On Combinatorial Analysis (2010-)

"Combinatorics seems to be the most fruitful source of easy-to-understand but hard-to-prove theorems, and it is also seems to be the place where insights from the infinite world most clearly illuminate the finite world." (John Stillwell, "Mathematics and Its History", 2010)

"Combinatorics with all its various aspects is a broad field of Mathematics which has many applications in areas like Topology, Group Theory and even Analysis. A reason for its wide range of applications might be that Combinatorics is rather a way of thinking than a homogeneous theory, and consequently Combinatorics is quite difficult to define. Nevertheless, let us start with a definition of Combinatorics which will be suitable for our purpose: Combinatorics is the branch of Mathematics which studies collections of objects that satisfy certain criteria, and is in particular concerned with deciding how large or how small such collections might be." (Lorenz J Halbeisen,"Combinatorial Set Theory: With a Gentle Introduction to Forcing", 2011) 

"Galois and Abel independently discovered the basic idea of symmetry. They were both coming at the problem from the algebra of polynomials, but what they each realized was that underlying the solution of polynomials was a fundamental problem of symmetry. The way that they understood symmetry was in terms of permutation groups. A permutation group is the most fundamental structure of symmetry. […] permutation groups are the master groups of symmetry: every kind of symmetry is encoded in the structure of the permutation group." (Mark C Chu-Carroll, "Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation", 2013)

"Combinatorics is the art and science of distilling a complex mathematical structure into simple attributes and developing from this a deeper understanding of the original structure." (Josephine Yu, "Tropical Combinatorics and Applications", 2016)

"Combinatorics is everything. All our worlds, the physical, mathematical, and even spiritual, are inherently finite and discrete, and so-called infinities, be their actual or potential, as well as the ‘continuum’, are ‘optical illusions’." (Doron Zeilberger, [interview with Enumerative Combinatorics and Applications, by Toufik Mansour", 2020)

"If you ask a mathematician what they love most about mathematics, certain answers invariably arise: beauty, abstraction, creativity, logical structure, connection (between disciplines and between people), elegance, applicability, and fun. [..] Combinatorics is the branch of mathematics best situated to embody and illustrate all of these virtues." (Stephen Melczer, "An Invitation to Analytic Combinatorics", 2021)