数学物理

We present several known solutions to the two-dimensional Ising model. This review originated from the ``Ising 100'' seminar series held at Bo\u{g}azi\c{c}i University, Istanbul, in 2024.

This paper analyzes the scattering matrix for two unbounded self-adjoint operators: the standard Laplace operator in three-dimensional space and a second operator that differs from the first by an infinite sum of zero-range potentials.

Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph…

Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of…

In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three…

The Thouless theory of quantum pumps establishes the conditions for quantized particle transport per cycle, and determines its value. When describing the pump from a moving reference frame, transported and existing charges transform, though…

This paper presents the axioms for a quantum Yang-Mills theory in the Minkowski spacetime. There are two routes of analytic continuation for the Schwinger functions, namely the Wightman functions and time-ordered products of field…

We apply the cobordism hypothesis with singularities to the case of affine Rozansky--Witten models, providing a construction of extended TQFTs that includes all line and surface defects. On a technical level, this amounts to proving that…

We construct extended TQFTs associated to Rozansky--Witten models with target manifolds $T^*\mathbb{C}^n$. The starting point of the construction is the 3-category whose objects are such Rozansky--Witten models, and whose morphisms are…

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $\mathfrak{osp}_{2m+1|2n}$ and $\mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a…

For each of the forty-eight exceptional algebraic solutions $u(x)$ of the sixth equation of Painlev\'e, we build the algebraic curve $P(u,x)=0$ of a degree conjectured to be minimal, then we give an optimal parametric representation of it.…

We provide an introductory exposition to the sheaf topos theoretic description of classical field theory motivated by the rigorous description of both $\bf{(i)}$ the variational calculus of (infinite dimensional) field-theoretic spaces, and…

Explicit formulas to calculate MV functions in a basis-free representation are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on analysis of the roots of minimal MV polynomial and covers defective MVs,…

This study proposes a simple and novel class of stretch-limiting constitutive relations for perfectly flexible elastic strings drawing from modern advances in constitutive theory for elastic bodies. We investigate strings governed by…

We obtain explicit formulas for capped descendent vertex functions of $\text{Hilb}^n(\mathbb{C}^2)$ for descendents given by the exterior algebra of the tautological bundle. This formula provides a one-parametric deformation of the…

The probabilistic characterization of non-Markovian responses to nonlinear dynamical systems under colored excitation is an important issue, arising in many applications. Extending the Fokker-Planck-Kolmogorov equation, governing the…

We study a Gaussian measure with parameter $q\in(0,1)$ on the dual of the unitary group of size $N$: we prove that a random highest weight under this measure is the coupling of two independent $q$-uniform random partitions $\alpha,\beta$…

We examine generalized global symmetries as a kind of compactly supported cohomology, and so are led to revisit questions about the locality of quantum field theory, following Segal. Physics naturally suggests a generalization of…

In this paper, we establish a correspondence between algebraic and analytic approaches to constructing representations of the braid group $B_n$, namely Katz-Long-Moody construction and multiplicative middle convolution for…

A novel approach to the finite dimensional representation theory of the entire Lorentz group $\operatorname{O}(1,3)$ is presented. It is shown how the entire Lorentz group may be understood as a semi-direct product between its identity…