Birational Geometry of Singular Moduli Spaces of O'Grady Type

Following Bayer and Macr\`{i}, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macr\`{i} map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland semistable objects on $X$. An essential ingredient of our proof is an isometry between the orthogonal complement of a Mukai vector inside the algebraic Mukai lattice of $X$ and the N\'{e}ron-Severi lattice of $M$ which generalises results of Yoshioka, as well as Perego and Rapagnetta. Moreover, this allows us to conclude that the symplectic resolution of $M$ is deformation equivalent to the 10-dimensional irreducible holomorphic symplectic manifold found by O'Grady.Comment: Final versio

Weak solutions for forward--backward SDEs--a martingale problem approach

In this paper, we propose a new notion of Forward--Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward--backward stochastic differential equations (FBSDEs). The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of an FBSDE. We first prove a general sufficient condition for the existence of the solution to the FBMP. In the Markovian case with uniformly continuous coefficients, we show that the weak solution to the FBSDE (or equivalently, the solution to the FBMP) does exist. Moreover, we prove that the uniqueness of the FBMP (whence the uniqueness of the weak solution) is determined by the uniqueness of the viscosity solution of the corresponding quasilinear PDE.Comment: Published in at http://dx.doi.org/10.1214/08-AOP0383 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Creativity: Generating Diverse Questions using Variational Autoencoders

Generating diverse questions for given images is an important task for computational education, entertainment and AI assistants. Different from many conventional prediction techniques is the need for algorithms to generate a diverse set of plausible questions, which we refer to as "creativity". In this paper we propose a creative algorithm for visual question generation which combines the advantages of variational autoencoders with long short-term memory networks. We demonstrate that our framework is able to generate a large set of varying questions given a single input image.Comment: Accepted to CVPR 201

Surface chemistry and growth of oxide supported metal nanoparticles

A model heterogeneous catalyst inspired by a real catalyst is synthesized for the purpose of understanding how it works. To make such model catalysts, we choose to evaporate metal atoms on metal oxide single crystals and measure them under UHV conditions. The study of model catalysts could advance the understandings of fundamental catalytic properties of real catalysts and helps to optimize or redesign industrial catalysts. In our experiments, many ultra-high vacuum (UHV) techniques have been employed to investigate the atomic and electronic structure of the surface as well as the interface of the prepared samples. In particular, the surface-sensitive tools such as electron energy loss spectra (EELS) and low energy ion scattering (LEIS) spectra provide us detailed information of the surface modification. In this dissertation, we confine our attention to three popular catalysts: Cu on ZnO, Au on ZnO and Cu on TiO2, which play primary roles on modern methanol industry. For instance, Au on ZnO and Cu on ZnO are important catalysts for methanol synthesis, water-shift reaction and methanol-steam reforming, while Cu on TiO2 possesses a high photocatalytic activity for photoreduction of CO2 into methanol. It becomes important for us to develop an understanding of which factors determine the functions of the prepared samples. Different metal growth models are observed for the above samples, due to the varied metal-oxide interactions. At the high substrate temperature, full encapsulation of metal nanoparticles takes place to all of the above samples, which dramatically changes the adsorption behavior and catalytic performance. It provides a strong indication that these thin encapsulation layers are very different from their bulk materials in both geometric and electronic sides. The charge transfer in the interface may be responsible for the modification of geometric and electronic structure of surface, and results in high-thermal stability of these ultra-thin films. The above reconstruction leads to an exceptional catalytic activity of CO oxidation through a different reaction kinetics and mechanism. The CO oxidation experiments show a direct relation between encapsulation rate and reaction rate, which indicates the active sites should be localized at these thin oxide films rather than the metal nanoparticles

Moduli Spaces of Sheaves on K3 Surfaces and Symplectic Stacks

We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.Comment: 36 page