We defined a new type of open Gromov-Witten invariants on hyperK\"aher manifolds with holomorphic

Suppose $f,g_1,\cdots,g_p$ are holomorphic functions over $\Omega\subset\cxC^n$. Then there raises a natural question: when can we find holomorphic functions $h_1,\cdots,h_p$ such that $f=\sum g_jh_j$? The celebrated Skoda ...

We present an application of Hodge theory towards the study of irreducible unitary representations of reductive Lie groups. We describe a conjecture about such representations and discuss some progress towards its proof.

Major efforts to sequence cancer genomes are now occurring throughout the world. Though the emerging data from these studies are illuminating, their reconciliation with epidemiologic and clinical observations poses a major ...

The Hilbert scheme of \(n\) points in the projective plane parameterizes degree \(n\) zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert ...

In this dissertation I discuss and investigate the analytic aspect of several elliptic and parabolic partial differential equations arising from Rimannian and complex geometry, including the generalized Ricci flow, Gaussian ...

In this thesis, we study the compatibility between local and global Langlands correspondences for \(GL_n\). This generalizes the compatibility between local and global class field theory and is related to deep conjectures ...

Let F be a field finitely generated and of transcendence degree 2 over \(\bar{\mathbb{Q}}\). We describe a correspondence between the smooth algebraic surfaces X defined over \(\bar{\mathbb{Q}}\) with field of rational ...

Microorganisms have been cooperating with each other for billions of years: by sharing resources, communicating with each other, and joining together to form biofilms and other large structures. These cooperative behaviors ...

We consider a world of nucleotide sequences and protocells. The sequences have the property of spontaneous self-replication. Some sequences - so-called replicases - have enzymatic activity in the sense of enhancing the ...