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This PhD thesis deals with a number of different problems in mathematical physics with the common thread that they have probabilistic aspects. The problems all stem from mathematical studies of lattice systems in statistical and quantum physics; however beyond that, the selection of the concrete problems is to a certain extent arbitrary. This thesis consists of an introduction and seven papers., Comment: PhD Thesis, University of Copenhagen. Comments welcome. Previous versions of most papers are already on arXiv

We briefly recount the long friendship that developed between Ludwig and us (Moshe Flato and I), since we first met at ICM 1966 in Moscow. That friendship extended to his school and family, and persists to this day. Its strong personal impact and main scientific components are sketched, including reflexions on what mathematical physics is (or should be)., Comment: 12 pages

In this short paper, the results of the paper by Drab et al. (Eur. Phys. J. E (2022) 45:79) are described in the framework of wave mechanics and mathematical physics based on common understandings. The attention is focused on properties of Boussinesq-type equations, solitons, and peakons. These concepts are supported by several experimental observations.

It is shown with the help of skew-symmetric forms that the mathematical physics equations, on which no additional conditions are imposed, have quantum properties. And this is due to the integrability properties of differential equations, which depends on the consistency of derivatives with respect to different variables and the consistency of equations, if the mathematical physics equations are a system of equations. It was found that such equations on the original tangent space turn out to be non-integrable. Their derivatives do not form a differential. The integrability of such equations is realized only on the structures of a cotangent integrable manifold. This happens using a degenerate, non-differential-preserving transformation that has quantum properties. When implementing degenerate transformations, mini structures (quanta) arise, from which integrable structures are formed. Such properties of integrability of mathematical physics equations and features of degenerate transformations reveal the quantum properties of mathematical physics equations and their ability to generate quantum structures., Comment: 21 pp

This paper surveys the long-standing connections and impact between Vaughan Jones's theory of subfactors and various topics in mathematical physics, namely statistical mechanics,quantum field theory,quantum information and two-dimensional conformal field theory., Comment: 22 pages, 1 figure. To appear in an issue of the Bulletin of the AMS, dedicated to the mathematical legacy of Vaughan Jones

These are the third 50 issues of This Week's Finds of Mathematical Physics, from April 9, 1997 to June 18, 2000. These issues discuss quantum gravity, topological quantum field theory, n-categories and other topics in mathematics and physics. They also contain a series of expository mini-articles on homotopy theory. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird; you can get a better-looking version at https://math.ucr.edu/home/baez/twf_latex/), Comment: 348 pages

We present new closed-form expressions for certain improper integrals of Mathematical Physics such as certain Ising, Box, and Associated integrals. The techniques we employ here include (a) the Method of Brackets and its modifications and suitable extensions to obtain the Mellin-Barnes representation. (b) The evaluation of the resulting Mellin-Barnes representations via the recently discovered Conic Hull method via the automated package $\textit{MBConichulls.wl}$. Finally, the analytic continuations of these series solutions are then produced using the automated package \texttt{Olsson.wl}, based on the method of Olsson. Thus, combining all these recent advances allows for closed-form evaluation of the hitherto unknown $B_3(s)$, $B_4(s)$, and related integrals in terms of multivariable hypergeometric functions. Along the way, we also discuss certain complications while using the Original Method of Brackets for these evaluations and how to rectify them. The interesting cases of $C_{5,k}$ are also studied. It is not yet fully resolved for the reasons we discuss in this paper., Comment: To appear: EPJ ST Special Issue: Frontier 23: Elementary particle physics, Dark matter and Astroparticle physics

This text offers a personal account of the scientific legacy of two giants of mathematical physics at the turn of the Millenium and their heritage in Canada, their land of adoption., Comment: 12 pages, Memorial plenary talk at the 34th International Colloquium on Group Theoretical Methods in Physics, Strasbourg, France, July 2022. A few minor edits in the revised version

Mathematical physics has many facets, of which we shall briefly give a (very partial) description, centered around those of main interest for Elliott and us (Moshe Flato and I). In our case these aspects had as a corollary a variety of "parascientific activities", in particular the foundation of IAMP (the International Association of Mathematical Physics) and of the journal LMP (Letters in Mathematical Physics), both of which were strongly impacted by Elliott, and Elliott's long insistence that publishers do not demand "copyright transfer" as a precondition for publication but are satisfied with "consent to publish", which is increasingly becoming standard. Since this article is testimony to the huge scientific impact of Elliott, the latter intertwined aspects constitute the core of the present contribution. The last part deals briefly with metaphysical and metamathematical considerations related to axioms., Comment: 12 pages, third version (October 2022) of our paper dedicated to our friend Elliott Lieb on the occasion of the ninetieth anniversary of his birth. Some additions in the first two sections, a new "metamathematical" Section 3, and journal reference added

These are the second 50 issues of This Week's Finds of Mathematical Physics, from April 23, 1995 to March 23, 1997. These issues discuss quantum gravity, topological quantum field theory and other topics in mathematics and physics. They also include two expository series: one on ADE classifications and one on categories and higher categories. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird)., Comment: 281 pages

These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird)., Comment: 241 pages

We give an account of the theory of factorization spaces, categories, functors, and algebras, following the approach of [Ras1]. We apply these results to give geometric constructions of factorization $\mathbb{E}_n$ algebras describing mixed holomorphic-topological twists of supersymmetric gauge theories in low dimensions. We formulate and prove several recent predictions from the physics literature in this language: We recall the Coulomb branch construction of [BFN1] from this perspective. We prove a conjecture from [CosG] that the Coulomb branch factorization $\mathbb{E}_1$ algebra $\mathcal{A}(G,N)$ acts on the factorization algebra of chiral differential operators $\mathcal{D}^{ch}(Y)$ on the quotient stack $Y=N/G$. We identify the latter with the semi-infinite cohomology of $\mathcal{D}^{ch}(N)$ with respect to $\hat{\mathfrak{g}}$, following the results of [Ras3]. Both these results require the hypothesis that $Y$ admits a Tate structure, or equivalently that $\mathcal{D}^{ch}(N)$ admits an action of $\hat{\mathfrak{g}}$ at level $\kappa=-\text{Tate}$. We construct an analogous factorization $\mathbb{E}_2$ algebra $\mathcal{F}(Y)$ describing the local observables of the mixed holomorphic-B twist of four dimensional $\mathcal{N}=2$ gauge theory. We apply the theory of equivariant factorization algebras of the prequel [Bu1] in this example: we identify $S^1$ equivariant structures on $\mathcal{F}(Y)$ with Tate structures on $Y=N/G$, and prove that the corresponding filtered quantization of $\iota^!\mathcal{F}(Y)$ is given by the two-periodic Rees algebra of chiral differential operators on $Y$. This gives a mathematical account of the results of [Beem4]. Finally, we apply the equivariant cigar reduction principle of [Bu1] to explain the relationship between these results and our account of the results of [CosG] described above., Comment: 153 pages, 5 figures, Part II of a series

This is an extended version of my 2018 Heinemann prize lecture describing the work for which I got the prize. The citation is very broad so this describes virtually all my work prior to 1995 and some afterwards. It discusses work in non-relativistic quantum mechanics, constructive quantum field theory and statistical mechanics., Comment: This is the final presubmission version which has been submitted to J. Math. Phys