Abstract: On February 12, 2002, Michel Devoret’s lecture entitled “The Quantum Computer: Miracle or Mirage” kicked off the 150th anniversary celebration of the Yale School of Engineering. In his abstract, Devoret asserted that while quantum mechanics had often been viewed as limiting the information one could extract from a physical system, “recent discoveries show that fundamental properties of quantum mechanics could actually be used to perform computations that would be impossible on a standard ‘classical’ computer.”

The question of whether the quantum computer is a miracle or a mirage remains, to this day, one of the most fascinating clear-cut open scientific questions of our time. In my lecture, I will explain the question and the discoveries of the 1990s that suggested that quantum computers could perform miracles, and I will present my theory of why the quantum computer is a mirage.

Quantum computation presents important challenges for applied mathematics and computer science involving modeling, computational complexity, statistics, and the relationships between theory, experiments, and engineering. I will discuss three lines of my research: correlated errors that may cause quantum computation to fail; computational-complexity arguments (beginning with joint work with Guy Kindler) suggesting that the error rates required for quantum fault tolerance may be beyond reach; and statistical analyses, with Yosi Rinott, Tomer Shoham, and Carsten Voelkmann, of experimental claims—particularly Google’s 2019 “quantum supremacy” experiment.

This pre-talk for the Geometric Analysis and Application Seminar on 10/13, provides background on the basic tools used in singularity analysis for mean curvature flow, and if time permitted also discusses Huisken’s classification of mean-convex shrinkers.

When does an algebraic differential equation have solutions which are themselves algebraic functions? This question goes back at least to Fuchs, in 1875. Its conjectural answer, due to Grothendieck and Katz in the linear case and Ekedahl–Shepherd-Barron–Taylor and Bost in general, situates it as one of number theory. I’ll explain these conjectures and some progress towards them, joint with Josh Lam, for isomonodromy differential equations. This class of differential equations includes, for example, the Painlevé VI equation. This work is in part an excuse to study “motivic” aspects of the space of all algebraic differential equations.

Given a discrete group of isometries of a real hyperbolic space, we can look at its limit set, i.e., the set of accumulation points of its orbits on the sphere at infinity. This is a compact metric space on which the group acts, and which enjoys many interesting geometric, topological, and dynamical features. For example, classical Kleinian groups provide many examples with limit sets that are quasicircles, Cantor sets, and Sierpinski carpets. In this talk, I will discuss how to construct examples whose limit sets are various low-dimensional trees of manifolds. In particular, we answer a question of M. Kapovich, by constructing convex cocompact groups of isometries of real hyperbolic spaces, whose limit sets are Čech cohomology 3-spheres not homeomorphic to S^3. These groups are right-angled Coxeter groups and our construction is flexible enough to produce infinitely many quasi-isometry classes. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.

The Lagrange spectrum is a classical object in number theory, defined as the set of values of liminf_{q→∞} q dist(qα,ℤ), where α runs through irrational numbers. It has a complicated structure, with the discrete part, Hall’s ray, and a transitional part in between. One can similarly define the Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. I will show that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between 0 and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum. The latter is shown to be equal to the interval between 0 and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.

Algebraic and Geometric Convergence of Discrete and Non-Faithful Representations
Abstract: A sequence of representations converges strongly if it converges algebraically and its images converge geometrically to the image of the algebraic limit. In the faithful case, dynamical, geometric and topological properties often vary continuously with respect to strong convergence, and Anderson–Canary showed that strong convergence is guaranteed provided the algebraic limit has no parabolic elements. In the non-faithful case, however, strong convergence may fail even in the absence of parabolic elements due to the presence of a degenerate end in the algebraic limit that does not embed in the geometric limit. Biringer–Souto constructed an example of this phenomenon using a surface group, and we found that their construction can be generalized to many families of 3-manifold groups. In this talk, we will explain these two obstructions through examples and construct sequences of representations of 3-manifold groups exhibiting a non-embedded degenerate end in the algebraic limit.

Abstract: I’ll explain a tweak to Clausen and Scholze’s theory of solid modules that is better-adapted to applications in geometry, as opposed to number theory. This theory replaces point-set topology when dealing with objects like the field of formal Laurent series. For instance, it can be used to streamline deformation theory in infinite type.

The homotopy groups of spheres classify continuous functions from an n-dimensional sphere to an m-dimensional sphere up to continuous deformation. They are of great interest as they control many classification problems in geometric topology, but are rich and complicated. I will explain some of what we know about them, and describe the telescopic approach to understanding them systematically. I will then explain how algebraic K-theory has refined our understanding of the telescopic approach, and can be used to obtain asymptotic lower bounds on the complexity of these homotopy groups that were previously out of reach.

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 Physics has long supplied mathematics with some of its central equations. I will argue that Transformers, the engine of large language models, are becoming a source of the same kind. Starting from raw code, under various scaling assumptions, one can interpret Transformers as an interacting particle system in continuous time; tools from PDE and kinetic theory then explain empirically observed phenomena such as representation collapse and “lost in the middle”. Unlike in physics, however, the laws here are designed rather than given, so mathematical understanding can be written back into the architecture. I will close with some thoughts on what this two-way exchange means for applied mathematics.

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