. In 1969, Strassen shocked the computational world with his subcubic algorithm for multiplying matrices. Attempting to understand the best possible algorithm for this problem, Strassen went on to develop his magnificent theory of asymptotic spectra in three papers between 1986-1991. Expressed in the great generality of partially ordered semirings, the centerpiece of this theory is a duality theorem between the asymptotic "rank" of elements and a topological space which is called asymptotic spectrum. This duality theorem is a vast generalization of linear programming duality (in which we have a semigroup rather than a semiring), and indeed also of certain versions of the Positivstellensatz, the duality theorem of polynomial inequalities over the reals. Focusing on understanding the structure of the asymptotic spectrum of matrix multiplication, the theory has provided surprising connectivity and convexity theorems for it. Strassen's theory has led to many subsequent results, especially new algorithmic, structural, and barrier results on matrix multiplication, and, more generally, for the semiring of tensors (which includes the matrix multiplication tensors). Perhaps even more impressively, the generality of Strassen's theory has been applied recently to the study of a variety of very different settings and parameters, in diverse fields including communication theory, graph theory, probability theory, quantum information theory and computational complexity. We feel that these developments call for an exposition of this growing field. This paper gives a comprehensive, self-contained, modern survey of Strassen's theory of asymptotic spectra and its various old and new application areas. For accessibility we provide many examples and high-level discussions of definitions and techniques. The paper contains some new ingredients. We disentangle some proofs to make them more modular, and make each part as general as possible. We introduce some new notions, which sometimes lead to simpler, more intuitive proofs, as well as to some stronger or more general theorems. One such consequence is our connectivity theorem for the asymptotic spectrum of any tensor network, greatly generalizing Strassen's connectivity theorem for the special case of matrix multiplication. Another consequence is progress on a conjecture of Strassen which generalizes Scho & uml;nhage's tau theorem.
Nearly twenty years ago, a paper of Graham, Lagarias, Mallows, Wilks, and Yan on the number theory of Apollonian circle packings sparked an interest in the number theory community, which was just developing tools to handle arithmetic problems involving so-called thin groups. At the time, these packings were the only naturally occurring example of such an arithmetic problem and, naturally, number theorists sprang upon the opportunity to discover all their rich properties, thinness notwithstanding. In 2010, building upon conjectures of Graham et al., the author and Katherine Sanden gave evidence toward the local to global conjecture for Apollonian circle packings, stating that in any integral packing, any large enough integer that satisfied certain congruence conditions modulo 24 must appear as a curvature in the packing. For thirteen years, most everyone believed this conjecture to be true. In this article, we will explore the history of this conjecture and its fascinating downfall after Haag, Kertzer, Rickards, and Stange proved that, in fact, infinitely many integral Apollonian packings fail to abide by the local to global principle, and come with extra quadratic and quartic obstructions.
This is a survey on the Farrell-Jones Conjecture about the algebraic K- and L-theory of groups rings and its applications to algebra, geometry, group theory, and topology.
The purpose of this paper is to unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of symplectic tiling billiards in a very special case and then show how this result is related to planar linkages and hyperbolic geometry.
This article concerns the results obtained in [Cabre, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)], which established the Holder regularity of stable solutions to semilinear elliptic equations in the optimal range of dimensions n <= 9. For expository purposes, we provide self-contained proofs of all results. They involve only basic analysis tools and are intended to be accessible to a broader mathematical audience beyond PDE specialists. Two of the results in the 2020 article relied on compactness arguments. Here we present, instead, quantitative proofs from the more recent paper [Cabre, to appear in Amer. J. Math, arXiv:2211.13033]. They allow us to quantify the Holder regularity exponent and significantly simplify the treatment of boundary regularity. We also comment on similar progress and open problems for related equations.
You have heard the buzz: AI and formalization will revolutionize mathematics. Computers will soon surpass humans in solving olympiad-style problems. Humans will transition from proving research-level theorems on their own to guiding computers. And maybe you even believe the hype. Other than pulling out your hair waiting for the day when computers take your job, what can you do? If you are like most career mathematicians, you are already overwhelmed with too many academic responsibilities, reserving any precious spare work hours for research. Where are you going to find the time to learn Lean or machine learning techniques? While I do not have the answers, I can share how I have embraced the potential of AI and formalization in mathematics through the eXperimental Lean Lab (XLL) at the University of Washington. I hope to inspire you to also get involved and play an active role in guiding the transformation of our field.
. The aim of this article is to give an introduction to Luna's Slice Theorem for nonspecialists. In the first section we give an idea of its significance without being too precise. It is followed by an introduction to compact transformation groups in Section 2. Then we show by examples how classification problems lead to the study of orbits of algebraic groups in Section 3. In order to explain and formulate Luna's Slice Theorem, we need some basics from algebraic transformation groups and algebraic geometry; this is done in Sections 4 and 5. The paper ends with some striking applications in Section 6.
This issue of the Bulletin is in honor of Luis A. Caffarelli, in celebration of his being awarded the Abel Prize 2023. Luis is one of the great mathematicians of our time, a master of analysis and nonlinear partial differential equations, who has been justly recognized and celebrated with many awards and wide acclaim.
Luis Caffarelli is one of the most outstanding mathematicians of our generation. His fundamental work on the regularity theory of nonlinear PDEs, fluid dynamics, and free boundary problems is associated with many important breakthroughs in these fields over the last five decades. For his exceptional work, Caffarelli was awarded numerous honors and distinctions such as the Bôcher Memorial Prize, the Rolf Schock Prize, the AMS Leroy Steele Prize, the Wolf Prize, the Shaw Prize, and, in 2023, the Abel Prize.
Andrew Ogg's mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg's celebrated Torsion Conjecture -- as it relates to modular curves -- can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg's Torsion Conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years.
In this paper we describe the work of Luis Caffarelli in the area of fluid mechanics and related topics. Not only has his work on fluid mechanics been very influential, but many of his contributions that do not directly relate to fluid mechanics, such as his important results on the fractional Laplacian or the regularity of solutions to linear parabolic equations with oscillating coefficients, have been used in the study of fluids in many important ways. Thus, any review of his work has to include his contributions to the general (partial) regularity theory of solutions of Navier-Stokes equations and other studies related to fluid motion.
This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and new Minkowski problems studied in convex geometry, PDEs, and harmonic analysis, and structured in a conceptual framework of the Brunn-Minkowski theory, its extensions, and related subjects.
This is a tour through the various interactions I had with Professor Singer, with an eye on the Dirac operator, and illustrated with some minor mathematical results.
What can singularities of algebraic varieties say about the decompositions of a positive integer into a sum of positive integers?
These notes explore three amazing formulas proved by Abel in his 1826 Paris memoir on what we now call Abelian integrals. We discuss the first two formulas from the point of view of symbolic computation and explain their connection to residues and partial fractions. The third formula arises from the first two and is related to the genus and lattice points in the Newton polygon.
The aim of the note is to illustrate some of the ideas introduced by Luis Caffarelli in his groundbreaking works on the regularity theory for elliptic free boundary problems, in a way which can be understood by non-experts.
These are lecture notes focusing on recent progress towards Bourgain's slicing problem and the isoperimetric conjecture proposed by Kannan, Lovasz and Simonovits (KLS).
We give a systematic treatment of index theory on Pin manifolds, based on the Clifford linear Dirac operator and differential KO-theory. This expository article is based on joint work with Mike Hopkins.
The interpolation problem is a natural and fundamental question whose roots trace back to ancient Greece. The story is long and rich, with many chapters, and a complete solution has been obtained only recently [Forum Math. Pi 11 (2023), Paper No. e25]. Exploring it leads us on a tour through a number of general themes in geometry. This concrete problem motivates fundamental concepts such as moduli spaces and their properties, deformation theory, normal bundles, and more. Questions about smooth objects lead us to consider singular (nonsmooth) objects; and, in fact, these smooth objects are studied by instead focusing on somehow simpler “nonsmooth” objects and then deforming them.
Genus one amplitude for topological strings on Calabi-Yau 3-folds can be computed using mirror symmetry: The partition function at genus one gets mapped to a holomorphic version of Ray-Singer torsion on the mirror Calabi-Yau. On the other hand it can be shown by a physical argument that this gives a curvature squared correction term to the gravitational action. This in paticular leads to an effective quantum gravity cutoff known as the species scale, which varies over moduli space of Calabi-Yau manifolds. This resolves some of the puzzles associated to the entropy of small black holes when there are a large number of light species of particles. Thus Ray-Singer torsion, via its connection to topological strings at genus one, provides a measure of light degrees of freedom of four dimensional N=2 supergravity theories. Based on a talk given on May 12th, 2023 at the Singer Memorial Conference, MIT.
