Related papers: Gamma conjecture I for blowing up $\mathbb{P}^n$ a…

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting…

We prove a generalization of the Conley conjecture: Every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes over the second fundamental group. In particular, we…

The BKMP Remodeling Conjecture \cite{Ma,BKMP09,BKMP10} predicts all genus open-closed Gromov-Witten invariants for a toric Calabi-Yau $3$-orbifold by Eynard-Orantin's topological recursion \cite{EO07} on its mirror curve. The proof of the…

We found an explicit description of all $GL(n,\RR)$-Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds.

In this paper, we focus on Hamilton's pinching conjecture formulated in Hamilton's paper "Three-manifolds with positive Ricci curvature". Let $(M, g)$ be a complete, connected, noncompact Riemannian $3$-manifold satisfying the…

The Manin conjecture is established for Ch\^atelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not…

Let $M$ be a quaternionic manifold, $\dim M=4k$, whose twistor space is a Fano manifold. We prove the following: (a) $M$ admits a reduction to $Sp(1) \times GL(k,H)$ if and only if $M=HP^k$, (b) either $b_2(M)=0$ or $M=Gr_2(k+2,C)$. This…

Let $W$ be a quasi-homogeneous polynomial of general type and $<J>$ be the cyclic symmetry group of $W$ generated by the exponential grading element $J$. We study the quantum spectrum and asymptotic behavior in Fan-Jarvis-Ruan-Witten theory…

We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2, and satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect…

A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…

We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automatically has contact to order three. We give applications to…

I argue that cosmic Gamma-ray Bursts (GRB) may be produced by collapses or mergers of stars made of `mirror' matter. The mirror neutrinos (which are sterile for our matter) produced at these events can oscillate into ordinary neutrinos. The…

In this paper, we prove a special case of Campana--Peternell's conjecture in dimension 4. Specifically, we show that a projective smooth fourfold $X$ with $c^2_1(X)\cdot c_2(X)\neq 0$ and strictly nef anti-canonical divisor $-K_X$ is a Fano…

We study Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth curve of degree six and genus three. We produce many new K-stable examples of such threefolds, and we describe all finite…

Let $P$ be a Delzant polytope in $\mathbb{R}^k$ with $n+k$ facets. We associate a closed Lagrangian submanifold $L$ of $\mathbb{C}^n$ to each Delzant polytope. We prove that $L$ is monotone if and only if and only if the polytope $P$ is…

Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain…

In this paper we study Lagrangian Floer theory on toric manifolds from the point of view of mirror symmetry. We construct a natural isomorphism between the Frobenius manifold structures of the (big) quantum cohomology of the toric manifold…

As an application of a recent characterization of complete flag manifolds as Fano manifolds having only ${\mathbb P}^1$-bundles as elementary contractions, we consider here the case of a Fano manifold $X$ of Picard number one supporting an…

Geometric Manin's conjecture for complex Fano varieties describes the structure of the moduli space of curves. We propose a version of this conjecture in characteristic $p$ and describe its connection to the Batyrev--Manin--Peyre--Tschinkel…

There are three astroparticle physics puzzles which fire the imagination: the origin of the ``Great Positron Producer'' in the galactic bulge, the nature of the gamma-ray bursts central engine and the mechanism of supernova explosions. We…