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Amitai Netser Zernik

Publications and source records attributed to Amitai Netser Zernik.

9 recordsLinked to original sources
arXiv2022

Open $\mathbb{CP}^1$ descendent theory I: The stationary sector

We define stationary descendent integrals on the moduli space of stable maps from disks to $(\mathbb{CP}^1,\mathbb{RP}^1)$. We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus $0$ disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.

math.SG↗
arXiv2018

Exact maximum-entropy estimation with Feynman diagrams

A classical longstanding open problem in statistics is finding an explicit expression for the probability measure which maximizes entropy with respect to given constraints. In this paper a solution to this problem is found, using perturbative Feynman calculus. The explicit expression is given as a sum over weighted trees.

math.CO↗
arXiv2017

Moduli of Open Stable Maps to a Homogeneous Space

For $L \hookrightarrow X$ a Lagrangian embedding associated with a real homogeneous space, we construct the moduli space of stable holomorphic discs mapping to $(X,L)$ as an orbifold with corners equipped with a group action. Some essential constructions involving orbifolds with corners are also discussed, including the existence of fibered products and pushforward and pullback of differential forms with values in a local system.

math.SG↗
arXiv2017

Equivariant Open Gromov-Witten Theory of $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$

We define equivariant open Gromov-Witten invariants for $\mathbb{R}\mathbb{P}^{2m} \hookrightarrow \mathbb{C}\mathbb{P}^{2m}$ as sums of integrals of equivariant forms over resolution spaces, which are blowups of products of moduli spaces of stable disc-maps modeled on trees. These invariants encode the quantum deformation of the equivariant cohomology of $\mathbb{R}\mathbb{P}^{2m}$ by holomorphic discs in $\mathbb{C}\mathbb{P}^{2m}$ and, for $m=1$, specialize to give Welschinger's signed count of real rational planar curves in the non-equivariant limit.

math.SG↗
arXiv2017

Fixed-point Localization for $\mathbb{RP}^{2m} \subset \mathbb{CP}^{2m}$

We derive a fixed-point formula for integrals on moduli spaces of stable maps to projective spaces of even dimension. This gives a formula for the equivariant open Gromov-Witten invariants of (RP^{2m},CP^{2m}) and the structure constants of the equivariant Fukaya A-infinity algebra of RP^2m in CP^{2m} with bulk deformations. The formula involves contributions from Givental's correlators for the closed theory and the descendent integrals of discs, and specializes to give a new expression for the Welschinger count of real rational curves in the plane passing through some real and conjugation invariant pairs of points.

math.SG↗
arXiv2016

Equivariant A-infinity algebras for nonorientable Lagrangians

We set up an algebraic framework for the study of pseudoholomorphic discs bounding nonorientable Lagrangians, as well as equivariant extensions of such structures arising from a torus action. First, we define unital cyclic twisted $A_\infty$ algebras and prove some basic results about them, including a homological perturbation lemma which allows one to construct minimal models of such algebras. We then construct an equivariant extension of $A_\infty$ algebras which are invariant under a torus action on the underlying complex. Finally, we construct a homotopy retraction of the Cartan-Weil complex to equivariant cohomology, which allows us to construct minimal models for equivariant cyclic twisted $A_\infty$ algebras. In a forthcoming paper we will use these results to define and obtain fixed-point expressions for the open Gromov-Witten theory of $\mathbb{RP}^{2n} \hookrightarrow \mathbb{CP}^{2n}$, as well as its equivariant extension.

math.SG↗
arXiv2013

Taylor Expansion Proof of the Matrix Tree Theorem - Part II

The All Minors Matrix Tree Theorem states that the determinant of any submatrix of a matrix whose columns sum to zero can be computed as a sum over certain oriented forests. We offer a particularly short proof of this result, which amounts to comparing Taylor series expansions.

math.CO↗
arXiv2013

Taylor Expansion Proof of the Matrix Tree Theorem - Part I

The Matrix-Tree Theorem states that the number of spanning trees of a graph is given by the absolute value of any cofactor of the Laplacian matrix of the graph. We propose a very short proof of this result which amounts to comparing Taylor expansions.

math.CO↗