Problem Lists

AIM workshop problem lists

Algebraic geometry

Syzygies and mirror symmetry AimPL
Equivariant techniques in stable homotopy theory AimPL
Rationality problems in algebraic geometry AimPL
Categorified Hecke algebras, link homology, and Hilbert schemes AimPL
Rational subvarieties in positive characteristic AimPL
Positivity of cycles AimPL
Equivariant derived algebraic geometry AimPL
Algebraic vision AimPL
Degenerations in algebraic geometry PDF
Projective modules and $A^1$-homotopy theory PDF
The minimal model program in characteristic $p$ AimPL
Singular learning theory AimPL
Relating test ideals and multiplier ideals AIMPL
Numerical invariants of singularities and higher-dimensional algebraic varieties PDF
Compact moduli spaces and birational geometry PDF

Algebraic number theory

Degree d points on algebraic surfaces AimPL
Rational and integral points on higher-dimensional varieties PDF
Arithmetic statistics over finite fields and function fields PDF
The Tate conjecture PDF
Rational and integral points on higher dimensional varieties PDF

Analysis

Riemann-Hilbert problems, Toeplitz matrices, and applications AimPL
Shape optimization with surface interactions AimPL
Problems on holomorphic function spaces and complex dynamics AimPL
The complex Monge-Ampere equation PDF
Carleson theorems and multilinear operators PDF
Beyond Kadison-Singer: paving and consequences src
Mapping theory in metric spaces AimPL
Stability, hyperbolicity, and zero localization of functions AimPL
Low eigenvalues of Laplace and Schrodinger operators PDF
Random analytic functions PDF

Analytic number theory

Delta symbols and the subconvexity problem AimPL
Sarnak's conjecture AimPL
Moments of zeta and correlations of divisor sums AimPL
Gromov-Witten invariants and number theory AimPL
Mock modular forms in combinatorics and arithmetic geometry PDF
Subconvexity bounds for L-functions PDF
Gaps between primes PDF

Arithmetic geometry

Arithmetic intersection theory on Shimura varieties AimPL
Brauer groups and obstruction problems: moduli spaces and arithmetic AimPL
Cohomological methods in abelian varieties AimPL
Deformation theory, patching, quadratic forms, and the Brauer group AimPL
Components of Hilbert schemes PDF
Topology and geometry of the moduli space of curves PDF

Biology

Modeling the eye as a window on the body AimPL
Systems approaches to drug discovery and development in oncology AimPL
Inference and prediction in neocortical circuits PDF
Geometric models of biological phenomena PDF

Combinatorics

High-dimensional phenomena in discrete analysis AimPL
Additive combinatorics and its applications AimPL
Hereditary discrepancy and factorization norms AimPL
Graph Ramsey theory src
Generalizations of chip-firing and the critical group PDF
Rational Catalan combinatorics AimPL
Geometry of large networks AimPL
Hypergraph Turan problem AimPL
The Caccetta-Haggkvist conjecture PDF
Recent trends in additive combinatorics PDF
The Perfect Graph Conjecture PDF

Computation

The computational complexity of polynomial factorization PDF
Numerical methods for optimal control in high dimensions PDF
Time-reversal communications in richly scattering environments PDF
Computational algebraic statistics PDF
Numerical probabilistic methods for high-dimensional problems in finance PDF
Future directions in algorithmic number theory PDF

Convex geometry

Sections of convex bodies PDF
Mahler's conjecture and duality in convex geometry PDF
Algorithmic convex geometry PDF
Fourier analytic methods in convex geometry PDF

Cryptography

Post-quantum group-based cryptography AimPL
Quantum algorithms for analysis of public-key crypto AimPL

Dynamical systems

Groups of dynamical origin AimPL
The Galois theory of orbits in arithmetic dynamics AimPL
Stochastic methods for non-equilibrium dynamical systems PDF
Postcritically finite maps in complex and arithmetic dynamics AimPL
Low dimensional structures in dynamical systems with variable time lags PDF

Functional analysis

Symmetry and convexity in geometric inequalities AimPL
Set theory and C*-algebras AimPL

Geometric group theory

Rigidity properties of free-by-cyclic groups AimPL
Geometry and topology of Artin groups AimPL
Amenability of discrete groups AimPL
$L^2$ invariants and their relatives for finitely generated groups AimPL
The geometry of the outer automorphism group of a free group PDF
Problems in geometric group theory PDF×18
The property of rapid decay PDF

Geometry

Symmetry-breaking of optimal shapes AimPL
Cluster algebras and braid varieties AimPL
Equilibrium states for dynamical systems arising from geometry AimPL
Discrete geometry and automorphic forms AimPL
Soft packings, nested clusters, and condensed matter AimPL
Spectral data for Higgs bundles AimPL
Geometric flows and Riemannian geometry AimPL
Configuration spaces of linkages PDF
Frame theory intersects geometry PDF
Geodesics PDF
Calibrations PDF
Sphere packings, lattices, and infinite dimensional algebra PDF
Moment maps and surjectivity in various geometries PDF
Amoebas and tropical geometry PDF
Conformal structure in geometry, analysis, and physics PDF
Holomorphic curves in contact geometry PDF

Infrastructure

Open source mathematics curriculum and assessment tools PDF
Open-source cyberinfrastructure supporting mathematics research Aimpl
PreTeXt for small documents Wiki
Macaulay2: expanded functionality and improved efficiency Aimpl
Gems of combinatorics Aimpl
Research on inquiry based learning in undergraduate real analysis PDF

Linear algebra

Spectral graph and hypergraph theory: connections and applications AimPL
Theory and applications of total positivity AimPL
Spectra of families of matrices described by graphs, digraphs, and sign patterns PDF

Logic

From ℵ2 to infinity Aimpl
Definability and decidability problems in number theory AimPL
Descriptive inner model theory PDF
High and low forcing AimPL
Descriptive inner model theory PDF
Definability and decidability problems in number theory PDF
Computable stability theory PDF
Model theory of metric structures PDF
Extensions of Hilbert's tenth problem PDF
Recent advances in core model theory PDF

Optimization

Theory and algorithms of linear matrix inequalities PDF

PDEs

Small scale dynamics in incompressible fluid flows AimPL
Computer assisted proofs for stability analysis of nonlinear waves AimPL
Nonlocal differential equations in collective behavior AimPL
Nonlinear PDEs in real and complex geometry AimPL
Stability criteria for multi-dimensional waves and patterns PDF
Stiff sources and numerical methods for conservation laws PDF
New connections between dynamical systems and PDE's PDF

Physics

Non-Hermitian quantum mechanics and symplectic geometry AimPL
Mathematical aspects of physics with non-self-adjoint operators PDF
Integrable systems in Gromov-Witten and symplectic field theory AimPL
Phase transitions PDF
Gravitational lensing in the Kerr spacetime geometry PDF
$L^2$ harmonic forms in geometry and string theory PDF
Variational Methods in Celestial Mechanics PDF

Probability

Boltzmann Machines AimPL
Stein's method and applications in high-dimensional statistics AimPL
Markov chain mixing times AimPL
Vector equilibrium problems and their applications to random matrix models AimPL
The Kardar-Parisi-Zhang equation and universality class AimPL
Free analysis PDF
Braid groups, clusters and free probability PDF

Representation theory

Analytic, arithmetic, and geometric aspects of automorphic forms AimPL
Representation stability AimPL
Sheaves and modular representations of reductive groups AimPL
Automorphic kernel functions PDF
Automorphic forms and harmonic analysis on covering groups PDF
Supercharacters and combinatorial Hopf algebras PDF
$p$-adic representations, modularity, and beyond PDF

Several complex variables

The Cauchy-Riemann equations in several variables AimPL
Emerging applications of complexity for CR mappings PDF
Complexity of mappings in CR geometry PDF

Special functions

Generalized Kostka polynomials PDF

Topology

Higher-dimensional contact topology AimPL
Quantum invariants and low-dimensional topology AimPL
Algebra, geometry, and combinatorics of link homology AimPL
Discrete and combinatorial homotopy theory AimPL
Smooth concordance classes of topologically slice knots AimPL
Surfaces of infinite type AimPL
Lipschitz metric on Teichmueller space PDF
Contact topology in higher dimensions PDF
Manifolds with nonnegative sectional curvature PDF
Representations of surface groups PDF
Moduli spaces of knots PDF

Other

Dynamical algebraic combinatorics PDF
Classifying fusion categories AimPL
Emerging applications of measure rigidity PDF