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One of the oldest and most fundamental concepts in mathematics is the line. Depending on exactly what mathematical structures we want to study (algebraic, geometric, topological, order-theoretic, etc.), we model lines nowadays by a variety of standard mathematical objects, such as the real line , the complex line , the projective line , the extended real line , the affine line , the continuum , the long line , etc. We also have discrete versions of the line, such as the natural numbers , the integers , and the ordinal , as well as compact versions of the line, such as the unit interval or the unit circle . Finally we have discrete and compact versions of the line, such as the cyclic groups and the discrete intervals and . By taking Cartesian products we then obtain higher-dimensional objects such as Euclidean space , the standard lattice , the standard torus , and so forth. These objects of course form the background on which a very large fraction of modern mathematics is set.
Broadly speaking, the line has three major families of structures on it:
- Geometric structures, such as a metric or a measure, completeness, scales (coarse and fine), rigid motions (translations and reflection), similarities (dilation, affine maps), and differential structures (tangent bundle, etc.);
- Algebraic structures, such group, ring, or field structures, and everything else that comes from those categories (e.g. subgroups, homomorphisms, involutions, etc.); and
- One-dimensional structures, such as order, a length space structure (in particular, path-connectedness structure), a singleton generator, the Archimedean property, the ability to use mathematical induction (i.e. well-ordering), convexity, or the ability to disconnect the line by removing a single point.
Of course, these structures are inter-related, and it is an important phenomenon that a mathematical concept which appears to be native to one structure, can often be equivalently defined in terms of other structures. For instance, the absolute value of an integer can be defined geometrically as the distance from 0 to , algebraically as the index of the subgroup of the integers generated by n, or one-dimensionally as the number of integers between 0 and (including 0, but excluding ). This equivalence of definitions becomes important when one wants to work in more general contexts in which one or more of the above structures is missing or otherwise weakened.
What I want to talk about today is an important toy model for the line (in any of its incarnations), in which the geometric and algebraic structures are enhanced (and become neatly nested and recursive), at the expense of the one-dimensional structure (which is largely destroyed). This model has many different names, depending on what field of mathematics one is working in and which structures one is interested in. In harmonic analysis it is called the dyadic model, the Walsh model, or the Cantor group model; in number theory and arithmetic geometry it is known as the function field model; in topology it is the Cantor space model; in probability it is the martingale model; in metric geometry it is the ultrametric, tree, or non-Archimedean model; in algebraic geometry it is the Puiseux series model; in additive combinatorics it is the bounded torsion or finite field model; in computer science and information theory it is the Hamming cube model; in representation theory it is the Kashiwara crystal model. Let me arbitrarily select one of these terms, and refer to all of these models as dyadic models for the line (or of objects derived from the line). While there is often no direct link between a dyadic model and a non-dyadic model, dyadic models serve as incredibly useful laboratories in which to gain insight and intuition for the “real-world” non-dyadic model, since one has much more powerful and elegant algebraic and geometric structure to play with in this setting (though the loss of one-dimensional structure can be a significant concern). Perhaps the most striking example of this is the three-line proof of the Riemann hypothesis in the function field model of the integers, which I will discuss a little later.
This problem lies in the highly interconnected interface between algebraic combinatorics (esp. the combinatorics of Young tableaux and related objects, including honeycombs and puzzles), algebraic geometry (particularly classical and quantum intersection theory and geometric invariant theory), linear algebra (additive and multiplicative, real and tropical), and the representation theory (classical, quantum, crystal, etc.) of classical groups. (Another open problem in this subject is to find a succinct and descriptive name for the field.) I myself haven’t actively worked in this area for several years, but I still find it a fascinating and beautiful subject. (With respect to the dichotomy between structure and randomness, this subject lies deep within the “structure” end of the spectrum.)
As mentioned above, the problems in this area can be approached from a variety of quite diverse perspectives, but here I will focus on the linear algebra perspective, which is perhaps the most accessible. About nine years ago, Allen Knutson and I introduced a combinatorial gadget, called a honeycomb, which among other things controlled the relationship between the eigenvalues of two arbitrary Hermitian matrices A, B, and the eigenvalues of their sum A+B; this was not the first such gadget that achieved this purpose, but it was a particularly convenient one for studying this problem, in particular it was used to resolve two conjectures in the subject, the saturation conjecture and the Horn conjecture. (These conjectures have since been proven by a variety of other methods.) There is a natural multiplicative version of these problems, which now relates the eigenvalues of two arbitrary unitary matrices U, V and the eigenvalues of their product UV; this led to the “quantum saturation” and “quantum Horn” conjectures, which were proven a couple years ago. However, the quantum analogue of a “honeycomb” remains a mystery; this is the main topic of the current post.