Jonas Schnitzer

Research Interests

My research interest is centered around Poisson geometry and deformation quantization, understood in a broad sense, and its application to (mathematical) physics.

Poisson Manifolds

A Poisson manifold is a smooth manifold $M$ equipped with a Lie bracket $\{ \cdot, \cdot \}$ on the smooth functions $C^\infty(M)$ , which is derivative in both slots. Examples of Poisson manifolds include dual spaces of finite-dimensional real Lie algebras and symplectic manifolds. Poisson manifolds can be considered as singular symplectic manifolds.

Deformation Quantization

A deformation quantization of a Poisson manifold $(M, \{ \cdot, \cdot \})$ is a formal associative deformation of the pointwise product of smooth functions along the direction of the Poisson bracket. This is referred to as a star product $\star$ on $C^\infty(M)[[\hbar]]$ , and is defined as follows:

$f \star 1 = 1 \star f = f$
$f \star g = fg + \mathcal{O}(\hbar)$
$[f,g]_\star = f \star g - g \star f = i \hbar \{ f,g \} + \mathcal{O}(\hbar^2)$

Here, $f, g \in C^\infty(M)$ , and the individual terms of the expansion $\star = \sum_{k \geq 0} \hbar^k C_k$ are given by bidifferential operators.

My Interests

I work on semi-local models of poisson-related geometries, such as Jacobi bundles and Dirac structures, quantization of Poisson manifolds and the interplay of (semi-)local approximations and quantization. Moreover, I am interested in symmetries in Poisson geometry as well as reduction theory combined with quantizations. Recently, I got interested in graded (symplectic) geometry and its interplay with quantization.

Education

I did my undergraduate education at the University of Würzburg, where I received my B.Sc. (2010-2014, thesis supervisor: Knut Hüper) and my M.Sc. (2014-2016, thesis advisors: Chiara Esposito, Stefan Waldmann). I received my Ph.D. under the supervision of Luca Vitagliano at the University of Salerno (IT). Here you can find my thesis. After that, I did a PostDoc in the differential geometry group at the mathematical institute of the University of Freiburg. At the moment, I am a PostDoc in the Mathematical Physics group at the University of Pavia under the supervision of Michele Schiavina. You can find my full CV here.

Some online talks

Global homotopies for differential Hochschild cohomologies: https://www.youtube.com/watch?v=5jVNqSvlRQk
Deformations of Lagrangian NQ-submanifolds: https://www.youtube.com/watch?v=5L2tsudF-ns
Strong homotopy structure of Poisson reduction: https://www.youtube.com/watch?v=ITre61nj_Rg
The Quantization of momentum maps and adapted formality morphisms: https://www.youtube.com/watch?v=Ys6jaabD23c
Some lectures on $L_\infty$ -algebras with J.P. Pridham and S. Barmeier: https://www.youtube.com/@deformation-theory-freiburg

Publications & Preprints

Publications:

Deformations of Lagrangian NQ-submanifolds, Adv. Math. 458, 109952 (2024) arXiv journal
No-go theorems for r-matrices in symplectic geometry, Commun. Anal. Mech. 16 (3), 448-456 (2024) journal
The Homotopy Class of twisted L_∞-morphisms, with A.Kraft, Homology Homotopy Appl. 26 (1), 201–227 (2024) arXiv journal
An Introduction to L_∞-Algebras and their Homotopy Theory for the working Mathematician, with A.Kraft, Rev. Math. Phys. 36 (01), 2330006 (2024) arXiv journal
The Strong Homotopy Structure of BRST Reduction, with C.Esposito, A.Kraft, Pacific J. Math. 325 (1), 47–83 (2023) arXiv, journal
Normal Forms for Dirac-Jacobi bundles and Splitting Theorems for Jacobi Structures, Math. Z. 303 (74), (2023) arXiv, journal
Weak Dual Pairs in Dirac-Jacobi Geometry, with A.G.Tortorella, Commun. Contemp. Math. 25 (8), 2250035 (2023) arXiv, journal
The Strong Homotopy Structure of Poisson Reduction, with C.Esposito, A.Kraft, J. Noncommutative Geom. 16 (3), 927-966 (2022) arXiv, journal
Characteristic (Fedosov-)class of a twist constructed by Drinfel’d, Lett. Math. Phys. 110, 2353–2361 (2020) arXiv, journal
Weakly Regular Jacobi Structures and Generalized Contact Bundles Ann. Global Anal. Geom. 56, 221–244 (2019) arXiv, journal
The local Structure of generalized contact bundles, with L.Vitagliano, Int. Math. Res. Not. (IMRN) 20, 6871–6925 (2020) arXiv, journal
A Universal Construction of Universal Deformation Formulas, Drinfel'd Twists and their Positivity, with C.Esposito, S.Waldmann, Pacific J. Math. 219 (2), 319-358 (2017) arXiv, journal

Preprints:

Quantization of the Momentum Map via $\mathfrak{g}$ -adapted Formalities, with C.Esposito, R. Nest, B. Tsygan , arXiv
Global Homotopies for Differential Hochschild Cohomologies, with M. Dippell, C.Esposito, S.Waldmann, arXiv