Algebraic Geometry

Derived Categories and Homological Mirror Symmetry

Studies equivalences between derived categories of coherent sheaves and Fukaya categories through homological mirror symmetry frameworks.

RESEARCH GAP FRONTIERS

Moduli Spaces of Stable Curves

Investigates the geometry and compactifications of moduli spaces parameterizing families of stable algebraic curves.

RESEARCH GAP FRONTIERS

Intersection Theory and Chow Rings

Develops computational and theoretical methods for understanding intersection products and Chow ring structures on varieties.

RESEARCH GAP FRONTIERS

Birational Geometry and Minimal Models

Explores birational transformations, divisorial contractions, and the minimal model program for higher-dimensional varieties.

RESEARCH GAP FRONTIERS

Toric Varieties and Polytope Combinatorics

Connects combinatorial properties of polytopes to geometric properties of toric varieties and their invariants.

RESEARCH GAP FRONTIERS

Abelian Varieties and Principally Polarized Spaces

Studies moduli spaces of abelian varieties with principal polarizations and their arithmetic properties.

RESEARCH GAP FRONTIERS

Hodge Theory and Period Domains

Investigates Hodge structures on cohomology, period matrices, and their applications to classification of varieties.

RESEARCH GAP FRONTIERS

Calabi-Yau Manifolds and String Theory

Examines geometric and arithmetic properties of Calabi-Yau varieties and their applications in mathematical physics.

RESEARCH GAP FRONTIERS

Enumerative Geometry and Gromov-Witten Invariants

Develops techniques for counting rational curves and computing Gromov-Witten invariants on algebraic varieties.

Singularities and Resolution Techniques

Studies classification and resolution of algebraic singularities including log resolution and symplectic resolutions.

Sheaf Cohomology and Vanishing Theorems

Applies cohomological techniques and vanishing theorems to study line bundles and divisors on varieties.

Algebraic K-Theory and Motivic Cohomology

Investigates K-theory groups, motivic cohomology theories, and their relationships to algebraic cycles.

Derived Algebraic Geometry and Stacks

Develops higher categorical approaches to algebraic geometry using derived schemes and higher stacks.

Rational Points on Varieties

Studies arithmetic properties and distribution of rational solutions to polynomial equations over number fields.

Arithmetic Surfaces and Arakelov Geometry

Combines complex geometry and arithmetic using Arakelov theory to study curves over number fields.

Fano Varieties and Del Pezzo Surfaces

Classifies Fano varieties and Del Pezzo surfaces while studying their geometric and arithmetic properties.

Algebraic Surfaces and Invariants

Investigates classification of surfaces using numerical invariants like Chern numbers and self-intersection indices.

Vector Bundles on Curves and Moduli

Studies moduli spaces of vector bundles on algebraic curves and their geometric properties.

Quantum Cohomology and Symplectic Geometry

Develops quantum cohomology ring structures using symplectic techniques and holomorphic curve counts.

Grothendieck Duality and Serre Duality

Explores duality theorems for sheaves and their applications to canonical divisors and line bundles.

Secant Varieties and Tensor Decomposition

Studies secant varieties to understand tensor rank and decompositions using algebraic geometry methods.

Symbolic Powers and Integral Closure

Investigates relationships between ordinary and symbolic powers of ideals through geometric methods.

Bezout Theory and Intersection Multiplicities

Develops precise intersection multiplicity formulas and classical Bezout bounds for algebraic varieties.

K3 Surfaces and Automorphisms

Studies automorphism groups of K3 surfaces and their action on cohomology and derived categories.

Stability Conditions and Wall Crossing

Examines Bridgeland stability conditions on derived categories and wall crossing phenomena in moduli spaces.

Tropical Geometry and Piecewise Linear Structures

Develops tropical geometry as a combinatorial shadow of complex geometry for polynomial equations.

Invariant Theory and Quotient Varieties

Studies geometric invariant theory to construct quotients of group actions and study their properties.

Representation Varieties and Character Schemes

Investigates varieties of representations and their geometric structures using algebraic geometry techniques.

Affine and Projective Schemes

Develops foundational theory of affine and projective schemes using commutative algebra and category theory.

Neron Models and Arithmetic Geometry

Studies minimal regular models of abelian varieties over discrete valuation rings.

Cremona Transformations and Birational Maps

Analyzes special birational transformations and their factorization properties.

Cohen-Macaulay Varieties and Canonical Modules

Studies Cohen-Macaulay varieties using canonical modules and applications to singularity theory.

Hilbert Schemes and Parameter Spaces

Investigates Hilbert schemes parameterizing subvarieties and their geometric properties.

Syzygies and Castelnuovo-Mumford Regularity

Studies syzygy modules and regularity bounds for sheaves on projective varieties.

Amplfication and Numerical Geometry

Develops numerical criteria for positivity of divisors and cone structures on varieties.

Goppa Codes and Algebraic Geometry Codes

Applies algebraic geometry to construct and analyze error-correcting codes from curves.

Pfaffian Varieties and Determinantal Equations

Studies varieties defined by determinantal and Pfaffian equations using geometric techniques.

Graded Rings and Projective Embeddings

Investigates relationships between graded ring structures and projective embeddings of varieties.

Fontaine-Laffaille Modules and p-Adic Geometry

Studies p-adic geometric objects and their relationship to Galois representations.

Supersingular Varieties in Characteristic p

Investigates geometric properties of supersingular varieties over algebraically closed fields of positive characteristic.

Secant Lines and Classical Enumerative Problems

Resolves classical enumerative problems by counting geometric configurations using intersection theory.

Orbifold Cohomology and Chen-Ruan Theory

Develops orbifold cohomology theories and their applications to singular quotient varieties.

Vanishing Ideals and Combinatorial Commutative Algebra

Studies ideals defined by combinatorial structures using algebraic geometry methods.

Categorical Resolutions and Noncommutative Geometry

Develops categorical approaches to resolutions of singularities using derived equivalences.

Tropical Varieties and Log Structures

Combines tropical geometry with logarithmic structures to study degenerations of varieties.

Berkovich Spaces and Non-Archimedean Geometry

Develops analytic geometry over non-Archimedean fields using Berkovich space theory.

Valuations and Singularity Exponents

Studies valuative approaches to measuring singularities and applications to birational geometry.

Cluster Algebras and Positivity

Investigates cluster algebra structures arising from algebraic geometry and their positivity properties.

Monodromy and Galois Groups of Covers

Studies monodromy representations and Galois groups for branched coverings of varieties.

Perfect Ideals and Resolutions

Analyzes minimal free resolutions and homological properties of perfect ideals.

Derived Stacks and Higher Categorical Methods

Investigation of derived algebraic stacks using infinity-categories and their applications to geometric invariant theory and moduli problems.

Motive Theory and Motivic Integration

Study of Chow motives, mixed motives, and motivic integration techniques for understanding geometric properties of varieties.

Bridgeland Stability and D-Brane Categories

Research on Bridgeland stability conditions on triangulated categories with applications to derived categories and D-brane physics.

Logarithmic Geometry and Log-Smooth Degenerations

Analysis of logarithmic structures on varieties and their role in understanding degenerations and compactifications.

Derived Invariants and Categorical Invariants

Investigation of derived categorical properties as invariants for algebraic varieties and their applications to classification problems.

Positivity and Valuative Methods

Study of positivity cones, Okounkov bodies, and valuative methods for understanding geometry of divisors on varieties.

Singularities in Positive Characteristic Geometry

Analysis of singularities in characteristic p greater than zero including F-singularities and tight closure theory.

Derived Representation Schemes and Stacks

Study of derived representation schemes using derived algebraic geometry with applications to character varieties.

Categorical Resolutions and Noncommutative Deformations

Research on categorical resolutions of singularities via noncommutative algebra and homological methods.

Frobenius Morphisms and Arithmetic Schemes

Investigation of Frobenius endomorphisms on schemes in positive characteristic and their arithmetic-geometric implications.

Moduli of Holomorphic Differential Forms

Study of moduli spaces parametrizing holomorphic differentials on curves and their geometric invariants.

Refined Gromov-Witten Invariants and Quantum Cohomology

Investigation of refined and relative Gromov-Witten invariants with connections to topological quantum field theories.

Spaces of Rational Curves and Unirationality

Study of moduli spaces of rational curves on varieties and their relationship to rationality and unirationality problems.

Sheaf Quantization and Microlocal Analysis

Application of microlocal analysis and sheaf quantization methods to algebraic geometry and derived categories.

Torus Equivariant Cohomology and Localization

Study of equivariant cohomology with torus actions and localization formulas in enumerative geometry.

Homological Mirror Symmetry and Fukaya Categories

Investigation of homological mirror symmetry conjectures relating derived categories to Fukaya categories of symplectic manifolds.

Instanton Invariants and Moduli of Vector Bundles

Study of gauge theory invariants and their relationship to moduli spaces of stable vector bundles on surfaces.

Bridgeland Stability on Surfaces and 3-Folds

Investigation of Bridgeland stability conditions on derived categories of surfaces and three-dimensional varieties.

Perfectoid Spaces and Adic Geometry

Study of perfectoid spaces and adic geometry with applications to p-adic Hodge theory and arithmetic geometry.

Multilinear Algebra and Tensor Rank Varieties

Investigation of varieties defined by tensor rank conditions with applications to tensor decomposition problems.

Lattice Polytopes and Toric Degenerations

Study of combinatorial structures of lattice polytopes and their geometric realizations as toric degenerations.

K-Stability and Kahler-Einstein Metrics

Investigation of K-stability of Fano varieties and their connections to existence of Kahler-Einstein metrics.

Periods of Algebraic Varieties and Hodge Conjecture

Study of period integrals of algebraic varieties and approaches to the Hodge conjecture via period domains.

Sato-Tate Distributions and L-Functions

Research on Sato-Tate distributions for varieties over number fields and their connections to L-function properties.

Categorical Structures on Derived Categories

Investigation of monoidal, braided, and symmetric structures on triangulated and derived categories of varieties.

Geometric Langlands Program and D-Modules

Study of the geometric Langlands correspondence using D-modules and derived categories on algebraic stacks.

Reduction Modulo Prime and Specialization

Analysis of how geometric properties behave under reduction modulo primes and specialization techniques.

Cohomological Field Theories and Correlators

Investigation of cohomological field theories and their correlation functions in the context of enumerative invariants.

Mixed Hodge Structures and Weight Filtrations

Study of mixed Hodge structures on singular varieties and their applications to understanding cohomology.

Étale Cohomology and Galois Representations

Research on étale cohomology groups and associated Galois representations of arithmetic varieties.

Elliptic Surfaces and Fibrations

Study of elliptic fibrations over curves and their singular fibers, Mordell-Weil groups, and height theory.

Stability and Destabilizing Sequences

Investigation of Gieseker and slope stability for sheaves and the role of destabilizing sequences in classification.

Canonical and Anticanonical Systems

Study of canonical divisors, anticanonical systems, and their roles in determining geometric properties of varieties.

Deformation Theory and Obstruction Spaces

Analysis of deformation spaces of geometric objects using cotangent complexes and obstruction theory.

Tautological Classes and Cycle Classes

Study of tautological rings on moduli spaces and the relationships between cycle classes and invariants.

Bridgeland Stability for 3-Fold Flopping Contractions

Investigation of stability conditions related to flops and divisorial contractions on three-dimensional varieties.

Motivic Nearby Cycles and Singularity Theory

Study of motivic vanishing cycles and nearby cycle functors in understanding singularities of varieties.

Cluster Algebras from Quivers and Geometry

Investigation of cluster algebra structures arising from quiver representations and their geometric realizations.

Intersection Cohomology and Perverse Sheaves

Study of intersection cohomology and perverse sheaf categories on singular stratified spaces.

Modular Forms and Arithmetic Surfaces

Research on connections between modular forms and arithmetic invariants of modular curves and surfaces.

Equivariant Derived Categories and Quotient Singularities

Study of equivariant derived categories related to quotient singularities and orbifold resolutions.

Chow Groups of Zero-Cycles

Investigation of Chow groups of zero-cycles and their connections to arithmetic geometry and rationality.

Brill-Noether Theory on Curves

Study of vector bundles of fixed rank and degree on curves via Brill-Noether locus and expected dimensions.

Spherical Double Affine Hecke Algebras

Investigation of spherical DAHA representations and their geometric realizations on varieties.

Prym Varieties and Spectral Curves

Study of Prym varieties of covers and their connections to integrable systems via spectral curves.

Convex Geometry of Weight Polytopes

Research on geometric properties of weight polytopes from representation theory and their algebraic realizations.

Hyperplane Arrangements and Cohomology

Study of cohomology rings of hyperplane arrangement complements and their algebraic structure.

Gromov-Witten Invariants of Orbifolds

Investigation of Gromov-Witten invariants for orbifold targets and their computational techniques.

Universal Formulas and Localization Techniques

Study of universal enumerative formulas derived via localization on moduli spaces of stable maps.

Sheaf Extensions and Ext-Algebras

Investigation of extension groups and Ext-algebra structures on categories of coherent sheaves.

Geometric Langlands Program and Automorphic Forms

Investigates the geometric interpretation of the Langlands correspondence through sheaves on moduli stacks and automorphic representations.

Higher Dimensional Fano Varieties Classification

Classifies and studies properties of Fano varieties in dimensions greater than three using modern birational techniques and MMP.

Derived Categories of Coherent Sheaves

Analyzes the structure and invariants of derived categories of coherent sheaves on algebraic varieties and their categorical properties.

Moduli of Sheaves and Bridgeland Stability

Constructs moduli spaces of semistable sheaves using Bridgeland stability conditions and studies their geometric properties.

Rational Curves and Unirational Varieties

Studies parametrization properties of varieties by rational curves and investigates rationality questions in algebraic geometry.

Equivariant Intersection Theory and Localization

Develops intersection theory with group actions using localization formulas and equivariant cohomology computations.

Foliations on Algebraic Varieties

Studies singular and regular foliations on algebraic varieties and their invariant subvarieties using geometric and algebraic methods.

Derived Stacks and Higher Categorical Structures

Investigates derived stacks using infinity categories and higher topos theory for understanding derived moduli problems.

Positivity Cones in Algebraic Varieties

Studies the geometry of cones of positive divisors and their role in determining fundamental properties of varieties.

Manin-Mumford Conjecture and Unlikely Intersections

Explores heights, unlikely intersections, and Diophantine problems in arithmetic geometry using tools from algebraic geometry.

Brill-Noether Theory on Singular Curves

Extends classical Brill-Noether theory to singular curves and studies the geometry of varieties with singular loci.

Logarithmic Schemes and Toroidal Geometry

Develops the theory of logarithmic schemes and their applications to compactifications and degeneration problems in algebraic geometry.

Frobenius Splitting and Tight Closure

Studies tight closure theory and Frobenius splitting methods for analyzing singularities in characteristic p geometry.

Perverse Sheaves and Intersection Cohomology

Develops perverse sheaf theory and intersection cohomology with applications to representation theory and singularity theory.

Nested Hilbert Schemes and Quot Schemes

Studies the geometry and invariants of nested Hilbert schemes, Quot schemes, and related parameter spaces for sheaves.

Matrix Factorizations and Singularity Theory

Investigates matrix factorizations as tools for studying hypersurface singularities and Landau-Ginzburg models.

Seshadri Constants and Positivity of Line Bundles

Analyzes Seshadri constants and their role in measuring local positivity of divisors on algebraic varieties.

Exceptional Collections and Semiorthogonal Decompositions

Studies exceptional collections in derived categories and semiorthogonal decompositions with applications to derived geometry.

Stability of Vector Bundles and Moduli

Investigates slope stability, Gieseker stability, and other stability notions for constructing moduli spaces of vector bundles.

Jet Schemes and Arc Spaces

Studies the geometry of jet schemes and arc spaces as tools for understanding singularities and motivic integration.

Motivic Integration and Stringy Invariants

Develops motivic integration theory and studies stringy invariants of singularities with connections to physics.

Derived Equivalences and Non-Commutative Geometry

Examines derived equivalences between algebraic varieties and applications to non-commutative algebraic geometry.

Iwahori-Whittaker Functions and Affine Hecke Algebras

Investigates geometric approaches to Iwahori-Whittaker functions and affine Hecke algebras through algebraic varieties.

Donaldson-Thomas Invariants and Counting

Studies Donaldson-Thomas invariants as wall-crossing invariants and their role in counting coherent sheaves on varieties.

Pandharipande-Thomas Stable Pairs Theory

Develops stable pairs invariants and their relationship to Gromov-Witten and Donaldson-Thomas theories.

Mirror Symmetry for Fano Varieties

Constructs mirror varieties for Fano manifolds and explores mirror duality predictions for invariants.

Higher Genus Curves and Noether-Lefschetz Locus

Studies Noether-Lefschetz loci and Hodge-theoretic constraints on algebraic cycles on varieties.

Derived Autoequivalences and Stability Conditions

Analyzes autoequivalence groups of derived categories and their connections to stability manifolds.

Non-Reduced Schemes and Scheme Structure

Studies nilpotent extensions and non-reduced algebraic structures appearing in deformation and obstruction theory.

Finite Morphisms and Galois Theory

Investigates finite morphisms between varieties using Galois theory and ramification invariants.

Étale Cohomology and L-Adic Sheaves

Develops étale cohomology theory and l-adic sheaves with applications to arithmetic and geometric questions.

Rigid Analytic Varieties and Formal Schemes

Studies rigid analytic geometry and formal schemes as tools for p-adic analysis and deformation theory.

Modular Curves and Shimura Varieties

Investigates modular curves and Shimura varieties as special cases of PEL moduli spaces in arithmetic geometry.

Gonality and Brill-Noether Divisors

Studies gonality of curves and the geometry of loci where special divisors of prescribed degrees exist.

Splitting Type of Vector Bundles

Analyzes splitting types of vector bundles and restrictions to curves for understanding bundle properties.

Cox Rings and Mori Dream Spaces

Studies Cox rings of varieties with finitely generated picard groups and geometric properties of Mori dream spaces.

Toric Embeddings and Moment Polytopes

Investigates toric embeddings, moment maps, and connections between polytope combinatorics and algebraic geometry.

Theta Functions and Jacobian Varieties

Studies theta divisors on Jacobian varieties and classical questions in the geometry of abelian varieties.

Intermediate Jacobians and Hodge Conjecture

Analyzes intermediate Jacobians and their role in studying algebraic cycles and the Hodge conjecture.

Kummer Surfaces and Hyper-Kahler Geometry

Studies Kummer surfaces arising from abelian surfaces and hyper-Kähler varieties in general.

Fano Threefolds and Mori Geometry

Classifies Fano threefolds and studies their geometric properties using the minimal model program.

Stable Maps and Virtual Moduli Classes

Constructs virtual fundamental classes for moduli spaces of stable maps and applications to enumerative geometry.

Moduli of Hypersurfaces and Pathologies

Studies moduli spaces of hypersurfaces and investigates geometric pathologies and singularities.

F-Singularities and Characteristic p Methods

Analyzes F-singularities, F-regularity, and characteristic p techniques for studying singularities in algebraic varieties.

Multiplier Ideals and Singularity Invariants

Develops multiplier ideal theory and its applications to studying singularities and test ideals.

Étale Cohomology and Galois Actions

Studies the structure of étale cohomology groups and their Galois module structures over algebraic varieties and schemes.

Intersection Homology and Perverse Sheaves

Investigates intersection homology theory and perverse sheaf categories for singular spaces and their topological properties.

Deformation Theory and Obstruction Spaces

Examines the deformation theory of algebraic varieties and schemes using tangent spaces and obstruction calculus.

Bridgeland Stability and D-brane Categories

Explores Bridgeland stability conditions on derived categories motivated by string theory and D-brane physics.

Grothendieck-Riemann-Roch Theorem Applications

Studies applications of the Grothendieck-Riemann-Roch theorem to characteristic classes and Chern characters on algebraic varieties.

Moduli of Abelian Surfaces and Siegel Varieties

Investigates the geometry of moduli spaces of principally polarized abelian surfaces and their compactifications.

Derived Autoequivalences and Stability

Studies autoequivalences of derived categories and their connections to geometric stability conditions and wall-crossing phenomena.

Crystal Bases and Quantum Groups

Examines the role of crystal bases from quantum group theory in algebraic geometry and representation-theoretic contexts.

Generic Singularities and Whitney Stratification

Studies generic singularities of algebraic varieties and Whitney stratification of singular loci.

Brill-Noether Theory and Rank Conditions

Investigates Brill-Noether theory for vector bundles on curves and conditions for existence of special line bundles.

Positivity Cones and Effective Divisors

Studies the structure of cones of positive divisors and the relationship to ample and effective divisors on varieties.

Categorical Symplectic Reduction

Explores symplectic reduction from a categorical perspective using derived algebraic geometry and stack techniques.

Monge-Ampère Equations and Kähler Geometry

Studies solutions to complex Monge-Ampère equations and their geometric implications for Kähler metrics on algebraic varieties.

Langlands Duality and Spectral Geometry

Investigates the geometric Langlands program and its connections to spectral geometry and automorphic forms.

Virtual Fundamental Classes and GW Theory

Studies construction and properties of virtual fundamental classes in Gromov-Witten theory and related enumerative invariants.

Minimal Generating Sets and Ideal Generators

Examines minimal generating sets of ideals defining algebraic varieties and computational approaches to ideal theory.

Stability of Bundles and Yang-Mills Equations

Studies slope stability of vector bundles and solutions to Yang-Mills equations on algebraic surfaces.

Tautological Rings and Psi Classes

Investigates tautological classes and psi classes in the cohomology of moduli spaces of curves.

Elliptic Fibrations and Picard Lattices

Studies elliptic fibrations on algebraic surfaces and the structure of their Picard lattices.

O-Minimality and Semi-Algebraic Geometry

Explores o-minimal structures and their applications to studying geometry of semi-algebraic sets and real algebraic varieties.

Symplectic Quotient Singularities

Examines singularities arising from symplectic quotients and their resolutions via symplectic desingularization.

Gromov Compactness and Curve Moduli

Studies Gromov compactness for pseudo-holomorphic curves and applications to compactifying moduli spaces.

Donaldson Invariants and Gauge Theory

Investigates Donaldson invariants of algebraic surfaces computed via gauge-theoretic instantons and their geometric meaning.

Homotopy Sheaves and Higher Topos Theory

Studies homotopy sheaves and derived sheaf theory using higher topos theory and infinity categories.

Moment Maps and Symplectic Reduction

Examines moment maps for group actions on algebraic varieties and their use in symplectic reduction.

Character Varieties and Representation Moduli

Studies character varieties parameterizing representations of fundamental groups and their geometric properties.

Nakayama Categories and Recollements

Investigates Nakayama categories and recollements of triangulated categories with applications to algebraic geometry.

Intersection Algorithms and Computational Methods

Studies effective computational algorithms for computing intersections and invariants of algebraic varieties.

Invariant Differential Forms and Canonical Divisors

Examines differential forms invariant under group actions and their relationships to canonical divisors.

Rational Curves and Pathologies

Studies families of rational curves on algebraic varieties and pathologies in their behavior.

Serre-Tate Theory and p-Adic Deformations

Investigates Serre-Tate theory describing deformations of abelian varieties in characteristic p.

Coherent Sheaves on Projective Spaces

Studies the category of coherent sheaves on projective spaces and their stability conditions.

Bott Periodicity and K-Theory

Examines Bott periodicity in algebraic K-theory and applications to understanding vector bundles.

Instanton Moduli and Gauge Fixing

Studies moduli spaces of instantons on algebraic surfaces using gauge-theoretic and algebraic techniques.

Exceptional Collections and Stability Chains

Investigates exceptional collections in derived categories and their connections to stability conditions.

Jacobian Fibrations and Height Functions

Studies Jacobian varieties of curve families and height functions in arithmetic geometry.

Quantum Corrections and Mirror Formulas

Examines quantum corrections to classical formulas in mirror symmetry and enumerative geometry.

Divisor Class Groups and Picard Schemes

Studies Picard schemes and divisor class groups of algebraic varieties and their fine structure.

Holomorphic Curves and Floer Homology

Investigates holomorphic curves in symplectic manifolds and their role in constructing Floer homology theories.

Weil Divisors and Cartier Divisors

Studies divisor theory on varieties with singularities including Weil divisors and their properties.

Tropical Intersection and Piecewise Linearity

Examines tropical intersection theory and piecewise linear structures arising in tropicalization.

Poisson Structures and Noncommutative Resolution

Studies Poisson structures on algebraic varieties and their resolutions via noncommutative algebras.

Spectral Sequences and Derived Functors

Investigates spectral sequences for derived functors and applications to computing cohomology of algebraic varieties.

Automorphism Groups and Dynamical Systems

Studies automorphism groups of algebraic varieties and dynamical properties of their actions.

Reduced Schemes and Nilpotents

Examines the role of nilpotent elements in scheme theory and reduced structures on varieties.

Morse Theory on Manifolds with Corners

Studies Morse theory on manifolds with boundary and its applications to algebraic geometry.

Fundamental Groups and Etale Covers

Investigates fundamental groups of algebraic varieties through étale covers and Galois theory.

Derived Stacks and Higher Categorical Structures

Investigation of derived algebraic stacks using infinity categories and higher topos theory to study homotopical properties of moduli problems and derived intersection theory.

Graded Bialgebras and Hopf Modules

Studies graded bialgebra structures and Hopf modules with geometric interpretations.

Hilbert Functions and Graded Modules

Examines Hilbert functions of graded modules and their relationships to dimensions of varieties.

Mirror Symmetry and Categorical Enumerative Invariants

Study of categorical aspects of mirror symmetry through derived equivalences, categorical resolutions, and relationships between enumerative invariants on mirror pairs of varieties.

Faltings Height and Arithmetic Intersection

Studies Faltings heights and arithmetic intersection theory on arithmetic varieties.

Singularities in Positive and Mixed Characteristic

Analysis of singularity types and their invariants in characteristic p and mixed characteristic rings, including F-singularities, tight closure, and applications to commutative algebra.

Non-Archimedean Geometry and Analytic Spaces

Development of analytic and geometric theories over non-Archimedean fields using Berkovich spaces and Huber''s adic spaces with applications to arithmetic and non-Archimedean dynamics.

Dg-Algebras and Noncommutative Resolutions

Exploration of differential graded algebras and noncommutative geometry as tools for constructing resolutions and studying derived categories of singular varieties and quotient singularities.