We prove two structure theorems for the Gromov-Witten theory of the quintic threefolds, which together give an effective algorithm for the all genus Gromov-Witten potential functions of quintics. By using these structure theorems, we prove Yamaguchi-Yau's Polynomial Ring Conjecture in this paper and prove Bershadsky-Cecotti-Ooguri-Vafa's Feynman rule conjecture in the subsequent paper.

It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle.

Fix $0\lt \alpha \lt 1$. Let $_\alpha(d)$ denote the maximum number of lines through the origin in $\mathbb{R}^d$ with pairwise common angle $\mathrm{arccos}\ \alpha$. Let $k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-\alpha)/(2\alpha)$. If $k \lt \infty$, then $N_\alpha(d) = \lfloor k(d-1)/(k-1)\rfloor$ for all sufficiently large $d$, and otherwise $N_\alpha(d) = d+ o(d)$. In particular, $N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1)\rfloor$ for every integer $k\ge 2$ and all sufficiently large $d$.

A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

In this paper, we establish the global $C^{2,\alpha}$ and $W^{2,p}$ regularity for the Monge-Ampère equation $\mathrm{det}\ D^2u = f$ subject to boundary condition $Du(\Omega) = \Omega^\ast$, where $\Omega$ and $\Omega^\ast$ are bounded convex domains in the Euclidean space $\mathbb{R}^n$ with $C^{1,1}$ boundaries, and $f$ is a Hölder continuous function. This boundary value problem arises naturally in optimal transportation and many other applications.

A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all of them. Erdős and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower with $r$ petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to $c^w$ for some constant $c$. In this paper, we improve the bound to about $\mathrm{log}\ w)^w$. In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is sharp up to lower order terms.

In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions on the ramification of $\pi$ if the central derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and L-functions, which generalizes the Birch and Swinnerton-Dyer conjecture. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $\pi$-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by one of us, which generalizes the Gross--Zagier formula to higher dimensional motives.

We answer in the negative Siegel's question whether all $E$-functions are polynomial expressions in hypergeometric $E$-functions. Namely, we show that if an irreducible differential operator of order three annihilates an $E$-function in the hypergeometric class, then the singularities of its Fourier transform are constrained to satisfy a symmetry property that generically does not hold. The proof relies on André's theory of $E$-operators and Katz's computation of the Galois group of hypergeometric differential equations.

We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic $\mathbb{Z}_p$-extension of the unramified quadratic extension of $\mathbb{Q}_p$ for $p\geq 5$ a prime.

Rubin's conjecture underlies Iwasawa theory of the anticyclotomic deformation of a CM elliptic curve over the CM field at primes $p$ of good supersingular reduction, notably the Iwasawa main conjecture in terms of $p$-adic $L$-function. As a consequence, we prove an inequality in the $p$-adic Birch andSwinnerton-Dyer conjecture for Rubin's $p$-adic $L$-function. Rubin'sconjecture is also an essential tool in our exploration of the arithmeticof Rubin's $p$-adic $L$-function, which includes a Bertolini--Darmon--Prasanna type formula.

The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group, then the only units of the group ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.