Pure Mathematics

Algebraic Geometry and Moduli Spaces

Investigation of geometric properties of algebraic varieties and the construction of parameter spaces classifying geometric objects up to equivalence.

RESEARCH GAP FRONTIERS

Arithmetic Geometry and Diophantine Equations

Study of integer and rational solutions to polynomial equations using geometric methods and tools from algebraic number theory.

RESEARCH GAP FRONTIERS

Homotopy Theory and Higher Category Theory

Exploration of categorical structures and topological properties using infinity categories and abstract homotopical algebra.

RESEARCH GAP FRONTIERS

Derived Algebraic Geometry

Extension of algebraic geometry using derived functors and homological algebra to study singularities and intersection theory.

RESEARCH GAP FRONTIERS

Noncommutative Geometry and Operator Algebras

Investigation of geometric structures arising from noncommutative rings and C-star algebras in quantum settings.

RESEARCH GAP FRONTIERS

Symplectic Geometry and Hamiltonian Dynamics

Study of symplectic manifolds and their dynamical systems using techniques from geometric topology and analysis.

RESEARCH GAP FRONTIERS

Differential Topology and Manifold Theory

Research on smooth manifolds, characteristic classes, and topological invariants using differential-geometric methods.

RESEARCH GAP FRONTIERS

Representation Theory of Finite Groups

Classification and analysis of linear representations of finite groups and their decomposition properties.

RESEARCH GAP FRONTIERS

Representation Theory of Lie Groups

Study of continuous representations of Lie groups and their connections to harmonic analysis and differential geometry.

Quantum Groups and Hopf Algebras

Investigation of deformations of universal enveloping algebras and quantum symmetries in algebra and topology.

Homological Algebra and Cohomology Theories

Development and application of derived functors, spectral sequences, and generalized cohomology operations.

Algebraic Topology and Stable Homotopy

Study of stable homotopy groups using spectral sequences, Adams operations, and chromatic techniques.

K-Theory and Index Theory

Investigation of topological and analytic K-theory with applications to differential operators and elliptic regularity.

Knot Theory and Low-Dimensional Topology

Analysis of knots and links using invariants from quantum groups, categorification, and geometric structures.

Gauge Theory and Yang-Mills Theory

Study of connections on principal bundles and their moduli spaces using differential geometric and analytical methods.

Floer Homology and Lagrangian Topology

Investigation of Floer cohomology theories and their applications to Lagrangian submanifolds and mirror symmetry.

Geometric Representation Theory

Study of representations through geometric methods including perverse sheaves, D-modules, and derived categories.

Lie Algebras and Enveloping Algebras

Investigation of structure theory, highest weight modules, and representation theory of Lie algebras.

Commutative Algebra and Scheme Theory

Study of commutative rings, ideals, modules and their geometric interpretation through algebraic schemes.

Singularities and Resolution Techniques

Analysis of singular varieties through blowups, deformation theory, and invariant resolutions.

Intersection Theory and Chow Groups

Study of intersection multiplicities and Chow groups to understand cycles and enumerative geometry.

Number Theory and Elliptic Curves

Investigation of rational points on elliptic curves using Mordell-Weil theory and descent methods.

Algebraic Number Theory and Class Field Theory

Study of algebraic integers, prime ideals, and abelian extensions of number fields through cohomological methods.

Analytic Number Theory and Zeta Functions

Analysis of distribution of primes and L-functions using complex analytic techniques and special functions.

RESEARCH GAP FRONTIERS

Modular Forms and Automorphic Representations

Study of modular forms and their generalizations as automorphic representations on adelic groups.

Galois Theory and Inverse Galois Problem

Investigation of Galois extensions, absolute Galois groups, and realizability of finite groups as Galois groups.

Iwasawa Theory and p-adic Analysis

Study of p-adic L-functions, Selmer groups, and growth of arithmetic invariants in towers of fields.

Birational Geometry and Minimal Models

Investigation of birational transformations and classification of varieties through minimal model program.

Enumerative Geometry and Intersection Counts

Study of counting problems in geometry using intersection theory, curves, and moduli space techniques.

Toric Geometry and Fan Theory

Investigation of algebraic varieties defined by combinatorial data from polytopes and fans.

Derived Categories and Triangulated Categories

Study of derived functors, triangulated structures, and stability conditions in homological algebra.

Sheaf Cohomology and Characteristic Classes

Investigation of cohomology of sheaves on spaces and topological invariants of vector bundles.

Hodge Theory and Kähler Geometry

Study of Hodge structures on cohomology and their applications to Kähler manifolds and mirror symmetry.

Categorical Logic and Topos Theory

Investigation of topoi as categorical generalizations of sets and their applications to logic and geometry.

Category Theory and Natural Transformations

Foundational study of categorical structures, functors, and universal properties across mathematics.

Infinity Categories and Homotopy Limits

Development of higher category theory with applications to derived geometry and stable homotopy theory.

Abstract Algebra and Group Theory

Study of group structures, subgroups, homomorphisms, and fundamental theorems of group theory.

Ring Theory and Noncommutative Rings

Investigation of ring properties, ideals, modules, and structure of noncommutative rings.

Field Theory and Galois Extensions

Study of field extensions, separability, algebraic closure, and fundamental theorem of Galois theory.

Combinatorial Algebra and Symbolic Computation

Investigation of algebraic structures arising from combinatorics and computational algebra methods.

Combinatorial Topology and Simplicial Complexes

Study of combinatorial structures and their topological properties using simplicial homology.

Graph Theory and Extremal Combinatorics

Investigation of extremal problems on graphs and hypergraphs using algebraic and probabilistic methods.

Matroid Theory and Geometric Combinatorics

Study of matroids as abstract generalizations of linear independence with connections to geometry.

Partition Functions and Combinatorial Species

Investigation of generating functions and formal power series in enumerative combinatorics.

Real Algebraic Geometry and Positivity

Study of real solutions to polynomial equations and semidefinite programming using algebraic methods.

Tropical Geometry and Polyhedral Methods

Investigation of tropical varieties arising from degenerations and their combinatorial structure.

Logarithmic Structures and Log Geometry

Study of logarithmic differential forms and log-smooth schemes in algebraic and tropical geometry.

Toroidal Embeddings and Compactifications

Investigation of toroidal compactifications of arithmetic and symmetric spaces.

Stacks and Orbifold Geometry

Study of algebraic stacks as generalizations of schemes with applications to quotient singularities.

Deformation Theory and Obstruction Theory

Investigation of infinitesimal deformations of geometric and algebraic structures using tangent spaces.

Perfectoid Spaces and Adic Geometry

Studies perfectoid spaces and their applications to p-adic Hodge theory and arithmetic geometry through adic techniques.

Chromatic Homotopy Theory and Morava K-theory

Investigates the chromatic filtration of stable homotopy groups using Morava K-theory and related invariants.

Condensed Mathematics and Analytic Geometry

Develops condensed mathematics framework to unify p-adic and complex analytic geometry through liquid vector spaces.

Motivic Cohomology and Algebraic Cycles

Studies motivic cohomology theories and their connections to algebraic cycle theory and Bloch-Kato conjecture.

Derived Symplectic Geometry

Explores shifted symplectic structures on derived stacks and their applications to quantization and mirror symmetry.

Microlocal Analysis and Sheaf Theory

Develops microlocal sheaf theory and microlocal kernel methods with applications to representation theory.

Prismatic Cohomology and p-adic Hodge Theory

Studies prismatic cohomology as a unified framework for p-adic Hodge theory and crystalline cohomology.

Khovanov Homology and Categorification

Investigates Khovanov homology, its generalizations, and categorification of quantum invariants.

Arithmetic Curves and Arithmetic Surfaces

Studies arithmetic invariants of curves and surfaces over number fields including Arakelov theory.

Fukaya Categories and Mirror Symmetry

Examines Fukaya categories of symplectic manifolds and their role in homological mirror symmetry.

Equivariant Homotopy Theory and Group Actions

Develops equivariant homotopy theory including equivariant stable homotopy and RO-grading systems.

Resurgent Transseries and Asymptotic Analysis

Studies resurgent functions and transseries with applications to asymptotic analysis and differential equations.

Cluster Algebras and Quiver Representations

Investigates cluster algebra structures, mutations, and connections to representation theory of quivers.

Foliations and Foliated Bundles

Studies geometric and topological properties of foliations including characteristic classes and transverse geometry.

Higher Dimensional Algebraic Varieties

Analyzes geometry and classification of varieties of dimension greater than three including Fano varieties.

Étale Cohomology and L-functions

Studies étale cohomology theories and their applications to L-function computations and Tamagawa numbers.

Instantons and Self-Dual Connections

Examines moduli spaces of instantons and self-dual Yang-Mills connections on four-manifolds.

Spectral Sequences and Computational Topology

Develops spectral sequence methods for computing homology and cohomology with computational applications.

Descent Theory and Faithfully Flat Descent

Studies descent theory including Galois descent and faithfully flat descent for schemes and stacks.

Homotopy Type Theory and Univalent Foundations

Develops homotopy type theory foundations for mathematics including univalent axioms and higher inductive types.

Stable Commutative Ring Spectra

Studies ring spectra and their algebraic properties including E-infinity structures and derived commutative algebra.

Orbifold Cohomology and Chen-Ruan Theory

Investigates Chen-Ruan orbifold cohomology and twisted sector geometry on orbifolds.

Schubert Calculus and Flag Varieties

Studies Schubert varieties, Schubert calculus, and intersection theory on flag manifolds.

Exceptional Collections and Derived Equivalences

Examines exceptional collections of sheaves and derived equivalences between triangulated categories.

Rational Homotopy Theory and Formality

Studies rational homotopy groups via differential graded algebras and formality of algebraic varieties.

Heegner Points and Birch-Swinnerton-Dyer

Investigates Heegner points on modular curves and their role in Birch-Swinnerton-Dyer conjecture.

Convex Geometry and Polytopes

Studies geometry of convex polytopes including face lattices, Ehrhart polynomials, and combinatorial properties.

Motivic Integration and Motivic Measures

Develops motivic integration theory and applies it to counting problems in geometry and number theory.

Étale Fundamental Groups and Galois Categories

Studies étale fundamental groups and Galois categories with applications to arithmetic geometry.

Configuration Spaces and Braid Groups

Investigates topology and geometry of configuration spaces including applications to braid groups.

Orbifold Fundamental Groups and Orbifold Covers

Studies fundamental groups of orbifolds and orbifold covering spaces with topological applications.

Geometric Langlands Program and Automorphic Forms

Explores geometric Langlands correspondence relating automorphic forms to perverse sheaves on stacks.

Stability Conditions on Triangulated Categories

Studies Bridgeland stability conditions on derived categories with applications to moduli spaces.

Quantum Invariants and Topological Quantum Field Theory

Investigates quantum invariants of links and manifolds through topological quantum field theory.

Multiplicative Structure on Cohomology Rings

Studies cup products and ring structures on cohomology including Poincaré duality algebras.

Homomorphisms Between Symmetric Groups

Analyzes homomorphisms and structural properties of symmetric groups and related permutation groups.

Asymptotic Invariants of Singularities

Studies asymptotic invariants of singularities including log canonical threshold and multiplier ideals.

Topological Data Analysis and Persistent Homology

Develops persistent homology and topological data analysis methods for studying point cloud geometry.

Universal Properties and Adjoint Functors

Studies universal properties, adjoint functors, and Kan extensions in categorical framework.

Morse Theory and Critical Point Analysis

Develops Morse theory for computing homology and studying critical point sets of smooth functions.

Langlands Duality and Quantum Groups

Examines connections between Langlands duality and quantum group representations.

Arithmetic of Dynamical Systems

Studies arithmetic properties of iterative dynamical systems and Fatou-Julia theory over number fields.

Spin Geometry and Clifford Algebras

Investigates spin manifolds, spinor bundles, and applications of Clifford algebras to geometry.

Quantum Cohomology and Gromov-Witten Theory

Studies quantum deformations of cohomology via Gromov-Witten invariants and curve-counting.

Nonstandard Analysis and Infinitesimals

Develops nonstandard analysis and infinitesimal methods for rigorous calculus and model theory.

Affine Lie Algebras and Vertex Algebras

Studies affine Kac-Moody algebras, vertex operator algebras, and their representation categories.

Derived Functors and Extension Groups

Investigates derived functors including Ext and Tor groups with applications to module theory.

Holomorphic Dynamics and Julia Sets

Studies dynamics of holomorphic maps including Julia sets, Fatou components, and Mandelbrot set.

Slice Knots and Concordance

Examines slice knots and knot concordance invariants including signature and Heegner invariants.

Chromatic Homotopy Theory and Periodicity

Investigates periodic phenomena in stable homotopy groups of spheres using chromatic filtration and complex-oriented cohomology theories.

Derived Functors and Spectral Sequences

Examines advanced computational techniques for homological algebra including spectral sequences, derived functors, and their convergence properties in categorical settings.

Cyclic Cohomology and Noncommutative Analysis

Develops cyclic cohomology theories for noncommutative algebras with applications to index theory and periodic cyclic homology.

Rational Homotopy Theory and Formality

Studies the rational homotopy category using differential graded algebras and minimal models to classify spaces up to rational equivalence.

Persistent Homology and Topological Data Analysis

Develops algebraic topological methods for analyzing multiscale features in point clouds and complex datasets through filtered simplicial complexes.

Mirror Symmetry and Homological Algebra

Explores the deep categorical equivalences between Fukaya categories and bounded derived categories arising from mirror symmetry phenomena.

Moment Maps and Geometric Invariant Theory

Analyzes the geometry of moment maps in symplectic and algebraic geometry with applications to stability conditions and quotient constructions.

Orbifold Cohomology and Chen-Ruan Theory

Develops orbifold cohomology theories that capture the homological structure of singular spaces with group actions.

Affine Hecke Algebras and Quantum Groups

Studies representations of affine Hecke algebras and their connections to quantum groups and double affine Hecke algebras.

Étale Cohomology and l-adic Sheaves

Investigates étale cohomology theories and their applications to number theory including Galois representations and Weil conjectures.

Intersection Cohomology and Perverse Sheaves

Studies intersection cohomology and perverse sheaf theory providing tools for analyzing singularities on algebraic varieties.

Quantum Invariants and Three-Manifolds

Develops quantum invariants of three-manifolds including Witten-Reshetikhin-Turaev invariants and their categorifications.

Heegaard Floer Homology and Contact Topology

Constructs and applies Heegaard Floer homology theories to study three and four-dimensional contact and symplectic manifolds.

Instanton Homology and Knot Invariants

Develops instanton Floer homology theories for studying knots and links in three-dimensional spaces.

Derived Noncommutative Algebraic Geometry

Extends derived algebraic geometry to noncommutative settings using dg-categories and derived stacks.

Rational Points and Brauer-Manin Obstructions

Studies existence of rational points on algebraic varieties using Brauer group obstructions and adelic approaches.

Arithmetic Intersections and Arakelov Theory

Develops intersection theory on arithmetic varieties including height pairings and arithmetic Chow groups.

Categorical Quantum Mechanics and Logic

Applies category theory to formalize quantum mechanics and quantum logic using monoidal and compact closed categories.

Fibered Categories and Stack Descent

Examines descent theory for fibered categories and its applications to constructing and understanding algebraic stacks.

Stable Equivariant Homotopy Theory

Studies equivariant stable homotopy categories with group actions and their computational methods using equivariant spectra.

Motivic Spectral Sequences and Norm Maps

Develops spectral sequences in motivic stable homotopy theory including norm maps and computational techniques for stable motivic groups.

Cohen-Macaulay Rings and Canonical Singularities

Analyzes Cohen-Macaulay properties and canonical singularities in commutative algebra and algebraic geometry.

Linkage and Liaison Theory in Algebra

Studies linkage and liaison theory for ideals and varieties providing methods for understanding determinantal varieties and complete intersections.

Tight Closure and Characteristic p Methods

Develops tight closure theory and reduction to characteristic p techniques for studying singularities and integral closure problems.

Homological Conjectures in Commutative Algebra

Investigates open conjectures in homological algebra including the Cohen-Macaulay conjecture and Serre''s depth formula.

Brane Transport and Categorical Invariants

Studies categorical frameworks for understanding brane transport and developing invariants in symplectic geometry.

Microlocal Analysis and Constructible Sheaves

Applies microlocal analysis techniques to study constructible and perverse sheaves on manifolds and singular spaces.

Bordism and Cobordism Theories

Studies bordism rings and generalized cobordism theories including complex bordism and bordism with singularities.

Loop Spaces and String Topology

Investigates algebraic structures on loop spaces and string topology operations including Chas-Sullivan products.

Donaldson Theory and Moduli of Connections

Develops Donaldson invariants of four-manifolds using gauge theory and moduli spaces of anti-self-dual connections.

Seiberg-Witten Theory and Monopole Equations

Studies Seiberg-Witten invariants and monopole equations for four-dimensional manifolds with applications to topology.

Monopole Floer Homology and Instanton Theory

Develops and computes monopole Floer homology theories and relates them to instanton invariants of three-manifolds.

Calabi-Yau Manifolds and Derived Categories

Studies derived categories of coherent sheaves on Calabi-Yau manifolds and stability conditions.

Fano Varieties and Stability Theory

Analyzes geometric and arithmetic properties of Fano varieties including K-stability and applications to moduli problems.

Moduli of Curves and Gromov-Witten Theory

Studies moduli spaces of curves and their birational geometry with connections to Gromov-Witten invariants.

Wall-Crossing and Stability Conditions

Investigates wall-crossing phenomena for stability conditions on triangulated categories with applications to enumerative geometry.

Quiver Representations and Cluster Algebras

Studies representations of quivers and their connections to cluster algebras and upper cluster algebras.

Noncommutative Projective Geometry

Develops noncommutative analogues of projective geometry using graded rings and noncommutative algebraic varieties.

D-modules and Microlocalization

Studies D-module theory and microlocalization techniques with applications to representation theory and algebraic geometry.

Tensor Categories and Fusion Categories

Investigates tensor categories and fusion categories with applications to conformal field theory and quantum invariants.

Vertex Algebras and Conformal Structures

Develops vertex algebra theory and its connections to conformal field theory and representation theory.

Kac-Moody Algebras and Invariant Theory

Studies representations of Kac-Moody algebras and their invariant-theoretic properties.

Schubert Varieties and Parabolic Subgroups

Analyzes Schubert varieties in flag manifolds and their combinatorial and geometric properties.

Torus Actions and GKM Theory

Studies torus actions on manifolds and develops GKM cohomology theories for spaces with fixed points.

Equivariant Localization and Fixed Point Formulas

Develops equivariant localization techniques and fixed point formulas including Atiyah-Bott localization.

Hypergeometric Functions and Special Functions

Studies special functions arising from algebraic geometry including hypergeometric functions and their monodromy.

Lefschetz Hyperplane Theorem and Vanishing Theorems

Investigates hard Lefschetz theorem and vanishing theorems for cohomology with applications to algebraic varieties.

Abelian Varieties and Theta Functions

Studies abelian varieties, polarizations, and theta functions with applications to algebraic geometry and number theory.

Jacobian Varieties and Curve Geometry

Analyzes Jacobian varieties of curves and their geometric properties including special divisors and Torelli maps.

Chromatic Homotopy Theory and Morava Stabilizer Groups

Studies the chromatic filtration of the stable homotopy groups of spheres using Morava E-theory and formal group laws.

p-adic Hodge Theory and Galois Representations

Develops the theory of p-adic representations and Hodge-Tate weights for understanding Galois cohomology.

Equivariant Stable Homotopy and RO(G)-Grading

Explores equivariant stable homotopy theory with RO(G)-graded cohomology and its computational techniques.

Fano Varieties and Extremal Contractions

Studies the classification and geometry of Fano varieties through extremal ray methods and the Mori program.

Calabi-Yau Manifolds and Mirror Symmetry

Investigates mirror symmetry between Calabi-Yau manifolds and its implications for enumerative geometry and derived categories.

Elliptic Cohomology and Topological Modular Forms

Develops elliptic cohomology theories and the spectrum of topological modular forms with applications to stable homotopy.

Cluster Algebras and Poisson Geometry

Studies cluster algebras arising from quivers and surfaces with connections to Poisson brackets and categorification.

Contact Topology and Legendrian Knots

Examines contact structures on manifolds and the study of Legendrian and transverse knots within contact geometry.

Motivic Integration and Nonsingular Varieties

Applies motivic measure theory to study arc spaces and singularities of varieties via motivic integration.

Spectral Sequences and Computational Homological Algebra

Develops computational techniques using spectral sequences for calculating derived functors and cohomology operations.

Affine Hecke Algebras and Kazhdan-Lusztig Theory

Studies affine Hecke algebras and Kazhdan-Lusztig polynomials with applications to representation theory and geometry.

Braid Groups and Configuration Spaces

Analyzes braid groups through configuration spaces of particles and their topological and algebraic structures.

Nonabelian Hodge Theory and Character Varieties

Studies character varieties of fundamental groups and their connections to Higgs bundles and nonabelian Hodge structures.

Operads and Homotopy Algebras

Develops the theory of operads and their applications to homotopy associative and commutative algebras.

Knot Invariants and Quantum Algebra

Constructs knot invariants from quantum groups and explores categorical interpretations of polynomial invariants.

Hyperkähler Geometry and Twistor Theory

Investigates hyperkähler manifolds using twistor construction and quaternionic-Kähler reduction techniques.

Crystalline Cohomology and p-adic Differential Equations

Studies crystalline cohomology as a Weil cohomology theory and solutions to p-adic differential equations.

Donaldson-Thomas Theory and Wall-Crossing Formulas

Develops Donaldson-Thomas invariants of 3-Calabi-Yau categories with applications to wall-crossing phenomena.

Arithmetic Statistics and L-function Families

Studies statistical properties of arithmetic objects through families of L-functions and Cohen-Lenstra heuristics.

Tannaka Duality and Reconstruction Theorems

Applies Tannaka duality to recover algebraic structures from their representation categories.

Factorization Algebras and Quantum Field Theory

Studies factorization algebras as algebraic structures encoding local observables in quantum field theory.

Lattice Polytopes and Ehrhart Polynomials

Analyzes lattice points in polytopes and their enumeration via Ehrhart polynomials and Stanley-Reisner rings.

Symplectic Reduction and Moment Maps

Studies geometric quotients of symplectic manifolds via moment map techniques and symplectic reduction.

Stability Conditions and Bridgeland Stability

Develops stability conditions on triangulated categories with applications to Donaldson-Thomas invariants.

Foliations and Characteristic Classes

Studies geometric properties of foliations through characteristic classes and secondary characteristic classes.

Moduli of Curves and Teichmüller Theory

Investigates the moduli space of Riemann surfaces and its connections to Teichmüller theory and mapping class groups.

Rational Homotopy Theory and Formal Models

Applies rational homotopy theory to study spaces through commutative differential graded algebras and formal models.

Quantum Cohomology and Gromov-Witten Invariants

Develops quantum cohomology rings and Gromov-Witten invariants for counting pseudo-holomorphic curves.

Derived Algebraic Geometry over Simplicial Rings

Extends algebraic geometry to derived schemes and derived algebraic geometry using simplicial commutative rings.

Heegaard Splittings and 3-Manifold Topology

Studies 3-manifolds through Heegaard splittings and the theory of mapping class groups of surfaces.

Canonical Metrics and Kähler-Einstein Equations

Investigates existence and uniqueness of canonical metrics on Kähler varieties via Kähler-Einstein equations.

Cyclotomic Fields and Iwasawa Main Conjecture

Studies cyclotomic fields and proves cases of the Iwasawa main conjecture relating p-adic L-functions to class numbers.

Affine Flag Varieties and Zastava Spaces

Analyzes affine flag varieties and Zastava spaces with applications to representation theory and geometric Langlands.

Tensor Categories and Braided Monoidal Structures

Develops tensor categories and braided monoidal categories with connections to topological quantum field theory.

Syzygies and Koszul Complexes

Studies syzygy modules and Koszul homological invariants measuring non-linearity in polynomial rings.

Formal Schemes and Formal Algebraic Geometry

Develops the theory of formal schemes and formal algebraic geometry for completing along ideal sequences.

Instanton Homology and Gauge Theory Invariants

Constructs instanton Floer homology from self-dual connections with applications to 3-manifold topology.

Lattice Models and Integrable Systems

Studies mathematical structures arising from lattice models and integrable systems in statistical mechanics.

Abelian Varieties and Principally Polarized Moduli

Investigates abelian varieties and the moduli of principally polarized abelian varieties with applications to arithmetic.

Moment-Angle Complexes and Polyhedral Products

Analyzes moment-angle complexes and polyhedral products arising from simplicial complexes and their cohomology.

Perfectoid Spaces and Integral p-adic Geometry

Study of perfectoid spaces and their applications to integral p-adic Hodge theory, tilting theory, and the solution of arithmetic problems using characteristic p techniques.

Vertex Algebras and Conformal Field Theory

Studies vertex operator algebras and their representation categories with connections to conformal field theory.

Chromatic Homotopy Theory and Chromatic Redshift

Investigation of chromatic filtration in stable homotopy theory, transchromatic phenomena, and the arithmetic geometry underlying chromatic tower computations.

Monsky-Washnitzer Cohomology and Rigid Geometry

Develops Monsky-Washnitzer cohomology and rigid analytic geometry as tools for p-adic analysis.

Motivic Homotopy Theory and A¹-Invariants

Development of homotopy theory over schemes using A¹-homotopy equivalence and motivic spaces to bridge algebraic geometry with topological invariants.

Cluster Varieties and Poisson Structures

Studies cluster varieties and their Poisson structures arising from cluster algebras and cluster coordinates.

Higher Algebra and Spectral Algebraic Geometry

Exploration of E-infinity ring spectra, derived algebraic geometry over the sphere spectrum, and applications to arithmetic and moduli problems.

Motivic Measures and Arc Spaces

Applies motivic measures on arc spaces to understand singularities and rational singularities of varieties.

Condensed Mathematics and Pyknotic Objects

Study of condensed abelian groups and solid abelian groups as foundations for a unified approach to p-adic and real geometry through solid-state mathematics.

Higher Direct Images and Kodaira Vanishing

Studies higher direct images of coherent sheaves and proves vanishing theorems like Kodaira and Nakano.

Arithmetic Quantum Chaos and L-functions

Analysis of spectral properties of automorphic forms, equidistribution of eigenfunctions, and connections between quantum chaos and analytic properties of L-functions.

Anabelian Geometry and Grothendieck's Conjecture

Investigation of how absolute Galois groups and fundamental groups determine geometric objects, addressing reconstruction problems and Grothendieck''s section conjecture.