Implicit Function Theorem. Part II

In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here.In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.Kazuhisa Nakasho - Yamaguchi University, Yamaguchi, JapanYasunari Shidama - Shinshu University, Nagano, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321–327, 2004.Kazuhisa Nakasho. Invertible operators on Banach spaces. Formalized Mathematics, 27 (2):107–115, 2019. doi:10.2478/forma-2019-0012.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269–275, 2004.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Hideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. Banach’s continuous inverse theorem and closed graph theorem. Formalized Mathematics, 20(4):271–274, 2012. doi:10.2478/v10037-012-0032-y.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.27211713

Inverse Function Theorem. Part I

In this article we formalize in Mizar [1], [2] the inverse function theorem for the class of C1 functions between Banach spaces. In the first section, we prove several theorems about open sets in real norm space, which are needed in the proof of the inverse function theorem. In the next section, we define a function to exchange the order of a product of two normed spaces, namely ↶ ≂ (x, y) ∈ X × Y ↦ (y, x) ∈ Y × X, and formalized its bijective isometric property and several differentiation properties. This map is necessary to change the order of the arguments of a function when deriving the inverse function theorem from the implicit function theorem proved in [6]. In the third section, using the implicit function theorem, we prove a theorem that is a necessary component of the proof of the inverse function theorem. In the last section, we finally formalized an inverse function theorem for class of C1 functions between Banach spaces. We referred to [9], [10], and [3] in the formalization.This study has been supported in part by JSPS KAKENHI Grant Numbers JP20K19863 and JP17K00182.Kazuhisa Nakasho - Yamaguchi University, Yamaguchi, JapanYuichi Futa - Tokyo University of Technology, Tokyo, JapanGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321–327, 2004.Kazuhisa Nakasho. Invertible operators on Banach spaces. Formalized Mathematics, 27(2):107–115, 2019. doi:10.2478/forma-2019-0012.Kazuhisa Nakasho and Yasunari Shidama. Implicit function theorem. Part II. Formalized Mathematics, 27(2):117–131, 2019. doi:10.2478/forma-2019-0013.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026.Hideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. Banach’s continuous inverse theorem and closed graph theorem. Formalized Mathematics, 20(4):271–274, 2012. doi:10.2478/v10037-012-0032-y.Laurent Schwartz. Th´eorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul diff´erentiel, tome 2. Analyse. Hermann, 1997.29191

The Seiberg-Witten equations on manifolds with boundary

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 249-252).In this thesis, we undertake an in-depth study of the Seiberg-Witten equations on manifolds with boundary. We divide our study into three parts. In Part One, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Here, we study the solution space of these equations without imposing any boundary conditions. We show that the boundary values of this solution space yield an infinite dimensional Lagrangian in the symplectic configuration space on the boundary. One of the main difficulties in this setup is that the three-dimensional Seiberg-Witten equations, being a dimensional reduction of an elliptic system, fail to be elliptic, and so there are resulting technical difficulties intertwining gauge-fixing, elliptic boundary value problems, and symplectic functional analysis. In Part Two, we study the Seiberg-Witten equations on a 3-manifold with cylindrical ends. Here, Morse-Bott techniques adapted to the infinite-dimensional setting allow us to understand topologically the space of solutions to the Seiberg-Witten equations on a semiinfinite cylinder in terms of the finite dimensional moduli space of vortices at the limiting end. By combining this work with the work of Part One, we make progress in understanding how cobordisms between Riemann surfaces may provide Lagrangian correspondences between their respective vortex moduli spaces. Moreover, we apply our results to provide analytic groundwork for Donaldson's TQFT approach to the Seiberg-Witten invariants of closed 3-manifolds. Finally, in Part Three, we study analytic aspects of the Seiberg-Witten equations on a cylindrical 4-manifold supplied with Lagrangian boundary conditions of the type coming from the first part of this thesis. The resulting system of equations constitute a nonlinear infinite-dimensional nonlocal boundary value problem and is highly nontrivial. We prove fundamental elliptic regularity and compactness type results for the corresponding equations, so that these results may therefore serve as foundational analysis for constructing a monopole Floer theory on 3-manifolds with boundary.by Timothy Nguyen.Ph.D