Our research delves into the fascinating world of algebraic geometry, with a particular focus on vector bundles, principal bundles, and sheaves over diverse algebraic varieties, including curves and projective spaces. We actively investigate concepts such as stability, semistability, and moduli spaces of these bundles, often exploring their connections to ramified covering maps, singularities, and automorphism groups. Our work also encompasses areas like Frobenius splittings, compactified Jacobians, and geometric invariant theory, examining these fundamental ideas across both characteristic zero and positive characteristic settings. ...more

Our research spans the rich interplay between complex analysis and geometric group theory. We investigate the intricate properties of complex manifolds and intrinsic metrics, such as the Kobayashi and Teichmüller metrics, exploring how Hörmander's estimates illuminate the geometry of Bers domains. Simultaneously, we delve into the topological and algebraic structures of braid groups and mapping class groups, with a focus on liftability problems and the Birman-Hilden property. This combined approach allows us to explore deep connections between surface homeomorphisms, unitary representations, and quantum invariants, fostering an interdisciplinary understanding of complex spaces and fundamental groups. ...more

Our investigations center on harmonic analysis in nilpotent Lie groups and time-frequency methods, connecting classical Fourier analysis with modern operator theory. We explore modulation spaces, Fourier multipliers, twisted convolution operators, and their Lp-Lq boundedness properties. The group also analyzes nonlinear Schrödinger equations with twisted Laplacians, maximal functions along hypersurfaces, and develops Hardy-Sobolev inequalities in non-commutative settings, advancing oscillatory integral techniques for dispersive PDEs. ...more

We extend functional analysis to quantum realms through operator algebras and operator spaces, investigating quantum approximation properties and convexity in C*-algebras. Our work focuses on completely bounded approximation, quantum convexity, matricial ranges, and operator system structures. This research bridges abstract functional analysis with quantum information theory, with applications in quantum error correction, quantum channel classification, and geometric analysis of quantum states. ...more

Our group employs advanced analytic techniques to study prime distributions, multiplicative functions, and L-function special values. We investigate multiple zeta values and their generalizations (including multiple Apéry-like sums), explicit averages beyond classical estimates, Möbius sums with coprimality conditions, and additive number theory extensions. Key work involves the Beurling-Selberg function's applications to prime number theory and developing new methods for bounding multiple zeta value tails, connecting analytic techniques with combinatorial structures. ...more