Discrete Lagrangian field theories on Lie groupoids

We present a geometric framework for discrete classical field theories, where fields are modeled as "morphisms" defined on a discrete grid in the base space, and take values in a Lie groupoid. We describe the basic geometric setup and derive the field equations from a variational principle. We also show that the solutions of these equations are multisymplectic in the sense of Bridges and Marsden. The groupoid framework employed here allows us to recover not only some previously known results on discrete multisymplectic field theories, but also to derive a number of new results, most notably a notion of discrete Lie-Poisson equations and discrete reduction. In a final section, we establish the connection with discrete differential geometry and gauge theories on a lattice.Comment: 37 pages, 6 figures, uses xy-pic (v3: minor amendment to def. 3.5; remark 3.7 added

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Hearing shapes of drums - mathematical and physical aspects of isospectrality

In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.Comment: 42 pages, 60 figure

Continuous and discrete aspects of Lagrangian field theories with nonholonomic constraints

This dissertation is a contribution to the differential-geometric treatment of classical field theories. In particular, I study both discrete and continuous aspects of classical field theories, in particular those with nonholonomic constraints. After some introductory chapters dealing with the geometric structures inherent in field theories and the discretization of field theories, the first part of the thesis is concerned with discrete field theories taking values in Lie groupoids. It is shown that many previously known discrete field theories are particular instances of Lie groupoid field theories, and the geometry of Lie groupoids is used to construct a unifying framework for this class. In two further chapters, the effect of symmetry upon this setup is described, with particular attention to the case of Euler-Poincaré reduction, which can be rephrased using concepts of discrete differential geometry. In the second part of the thesis, nonholonomic constraints for field theories are described. A number of differential-geometric results that characterize the nature of nonholonomic constraints are derived: in particular, a version of the De Donder-Weyl equation suitable for constrained field theories is discussed and a so-called momentum lemma is derived (describing the influence of symmetry upon the nonholonomic framework). In the last chapter, a physical example of a nonholonomic field theory is given, based on the theory of Cosserat media. This example is treated using the theory of the preceding chapters. Furthermore, a geometric numerical integration scheme is derived and used to give a quantitative insight into the dynamics

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Enumerative Applications of Integrable Hierarchies

Countably infinite families of partial differential equations such as the Kadomtzev - Petviashvili (KP) hierarchy and the B-type KP (BKP) hierarchy have received much interest in the mathematical and theoretical physics community for over forty years. Recently there has been much interest in the application of these families of partial differential equations to a variety of problems in enumerative combinatorics. For example, the generating series for monotone Hurwitz numbers, studied by Goulden, Guay-Paquet and Novak, is known to be a solution to the KP hierarchy. Using this fact along with some additional constraints we may find a second order, quadratic differential equation for the generating series for simple monotone Hurwitz numbers (a specialization of the problem considered in corresponding to factorizations of the identity). In addition, asymptotic analysis can be performed and it may be shown that the asymptotic behaviour of the simple monotone Hurwitz numbers is governed by the map asymptotics constants studied by Bender, Canfield and Gao. In their enumerative study of various families of maps, Bender, Canfield and Gao proved that for maps embedded in an orientable surface the asymptotic behaviour could be completely determined up to some constant depending only on genus and that similarly for maps embedded on a non-orientable surface the asymptotic behaviour could be determined up to a constant depending on the Euler characteristic of the surface. However, the only known way of computing these constants was via a highly non-linear recursion, making the determination of these constants very difficult. Using the integrable hierarchy approach to enumerative problems, Goulden and Jackson derived a quadratic recurrence for the number of rooted triangulations on an orientable surface of fixed genus. This result was then used by Bender, Richmond and Gao to show that the generating series for the orientable map asymptotics constants was given by a solution to a nonlinear differential equation called the Painlev\'e I equation. This gave a method for computing the orientable map asymptotics constants which was significantly simpler than any previously known method. A remaining open problem was whether a suitable integrable hierarchy could be found which could be applied to the corresponding problems in the non-orientable case. Using the BKP hierarchy of partial differential equations applied to the enumeration of rooted triangulations on all surfaces (orientable or non-orientable) we find a cubic recursion for the number of such triangulations and, as a result, we find a nonlinear differential equation which determines the non-orientable map asymptotics constants. In this thesis we provide a detailed development of both the KP and BKP hierarchies. We also discuss three different applications of these hierarchies, the two mentioned above (monotone Hurwitz numbers and rooted triangulations on locally orientable surfaces) as well as orientable bipartite quadrangulations