Areas of Research:
1. Algebraic Geometry Research
We study the algebraic geometry of vector bundles, principal bundles, and sheaves on spaces like projective varieties and curves. Our research emphasizes their moduli, stability conditions, and related topics such as ramification, degeneration, and automorphism groups. A key feature is our comparative analysis of phenomena in characteristic zero versus positive characteristic, employing tools from geometric invariant theory and Frobenius splitting to explore how arithmetic influences geometry.
2. Complex Analysis Research
Our work operates at the intersection of complex analysis and geometric group theory. A primary focus is analyzing complex manifolds using intrinsic metrics and Hörmander's estimates for the ∂ˉ-operator, which provides deep insights into Teichmüller theory and deformation spaces. Concurrently, we investigate the algebraic and topological properties of braid groups and mapping class groups, studying their links to moduli spaces of surfaces and low-dimensional topology. Our goal is to unify these perspectives to understand the fundamental group actions and symmetries that define complex surfaces.
3. Harmonic Analysis Research
Our research is in harmonic analysis, focusing on nilpotent Lie groups and time frequency methods. We investigate the properties of modulation spaces, Fourier multipliers, and twisted convolution operators, particularly their boundedness between various function spaces. These tools are applied to solve nonlinear Schrödinger equations involving twisted Laplacians, analyze maximal functions along hypersurfaces, and establish sharp Hardy-Sobolev inequalities. Our approach integrates non-commutative pseudo-differential calculus and oscillatory integrals to understand dispersive PDEs in non-Euclidean settings.
4. Non-commutative Functional Analysis Research
Our research applies functional analysis to quantum mechanics using operator algebras and operator spaces. We study the properties of non-commutative objects, focusing on quantum approximation properties and non-commutative convexity. Key areas of our work include completely bounded maps, matricial ranges, and operator system structures. This research connects abstract mathematics with quantum information theory, providing ools for analyzing quantum channels and understanding the geometry of quantum state spaces. Our goal is to create a versatile mathematical framework for quantum computation and operator theory.
5. Analytic Number Theory Research
Our group uses analytic number theory to study prime numbers, multiplicative functions, and L-functions, with a special focus on multiple zeta values and their generalizations. We develop new methods to analyze averages and sums, including refined Möbius function summations.A key part of our research involves using Beurling–Selberg extremal functions to bound prime-counting errors and developing new techniques for bounding tails of multiple zeta value series. Ultimately, we aim to bridge analytic number theory with arithmetic geometry and combinatorics.