Current Developments in Mathematics

The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the “philosophy of deformations” both as a conceptual and computational tool, and suggest possible connections to Lusztig’s “philosophy of generations”. On the geometric side one can understand deformations in terms of localisation in equivariant cohomology. Recently Treumann and Leslie–Lonergan have added Smith theory, which provides a useful tool when considering $\operatorname{mod} p$ coefficients. In this context, we make contact with some remarkable work of Hazi. Using recent work of Abe on Soergel bimodules, we are able to reprove and generalise some of Hazi’s results. Our aim is to convince the reader that the work of Hazi and Leslie–Lonergan can usefully be viewed as some kind of localisation to “good” reflection subgroups. These are notes for my lectures at the 2019 Current Developments in Mathematics at Harvard.

We briefly outline the proof given by Chang–Guo–Li of the Yamaguchi–Yau polynomiality and the Bershadsky–Cecotti–Ooguri–Vafa Feynman rule conjectures for the Gromov–Witten invariants of the quintic Calabi–Yau threefolds.

This article gives an overview of the recent proof of the 2‑to‑2 Games Conjecture in [68, 39, 38, 69] (with additional contributions from [75, 18, 67]). The proof requires an understanding of expansion in the Grassmann graph.

This is an expanded version of the notes of lectures given at the conference “Current Developments in Mathematics 2019” held at Harvard University on November 22–23, 2019. We present an overview of some recent developments.

Random constraint satisfaction problems, such as the random $K\textrm{-SAT}$ model and colourings of random graphs naturally emerge in the study of combinatorics and theoretical computer science. Ideas from statistical physics describe a series of phase transitions these models undergo as the density of constraints increases. Throughout we will focus on the example of the maximal independent set of a random regular graph as a simple example of this phenomena and then conclude by describing the additional technical challenges needed to establish the Satisfiability Conjecture for large $K$.

This is a review of old and new results and methods related to the Yau conjecture on the zero sets of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau’s conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili’s conjecture. We also discuss two-dimensional methods to study zero sets.

This is an expository article on the averaged version of Colmez’s conjecture, relating Faltings heights of CM abelian varieties to Artin $L$-functions. It is based on the author’s lectures at the Current Developments in Mathematics conference held at Harvard in 2018.

We survey some conjectures and recent developments in the Langlands program, especially on the conjectures linking motives and Galois representations with automorphic forms and $L$-functions. We then give an idiosyncratic discussion of various recent modifications of the Taylor–Wiles method due in part to the author in collaboration with David Geraghty, and then give some applications. The final goal is to give some hints about ideas in recent joint work with George Boxer, Toby Gee, and Vincent Pilloni. This is an expanded version of the author’s talk in the 2018 Current Developments in Mathematics conference. It will be presented in three parts: exposition, development, and capitulation.

This article surveys recent results due to the author and his collaborators on rigidity properties of actions of certain countably infinite groups on compact manifolds. We specifically focus on the results of [12–14, 16, 18]. We primarily focus on groups such as $\Gamma = \operatorname{SL}(n, \mathbb{Z})$ (for $n \geq 3$) and more general lattices $\Gamma$ in (typically higher-rank) semisimple Lie groups. The actions considered will be either on low-dimensional manifolds (where the dimension is small relative to certain algebraic data associated with the acting group) or actions on tori $\mathbb{T}^d$ and nil-manifolds $N / \Lambda$.

According to the Allard regularity theory, the set of singular points (i.e. non $C^{1,\alpha}$-embedded points) of an integral $n$-varifold with generalized mean curvature locally in $L^p$ for some $p \gt n$ is a nowhere dense (closed) subset of the support of the varifold. A well-known codimension $1$ example due to Brakke shows that not much can be said about the Hausdorff measure of the singular set; it need not have zero $n$-dimensional measure. We survey recent work that shows nonetheless, that in codimension $1$, all is well whenever those parts of the varifold that are regular (in certain specific ways) satisfy further hypotheses, namely: (a) that the orientable portions of the $C^{1,\alpha}$ embedded part and the $C^2$ immersed part are respectively stationary (or equivalently CMC) and stable with respect to the area functional for volume preserving deformations, and (b) that there is appropriate control on two types of singularities—called classical and touching singularities—that are formed by $C^{1,\alpha}$ embedded pieces of the varifold coming together in a regular fashion. This work builds on and extends the recent codimension $1$ theory for zero mean curvature stable varifolds with no classical singularities, and the earlier fundamental curvature estimates of Schoen–Simon–Yau and of Schoen–Simon. We include a brief discussion of these previous works and their role in (different approaches to) the existence theory for minimal hypersurfaces in compact Riemannian manifolds. The main focus of the survey is on the novel aspects of the CMC regularity and compactness theory and the associated curvature estimates.