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1. Determinant of F_p-hypergeometric solutions under ample reduction

   Alexander Varchenko (January 2022, Contemporary mathematics)

   We consider the KZ differential equations over C in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field F_p. We study the polynomial solutions of these differential equations over F_p, constructed in a previous work joint with V. Schechtman and called the F_p-hypergeometric solutions. The dimension of the space of F_p-hypergeometric solutions depends on the prime number p. We say that the KZ equations have ample reduction for a prime p, if the dimension of the space of F_p-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over C. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis F_p-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials (z_i−z_j)^{M_i+M_j} are replaced with (z_i−z_j)^{Mi+Mj−p} and the Euler gamma function Γ(x) is replaced with a suitable F_p-analog defined on F_p 

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2. Constructing linear systems with particular kinds of solution sets

   Smith, J.; Lee, I.; Zandieh, M.; Andrews-Larson, C. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S.; & Higgins, A. (Ed.)

   Systems of equations is a core topic in linear algebra courses. Solving systems with no or infinitely many solutions tends to be less intuitive for students. In this study, we examined two students’ reasoning about the relationship between the structure of a system of linear equations and its solution set, particularly when creating systems with a certain number of equations and unknowns. Using data from a paired teaching experiment, we found that both students favored the notion of parallel planes, both geometrically and numerically, in the case of a system having no solution or infinitely many solutions. We also found that algebraic or numerical approaches were used as the main way of developing systems with a unique solution, especially in systems with more than two equations and two unknowns. In particular, one student gravitated toward geometric approaches and the other toward algebraic and numerical approaches. 

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3. Constructing linear systems with particular kinds of solution sets.

   Smith, J.; Lee, I.; Zandieh, M.; & Andrews-Larson, C. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S.; Higgins, A. (Ed.)

   Systems of equations is a core topic in linear algebra courses. Solving systems with no or infinitely many solutions tends to be less intuitive for students. In this study, we examined two students’ reasoning about the relationship between the structure of a system of linear equations and its solution set, particularly when creating systems with a certain number of equations and unknowns. Using data from a paired teaching experiment, we found that both students favored the notion of parallel planes, both geometrically and numerically, in the case of a system having no solution or infinitely many solutions. We also found that algebraic or numerical approaches were used as the main way of developing systems with a unique solution, especially in systems with more than two equations and two unknowns. In particular, one student gravitated toward geometric approaches and the other toward algebraic and numerical approaches. 

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4. Constructing linear systems with particular kinds of solution sets

   Smith, J. (January 2022, 24th conference on research in undergraduate mathematics education)

   S. S. Karunakaran; A. Higgins (Ed.)

   Systems of equations is a core topic in linear algebra courses. Solving systems with no or infinitely many solutions tends to be less intuitive for students. In this study, we examined two students’ reasoning about the relationship between the structure of a system of linear equations and its solution set, particularly when creating systems with a certain number of equations and unknowns. Using data from a paired teaching experiment, we found that both students favored the notion of parallel planes, both geometrically and numerically, in the case of a system having no solution or infinitely many solutions. We also found that algebraic or numerical approaches were used as the main way of developing systems with a unique solution, especially in systems with more than two equations and two unknowns. In particular, one student gravitated toward geometric approaches and the other toward algebraic and numerical approaches. 

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5. Counting the number of stationary solutions of partial differential equations via infinite dimensional sampling

   https://doi.org/10.1098/rsta.2024.0239  

   Kolodziejczyk, Martin; Ottobre, Michela; Simpson, Gideon (June 2025, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   This paper is concerned with the problem of counting solutions of stationary nonlinear Partial Differential Equations (PDEs) when the PDE is known to admit more than one solution. We suggest tackling the problem via a sampling-based approach. The method allows one to find solutions that are stable, in the sense that they are stable equilibria of the associated time-dependent PDE. We test our proposed methodology on the McKean–Vlasov PDE, more precisely on the problem of determining the number of stationary solutions of the McKean–Vlasov equation. This article is part of the theme issue ‘Partial differential equations in data science’. 

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