Asymptotic Spectrum and Approximation Approaches to Direct-sum Problems

What is the cost of a task if we have to perform it many times? Can we achieve economies of scale? Such "direct-sum problems" play a central role in many areas of mathematics, physics and computer science. Protagonistic problems of this kind are fast matrix multiplication in algebraic complexity theory (Strassen 1969) and Shannon capacity in graph theory (Shannon 1956). Despite much effort, both problems are wide open.

I will give a brief introduction to the asymptotic spectrum duality approach to direct-sum problems, which Strassen originally introduced to study matrix multiplication (asymptotic tensor rank), and which has recently found applications in various areas, including Shannon's capacity (see e.g. the survey Wigderson-Zuiddam 2025).

Next, I will discuss recent approximation (or limit) approaches to problems of this kind. For instance, to determine say the Shannon capacity of a "hard" graph, we construct a sequence of easier to analyse graphs "converging" to it---and similarly for matrix multiplication. What are the right notions of convergence, and how can such sequences be constructed? We will see some examples, including a natural notion of asymptotic spectrum distance arising from Strassen's duality, and another route via polynomials (algebraic geometry).

Based on joint work with David de Boer, Pjotr Buys, Sven Polak, Matthias Christandl, Koen Hoeberechts, Harold Nieuwboer, and Péter Vrana.