Topology Geometry

Persistent Homology and Topological Data Analysis

Studies multiscale topological features of point clouds and datasets using filtrations to extract robust invariants for machine learning applications.

RESEARCH GAP FRONTIERS

Khovanov Homology and Quantum Invariants

Develops categorical invariants for knots and links through Khovanov homology and explores connections to quantum field theory and representation theory.

RESEARCH GAP FRONTIERS

Symplectic Geometry and Hamiltonian Dynamics

Investigates symplectic manifolds, Lagrangian submanifolds, and integrable systems using tools from microlocal analysis and Floer theory.

RESEARCH GAP FRONTIERS

Floer Homology and Mirror Symmetry

Explores Lagrangian Floer homology and its role in mirror symmetry conjectures relating symplectic and algebraic geometry.

RESEARCH GAP FRONTIERS

Geometric Measure Theory and Minimal Surfaces

Studies rectifiability, varifolds, and regularity theory of minimal surfaces and Plateau''s problem using measure-theoretic methods.

RESEARCH GAP FRONTIERS

Kahler-Einstein Metrics and Extremal Geometry

Investigates existence and uniqueness of special metrics on Kahler manifolds through algebraic and differential geometric techniques.

RESEARCH GAP FRONTIERS

Derived Algebraic Geometry and Higher Stacks

Develops higher categorical foundations for algebraic geometry using derived schemes, infinity categories, and derived algebraic geometry.

RESEARCH GAP FRONTIERS

Topological Quantum Field Theory

Constructs TQFTs and extended TQFTs using categorical methods to compute topological invariants and understand quantum phenomena.

Gauge Theory and Yang-Mills Equations

Analyzes moduli spaces of connections on principal bundles and their application to 4-manifold topology and invariants.

Instanton Floer Homology and 3-Manifolds

Develops instanton gauge theory on 3-manifolds to compute Floer homology invariants distinguishing smooth structures and knot concordance.

Heegaard Floer Homology and Knot Theory

Studies Heegaard Floer invariants for 3-manifolds and knots, including applications to surgery formulas and concordance groups.

Contact Geometry and Legendrian Knots

Explores contact structures, Legendrian and transverse knots using symplectic techniques and contactomorphism groups.

Four-Manifold Topology and Donaldson Invariants

Investigates exotic smooth structures on 4-manifolds using gauge theory, Donaldson invariants, and Seiberg-Witten equations.

Seiberg-Witten Theory and Smooth Invariants

Applies Seiberg-Witten equations to compute invariants of 4-manifolds and study relationships with algebraic surfaces.

Calabi-Yau Manifolds and String Theory

Studies geometry and moduli spaces of Calabi-Yau varieties and their physical applications in string theory compactifications.

Riemannian Curvature and Positive Scalar Curvature

Investigates manifolds with positive scalar curvature bounds using spinor methods, Dirac operators, and geometric analysis.

Ricci Flow and Geometrization

Analyzes Ricci flow dynamics for geometric evolution equations with applications to uniformization and 3-manifold topology.

Kahler Geometry and Balanced Metrics

Studies special Kahler metrics including balanced metrics, constant scalar curvature Kahler metrics, and their extremal properties.

Orbifold Geometry and Singularities

Develops differential geometry of orbifolds and investigates resolution of singularities through topological and algebraic methods.

Higher Category Theory and Topological Invariants

Uses infinity categories and higher categorical structures to construct and study generalized topological invariants.

Knot Concordance and 4-Genus

Investigates concordance invariants and 4-genus of knots using Floer homology, Khovanov homology, and algebraic techniques.

Thurston Norm and Fibered Knots

Studies Thurston norm, fibered structures on 3-manifolds, and relationships to algebraic surfaces and symplectic geometry.

Homological Algebra and Derived Functors

Develops derived functors, spectral sequences, and homological methods for studying algebraic and topological structures.

Cohomology Rings and Intersection Forms

Analyzes cup product structures and intersection forms on manifolds with applications to classification and obstructions.

Characteristic Classes and Vector Bundles

Studies Chern classes, Stiefel-Whitney classes, and Pontryagin classes for vector bundles and topological applications.

Cobordism Theory and Adams Spectral Sequence

Investigates cobordism rings, bordism groups, and their computation using Adams spectral sequences and homotopy theory.

Homotopy Type Theory and Univalent Foundations

Develops computational foundations for topology using homotopy type theory and constructs topological invariants in formal systems.

Algebraic Topology and Model Categories

Uses model category structures and abstract homotopy theory to study topological and algebraic invariants systematically.

Stratified Spaces and Intersection Homology

Studies intersection homology, perverse sheaves, and topology of singular spaces using stratification techniques.

Manifold Learning and Topological Methods

Applies topological data analysis and persistent homology to machine learning for understanding high-dimensional manifolds.

Affine Differential Geometry and Convex Bodies

Studies affine geometry of hypersurfaces, convex bodies, and their topological and geometric invariants.

Discrete Differential Geometry and Polyhedral Surfaces

Develops discrete analogues of differential geometry for meshes, polyhedral surfaces, and computational geometry.

Geometric Group Theory and Word Metrics

Investigates finitely generated groups through geometric properties, Cayley graphs, and quasi-isometric invariants.

CAT(0) Spaces and Hyperbolic Geometry

Studies non-positive curvature spaces, hyperbolic groups, and their geometric and topological properties.

Right-Angled Artin Groups and Raags

Analyzes right-angled Artin groups, cube complexes, and their applications to geometric group theory.

Teichmuller Theory and Moduli Spaces

Studies Teichmuller spaces of Riemann surfaces and moduli space compactifications using geometric and algebraic methods.

Kleinian Groups and 3-Dimensional Hyperbolic Geometry

Investigates Kleinian groups, hyperbolic 3-manifolds, and deformation spaces of geometric structures.

Conformal Geometry and Mobius Transformations

Studies conformal structures, Mobius geometry, and conformal invariants on manifolds and metric spaces.

Metric Geometry and Optimal Transport

Applies optimal transport theory and Wasserstein distances to study geometric and topological properties of metric spaces.

Riemannian Submersions and Fiber Bundles

Studies geometry of submersions, fiber bundles with connection, and their topological and metric properties.

Spin Geometry and Dirac Operators

Investigates spin structures, spinor fields, Dirac operators, and their spectral properties and topological applications.

Index Theory and Heat Kernels

Studies Atiyah-Singer index theorem, heat kernel methods, and spectral geometry for elliptic operators.

Orbifold Cohomology and Chen-Ruan Theory

Develops orbifold cohomology and Chen-Ruan twisted sectors for algebraic and symplectic orbifolds.

Gromov-Witten Invariants and Enumerative Geometry

Computes Gromov-Witten invariants and studies enumerative problems through pseudo-holomorphic curves and localization.

Quantum Cohomology and Small Quantum Rings

Studies quantum deformations of cohomology rings and their structure through Gromov-Witten theory.

Fukaya Categories and Homological Mirror Symmetry

Constructs Fukaya categories from symplectic manifolds and investigates homological mirror symmetry conjectures.

Knot Homologies and Categorification

Develops categorifications of quantum invariants including Khovanov, Heegaard Floer, and instanton homologies.

Sutured Manifold Theory and Bordered Homology

Studies sutured manifolds and develops bordered versions of Heegaard Floer homology with gluing formulas.

Monopole Floer Homology and Link Invariants

Applies monopole equations to define Floer homology for links and study concordance invariants.

Knot Genus and Surfaces in 4-Space

Investigates minimum genus surfaces and ribbon surfaces using topological and gauge-theoretic methods.

Topological Complexity and Motion Planning

Studies the minimum number of continuous rules required to determine motion in configuration spaces and robotic path planning problems.

Floer Homology for Legendrian Submanifolds

Develops invariants of Legendrian submanifolds in contact manifolds using Floer-theoretic methods and their applications to knot classification.

Quasigeodesic Stability and Hyperbolic Groups

Investigates when quasigeodesics in hyperbolic spaces are uniformly close to geodesics and implications for group structure.

Knot Floer Homology and Surgery Formulas

Develops computational techniques for knot Floer homology under Dehn surgery and related topological operations.

Instanton Moduli Spaces and Representation Varieties

Studies the geometry and topology of moduli spaces of Yang-Mills instantons and connections to character varieties.

Tropical Geometry and Amoebas

Explores the piecewise-linear structures arising from algebraic varieties through tropicalization and logarithmic limits.

Geodesic Laminations and Foliations

Studies the structure and classification of measured geodesic laminations on hyperbolic surfaces and their dynamics.

Hochschild Homology and Cyclic Structures

Investigates cyclic homology theories and their applications to differential geometry and noncommutative topology.

Moduli of Algebraic Curves and Stacks

Studies the stack-theoretic foundations of moduli spaces of curves and their compactifications using geometric invariant theory.

Mapping Class Groups and Surface Bundles

Examines the structure of surface diffeomorphism groups and their homological properties in fibrations.

Anosov Diffeomorphisms and Dynamical Systems

Studies the topological and smooth rigidity of hyperbolic dynamical systems and partially hyperbolic structures.

Moment Maps and Symplectic Reduction

Investigates group actions on symplectic manifolds via moment maps and their quotient structures.

Thurston Geometries and Geometric Structures

Classifies and studies geometric structures on three-dimensional manifolds and their deformation spaces.

Derived Intersection Theory and Excess Intersections

Develops algebraic geometry techniques for computing intersections in derived schemes with excess dimension.

Lens Spaces and Surgery Invariants

Studies surgery descriptions of lens spaces and their invariants under different surgery presentations.

Heegaard Splittings and Thin Position

Analyzes Heegaard splittings of three-manifolds through thin position and complexity reduction techniques.

Symplectic Fillings and Contact Boundaries

Classifies symplectic four-manifolds with prescribed contact three-manifold boundaries using obstruction theory.

Quantum Representations and Jones Polynomial

Studies connections between quantum group representations and quantum invariants of links through categorification.

Stable Concordance and Link Homologies

Investigates concordance invariants derived from link homology theories and their stable classifications.

Orbibundle Structures and Singular Fibrations

Studies fiber bundles with orbifold structures and their characteristic classes in singular geometry.

Weinstein Manifolds and Exact Symplectic Geometry

Explores exact symplectic manifolds with Liouville forms and their applications to contact topology.

Plane Partitions and Toric Geometry

Studies combinatorial objects arising from toric varieties and their topological invariants.

Manifold Rigidity and Borel Conjecture

Investigates topological rigidity of manifolds with respect to their fundamental groups using surgery theory.

Persistence Modules and Algebraic Stability

Develops stability theory for persistence modules in topological data analysis with algebraic foundations.

Branched Coverings and Riemann Surfaces

Studies branched covering maps between surfaces and their moduli spaces through Galois theory.

Geometric Deformation Theory and Milnor Fibers

Analyzes singularities through deformation families and the topology of Milnor fiber complements.

Turaev-Viro and State Sum Invariants

Studies topological invariants of three-manifolds defined through state sum models and triangulations.

Holomorphic Disks and Pseudoholomorphic Curves

Investigates moduli spaces of pseudoholomorphic curves in symplectic manifolds and their enumerative properties.

Sutured Instanton Homology and Tangles

Develops gauge-theoretic invariants for sutured manifolds and their applications to tangle classification.

Fibered Links and Seifert Surfaces

Studies fibered knots and links with prescribed Seifert surface structures and their monodromy.

Resolution of Singularities and Blow-ups

Analyzes geometric desingularization techniques and their applications to algebraic topology.

Automorphism Groups and Nielsen Realization

Studies realization of finite subgroups of surface automorphisms as isometries of Riemannian metrics.

Contact Homology and Legendrian Invariants

Develops differential graded algebra structures encoding contact and Legendrian geometry.

Casson Handles and Four-Dimensional Topology

Studies Casson handles as fundamental building blocks in exotic smooth structures on four-manifolds.

Dehn Surgery and Three-Manifold Invariants

Investigates relationships between Dehn surgery operations and topological invariants of three-manifolds.

Spherical Double Covers and Branched Covers

Studies topology of branched covering spaces with focus on double covers and their invariants.

Controlled Topology and Assembly Maps

Develops controlled versions of algebraic topology for non-compact and singular spaces.

Toric Varieties and Polytope Combinatorics

Explores connections between combinatorial polytope data and topological properties of toric varieties.

Lattice Homology and Q-Homology Planes

Studies lattice-based invariants of rational surface singularities and their topological applications.

Wrapped Fukaya Categories and Stops

Investigates Fukaya categories with stops and their homological mirror symmetry correspondences.

Seifert Fibered Spaces and Orbifold Singularities

Classifies three-dimensional Seifert fibered spaces using orbifold singularity structures.

L-Spaces and Non-Thurston Structures

Studies three-manifolds with L-space properties that do not admit standard geometric structures.

Cylindrical Manifolds and Weinstein Structures

Develops symplectic geometry of manifolds with cylindrical end structures and Liouville vector fields.

Reshetikhin-Turaev Functors and TQFT

Studies rigorous constructions of topological quantum field theories from quantum group data.

Concordance Homomorphisms and Kernels

Investigates properties of homomorphisms from knot groups to concordance groups.

Thurston Obstructions and Rational Maps

Studies Thurston obstructions preventing realization of post-critically finite maps as rational functions.

Knotted Surfaces in Four-Dimensional Space

Analyzes smooth and topological embeddings of surfaces into four-space and their invariants.

Spectral Sequences and Page Computations

Develops computational methods for spectral sequences arising from filtrations and bundle structures.

Factorization Homology and Chiral Algebras

Investigates topological field theory invariants through factorization algebras and vertex algebra structures.

Knot Genus and Optimal Surfaces

Studies minimal genus surfaces bounded by knots in three-manifolds using topological methods.

Symplectic Capacities and Embedding Problems

Studies obstructions to symplectic embeddings using capacities and refined invariants derived from holomorphic curve theory.

Lagrangian Cobordism and Exact Sequences

Investigates cobordism relations between Lagrangian submanifolds and their applications to symplectic topology.

Cluster Algebras and Positivity Phenomena

Explores connections between cluster algebras and geometric structures including Grassmannians and positivity in vector spaces.

Toric Geometry and Moment Polytopes

Analyzes the correspondence between toric varieties and convex polytopes via moment maps and combinatorial methods.

Holomorphic Discs and Boundary Value Problems

Studies regularity and existence results for holomorphic discs with Lagrangian boundary conditions in symplectic manifolds.

Geometric Quantization and Prequantum Bundles

Develops the theory of prequantum line bundles and their role in quantizing symplectic and Kahler manifolds.

Equivariant Cohomology and GKM Theory

Applies Goresky-Kottwitz-MacPherson theory to compute equivariant cohomology of spaces with group actions.

Microlocal Sheaf Theory and Wrapped Sheaves

Uses microsupport and wrapped sheaves to study symplectic invariants and constructible sheaves on cotangent bundles.

Weinstein Structures and Liouville Manifolds

Investigates Weinstein domains and Liouville manifolds as fundamental objects in exact symplectic topology.

Symplectic Resolutions and Crepant Categories

Studies derived equivalences arising from symplectic resolutions and applications to categorical representation theory.

Geometric Invariant Theory and Quotient Singularities

Analyzes stability conditions and moduli spaces in GIT quotients and their singularities in algebraic geometry.

Nonnegative Sectional Curvature and Soul Theory

Develops the geometry and topology of Riemannian manifolds with nonnegative sectional curvature using soul theorems.

Controlled Topology and Surgery Theory

Applies controlled versions of algebraic topology to study manifold structures parametrized by control maps.

Hochschild Homology and Derived Stacks

Computes Hochschild invariants of derived algebraic structures and their geometric applications.

Holomorphic Curves in Almost Complex Manifolds

Develops pseudoholomorphic curve techniques in almost complex geometry beyond the symplectic setting.

Filtrations in Heegaard Floer Theory

Studies refinements of Heegaard Floer homology through algebraic and topological filtrations on chain complexes.

Khovanov-Rozansky Homology and Higher Representation Theory

Connects categorified link invariants to representations of quantum groups and higher categorical structures.

Symplectic Field Theory and Pseudoholomorphic Foliations

Develops cylindrical contact homology using pseudoholomorphic foliations in symplectic cobordisms.

Instantons and Gauge Theory on 4-Manifolds

Studies moduli spaces of anti-self-dual connections and their applications to 4-dimensional topology.

Orbifold Singularities and Resolution Types

Classifies resolutions of orbifold singularities and their role in algebraic and symplectic geometry.

Hofer Geometry and Bi-invariant Metrics

Studies the Hofer metric on symplectomorphism groups and its geometric consequences for rigidity.

Thurston-Bennequin Invariant and Contact Topology

Analyzes the Thurston-Bennequin and rotation numbers of Legendrian curves in contact 3-manifolds.

Morse Theory on Infinite-Dimensional Spaces

Develops Morse theory for functionals on loop spaces and path spaces with applications to geodesics.

Polyhedral Geometry and Polytope Algebra

Studies combinatorial and algebraic properties of polytopes including face lattices and Ehrhart polynomials.

Exotic Smooth Structures on 4-Manifolds

Constructs and classifies distinct smooth structures on topological 4-manifolds using gauge theory.

Aspherical Manifolds and Group Cohomology

Studies topology and cohomology of aspherical spaces with connections to group actions and classifying spaces.

Nakajima Quiver Varieties and Symplectic Resolutions

Analyzes Nakajima quiver variety constructions and their role as symplectic resolutions of singularities.

Legendrian Contact Homology and Augmentations

Studies augmentation varieties and generating families for Legendrian submanifolds in contact manifolds.

Positive Loop Spaces and Totally Positive Matrices

Investigates totally nonnegative structures in loop groups and their geometric topology.

Cyclic Polytopes and Neighborliness

Studies extremal combinatorial properties of cyclic polytopes and neighborly polytope constructions.

Hyperplane Arrangements and Matroid Geometry

Analyzes topology and cohomology of hyperplane arrangement complements using matroid-theoretic methods.

Symplectic Groupoids and Poisson Structures

Studies symplectic groupoids as geometric implementations of Poisson manifolds and their deformation theory.

Knot Invariants from Skein Relations

Develops and computes knot invariants defined recursively through skein relations and quantum algebra.

Orbifold Bordism and Twisted Spin Structures

Studies bordism rings of orbifolds with twisted spin structures and their characteristic class theory.

Instanton Moduli and Representation Varieties

Connects moduli spaces of instantons to representation varieties of fundamental groups via character varieties.

Symplectic Toric Varieties and Delzant Polytopes

Establishes the correspondence between toric symplectic manifolds and Delzant lattice polytopes.

Graded Modules and Topological Hochschild Homology

Develops topological Hochschild homology for ring spectra with applications to algebraic K-theory.

Wrapping and Non-Exact Symplectic Manifolds

Studies wrapped Fukaya categories for non-exact symplectic manifolds with proper convexity at infinity.

Matroid Realization and Oriented Matroids

Investigates realizability problems for matroids and oriented matroids using real algebraic geometry.

Bott Periodicity and K-Theory Computations

Applies Bott periodicity to compute K-theory of classifying spaces and vector bundles over manifolds.

Moment-Angle Complexes and Torus Actions

Studies homotopy and cohomology of moment-angle complexes determined by simplicial complexes.

Symplectic Spinors and Quantization

Develops the theory of symplectic spinors and their role in geometric quantization schemes.

Mapping Class Groups and Torelli Groups

Studies subgroups of mapping class groups of surfaces including Torelli and Heegaard groups.

Gromov-Hausdorff Convergence and Metric Limits

Analyzes limits of metric spaces under Gromov-Hausdorff distance with applications to Ricci flows.

Morse-Floer Theory for Symplectic Quotients

Develops Floer theory for symplectic quotients by Hamiltonian group actions.

Legendrian Fronts and Caustics

Studies singularities of Legendrian fronts and their caustics in contact geometry.

Stiefel-Whitney Classes and Vector Bundle Obstruction

Uses Stiefel-Whitney classes to determine orientability and obstruction theory for vector bundles.

Symplectic Fillings and Tight Contact Structures

Characterizes tight contact structures through their symplectic fillings and related invariants.

Foliations and Godbillon-Vey Invariant

Studies secondary characteristic classes of foliations including the Godbillon-Vey invariant.

Convexity in Symplectic Geometry and Maximally Graded Spaces

Investigates convexity properties of moment maps and their quantitative aspects in symplectic manifolds.

Symplectic Four-Manifolds and Elliptic Fibrations

Investigates classification and structure of symplectic four-dimensional manifolds through elliptic and Lefschetz fibration techniques.

Equivariant Cohomology and Torus Actions

Analyzes topological properties of spaces with group actions using equivariant cohomology theories and moment map constructions.

Legendrian Weaves and Cluster Algebras

Explores combinatorial structures of Legendrian surfaces and their connections to cluster algebra mutations and representations.

Instantons and Knot Invariants

Develops gauge-theoretic approaches to computing and understanding knot invariants via instanton Floer homology variants.

Tropical Geometry and Curve Counts

Studies enumerative geometry problems through tropical geometry techniques including correspondence theorems and chamber structures.

Alexandrov Spaces and Curvature Bounds

Analyzes metric geometry of singular spaces with synthetic curvature bounds and regularity properties of Alexandrov spaces.

Holomorphic Curves and Pseudoholomorphic Techniques

Applies pseudoholomorphic curve theory to study symplectic topology, quantum invariants, and enumerative invariants of target spaces.

Thurston Polytopes and Veering Triangulations

Investigates structures on three-manifolds using veering triangulations and polytopes arising from Thurston norm computations.

Hyperplane Arrangements and Formality

Studies rational homotopy theory and formality properties of arrangement complements and related configuration spaces.

Braid Groups and Configuration Spaces

Analyzes algebraic topology and homological properties of braid groups through configuration space and loop space techniques.

Symplectic Capacities and Rigidity

Studies symplectic rigidity phenomena through capacities and applications to embedding and rigidity problems in symplectic topology.

Monopole Equations and Three-Manifold Invariants

Develops monopole Floer homology and related invariants to characterize three-dimensional topological properties and cobordisms.

Cluster Structures and Positivity Geometry

Explores cluster algebras in geometric contexts including total positivity, cluster varieties, and their topological realizations.

Legendrian Surfaces and Exact Sequences

Studies higher-dimensional Legendrian submanifolds and their contact geometric properties via homological algebra methods.

Weinstein Structures and Liouville Domains

Investigates Weinstein manifolds and Liouville domains as fundamental objects in symplectic geometry with applications to flexible and rigid phenomena.

Bridgeland Stability and Wall-Crossing

Studies Bridgeland stability conditions on derived categories and applications to wall-crossing formulas in enumerative geometry.

Hyperbolic Volume and Simplicial Complexes

Examines hyperbolic structures on manifolds and their volumes through simplicial approximations and combinatorial bounds.

Quasiconformal Maps and Teichmuller Space

Studies Teichmuller spaces and moduli of Riemann surfaces using quasiconformal homeomorphisms and extremal quasiconformal maps.

Symplectic Toric Geometry and Delzant Theory

Analyzes completely integrable Hamiltonian systems and symplectic toric manifolds using Delzant polytope constructions.

Categorical Resolutions and Tilting Theory

Studies categorical resolutions of singularities and tilting complexes in derived categories with geometric applications.

Branched Coverings and Orbifold Invariants

Analyzes branched covers of manifolds and orbifold structures with applications to invariant computation and classification.

Wrapped Microlocal Theory and Sheaves

Develops wrapped microlocal sheaf theory connecting symplectic geometry to constructible sheaves and categorical invariants.

Lagrangian Correspondences and Functors

Studies Lagrangian correspondences as morphisms in symplectic topology inducing functors between Floer-type theories.

Simplicial Commutative Algebra and Derived Schemes

Applies simplicial methods and derived algebraic geometry to study homotopy-coherent algebraic structures and their moduli.

Thurston Geometries and Orbifold Fibrations

Studies geometric structures modeled on eight Thurston geometries and fibrations of orbifolds in three dimensions.

Pseudoholomorphic Discs and Weak Fillings

Analyzes pseudoholomorphic disc moduli spaces to obstruct weak fillings and study symplectic cobordism invariants.

Polygon Spaces and Inverse Kinematics

Investigates topology of configuration spaces arising from linkages and robot arms using algebraic topology methods.

Holonomy and Special Riemannian Geometry

Studies exceptional holonomy groups including G2 and Spin(7) structures and their geometric properties and moduli.

Symplectic Homology and Conical Singularities

Develops symplectic homology theories for symplectic manifolds with conical singularities and applications to fixed points.

Knot Floer Homology and Concordance

Applies knot Floer homology to study knot concordance groups, genus bounds, and cobordism maps in four dimensions.

Foliations and Transverse Geometry

Studies foliated manifolds and their transverse geometric structures including transverse contact geometry and holonomy.

Homological Mirror Symmetry and Noncommutative Geometry

Establishes equivalences between Fukaya categories and derived categories via mirror symmetry in noncommutative geometric contexts.

Asymptotic Expansion and Eta Invariants

Studies spectral properties of differential operators on manifolds using heat kernel expansions and eta invariant computations.

Quiver Varieties and Geometric Representation Theory

Analyzes representation theory of quivers through geometry of quiver varieties and symplectic resolutions.

Enumerative Invariants and Counting Curves

Studies curve counting problems using Gromov-Witten invariants, Donaldson-Thomas invariants, and related invariants.

Geodesic Flows and Dynamical Systems

Analyzes dynamical properties of geodesic flows on manifolds with focus on ergodicity and topological entropy.

Character Varieties and Representation Spaces

Studies character varieties of fundamental groups with geometric and topological methods including singular loci structure.

Submanifold Geometry and Normal Bundles

Investigates extrinsic geometry of submanifolds through normal bundle structures and higher fundamental forms.

Bordered Floer Homology and Pairing Theorems

Develops bordered variants of Floer homology with pairing theorems relating invariants across manifold decompositions.

Fibered Links and Open Book Decompositions

Studies fibered links in three-manifolds through open book decompositions and their properties in contact topology.

Knot Genus and Seifert Surfaces

Investigates minimal genus Seifert surfaces and topological genus bounds using homological and gauge-theoretic methods.

Categorical Morse Theory and Conormal Sheaves

Applies conical Lagrangian geometry and conormal sheaves to study Morse theory categorically in microlocal analysis.

Orbifold Singularities and Resolution Invariants

Studies topological and algebraic invariants of orbifold singular points and minimal resolutions in higher dimensions.

Lens Spaces and Surgery Diagrams

Analyzes three-manifolds via Dehn surgery and surgery diagrams with applications to invariant computation and classification.

Spin Structures and Dirac Spectra

Studies spin manifolds and spectral properties of Dirac operators with applications to topological invariants and rigidity.

Immersions and Immersion Theory

Investigates space of immersions of manifolds using obstruction theory and higher homotopy groups of Stiefel manifolds.

Geometric Flows and Parabolic PDEs

Studies evolution equations on manifolds including mean curvature flow, Kaczmarz flow, and their topological applications.

Equivariant Cohomology and Moment Graph Algebras

This research investigates the algebraic structures arising from group actions on topological spaces, focusing on moment graphs and their combinatorial-geometric invariants in the study of flag varieties and Schubert varieties.

Legendrian Surfaces and Augmentation Categories

This research explores the higher-dimensional analogues of contact topology through the study of Legendrian submanifolds and their invariants via augmentation categories and sheaf-theoretic methods.

Algebraic Cycles and Chow Groups

Analyzes algebraic cycles on varieties through Chow groups and motivic cohomology with applications to intersection theory.