original sources

arXiv2024

Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten theory of root stacks

Using recent results of Battistella, Nabijou, Ranganathan and the author, we compare candidate mirror algebras associated with certain log Calabi-Yau pairs constructed by Gross-Siebert using log Gromov-Witten theory and Tseng-You using orbifold Gromov- Witten theory of root stacks. Although the structure constants used to defined these mirror algebras do not typically agree, we show that any given structure constant involved in the construction the algebra of Gross and Siebert can be computed in terms of structure constants of the algebra of Tseng and You after a sequence of log blowups. Using this relation, we provide another proof of associativity of the log mirror algebra, and a proof of the weak Frobenius Structure Theorem in full generality. Along the way, we introduce a class of twisted punctured Gromov-Witten invariants of generalized root stacks induced by log étale modifications, and use this to study the behavior of log Gromov-Witten invariants under ramified base change.

math.AG↗

arXiv2022

Birational Invariance in Punctured Log Gromov-Witten Theory

Given a log smooth scheme $(X,D)$, and a log \'etale modification $(\tilde{X},\tilde{D}) \rightarrow (X,D)$, we relate the punctured Gromov-Witten theory of $(\tilde{X},\tilde{D})$ to the punctured Gromov-Witten theory of $(X,D)$, generalizing results of Abramovich and Wise in the non-punctured setting in "Birational invariance in log Gromov-Witten Theory". Using the main comparison results, we show a form of log \'etale invariance for the logarithmic mirror algebras and canonical scattering diagrams constructed in "Intrinsic Mirror Symmetry" and "The Canonical Wall Structure and Intrinsic Mirror Symmetry" respectively.

math.AG↗

arXiv2021

Thin-shell theory for rotationally invariant random simplices

For fixed functions $G,H:[0,\infty)\to[0,\infty)$, consider the rotationally invariant probability density on $\mathbb{R}^n$ of the form \[ μ^n(ds) = \frac{1}{Z_n} G(\|s\|_2)\, e^{ - n H( \|s\|_2)} ds. \] We show that when $n$ is large, the Euclidean norm $\|Y^n\|_2$ of a random vector $Y^n$ distributed according to $μ^n$ satisfies a Gaussian thin-shell property: the distribution of $\|Y^n\|_2$ concentrates around a certain value $s_0$, and the fluctuations of $\|Y^n\|_2$ are approximately Gaussian with the order $1/\sqrt{n}$. We apply this thin shell property to the study of rotationally invariant random simplices, simplices whose vertices consist of the origin as well as independent random vectors $Y_1^n,\ldots,Y_p^n$ distributed according to $μ^n$. We show that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior, providing a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA, Lat. Am. J. Probab. Math. Stat. 16, 141--177 (2019)]. Finally, by relating the volumes of random simplices to random determinants, we show that if $A^n$ is an $n \times n$ random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants $c_0,c_1\in(0,\infty)$ and an absolute constant $C\in(0,\infty)$ such that \[\sup_{ s \in \mathbb{R}} \left| \mathbb{P} \left[ \frac{ \log \mathrm{det}(A^n) - \log(n-1)! - c_0 }{ \sqrt{ \frac{1}{2} \log n + c_1 }} < s \right] - \int_{-\infty}^s \frac{e^{ - u^2/2} du}{ \sqrt{ 2 π}} \right| < \frac{C}{\log^{3/2}n}, \] sharpening the $1/\log^{1/3 + o(1)}n$ bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42 (1) (2014), 146--167].

math.PR↗

arXiv2020

Stability and asymptotic analysis of the Föllmer-Schweizer decomposition on a finite probability space

First, we consider the problem of hedging in complete binomial models. Using the discrete-time Föllmer-Schweizer decomposition, we demonstrate the equivalence of the backward induction and sequential regression approaches. Second, in incomplete trinomial models, we examine the extension of the sequential regression approach for approximation of contingent claims. Then, on a finite probability space, we investigate stability of the discrete-time Föllmer-Schweizer decomposition with respect to perturbations of the stock price dynamics and, finally, perform its asymptotic analysis under simultaneous perturbations of the drift and volatility of the underlying discounted stock price process, where we prove stability and obtain explicit formulas for the leading order correction terms.

q-fin.MF↗