Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph $G_{n,p}$. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in $G_{n,p}$. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in $G_{n,p}$ for all $p=p(n)$ satisfying $n^{-1}logn\ll p\ll 1$.

Using the Donaldson–Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kähler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfaces, and we obtain bounds for their Kobayashi hyperbolic metric.

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace $K\subseteq {\wedge }^{2}V$ as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, and the degree of growth and virtual nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves, (2) nilpotent fundamental groups of compact Kähler manifolds, and (3) the Torelli group of a free group.

We consider the problem of counting polynomial curves on analytic or definable subsets over the field $\mathbb{C}((t))$ as a function of the degree r. A result of this type could be expected by analogy with the classical Pila–Wilkie counting theorem in the Archimedean situation.

Some non-Archimedean analogues of this type have been developed in the work of Cluckers, Comte, and Loeser for the field ${\mathbb{Q}}_p$, but the situation in $\mathbb{C}((t))$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree r on the transcendental part of a subanalytic set over $\mathbb{C}((t))$ is automatically finite, but we give examples that show their number may grow arbitrarily quickly even for analytic sets. Thus no analogue of the Pila–Wilkie theorem can be expected to hold for general analytic sets. On the other hand, we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in r, thus showing that the analogue of the Wilkie conjecture does hold in this context.

We study fully nonlinear geometric flows that deform strictly k-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the k-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve k-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.

We introduce an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature $\le -1$, by the hyperbolic metric.

Consider the Hopf–Cole solution $h(t,x)$ of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding $t\to \mathrm{\infty }$ as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of $h(2t,0)+\frac{t}{12}$, with speed $t^2$ and an explicit rate function ${\mathrm{\Phi }}_{-}(z)$. This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be “classical,” we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.

If f is a function supported on the truncated paraboloid in ${\mathbb{R}}^3$ and E is the corresponding extension operator, then we prove that for all $p>3+3∕13$, $\Vert Ef{\Vert }_{L^p({\mathbb{R}}^3)}\le C\Vert f{\Vert }_{L^{\mathrm{\infty }}}$. The proof combines Wolff’s two ends argument with polynomial partitioning techniques. We also observe some geometric structures in the wave packets.

We give an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin–Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions $K_1,K_2$ over a non-Archimedean local field F. Our formula works for all cases: $K_1$ and $K_2$ can be either the same or different, ramified or unramified over F. This formula translates the linear arithmetic fundamental lemma (linear AFL) into a comparison of integrals. As an example, we prove the linear AFL for ${\mathrm{GL}}_2(F)$.