Research Areas and Core Pillars of ISQGD

1. Quantization Theory

   Quantization Theory is a central theme within the intellectual scope of the international society in Quantization, Geometry, and Dynamics. It encompasses both classical and modern mathematical approaches to quantization arising in applied mathematics, geometry, analysis, probability, and mathematical physics.

   In applied mathematical quantization, the subject includes constrained and unconstrained quantization problems; the study of optimal n-means for continuous, discrete, singular, and self-similar distributions; and detailed investigations of quantization error, including error analysis, high-resolution asymptotics, convergence rates, and stability results. It also involves the study of quantization dimension, quantization coefficients, and their relationships with local dimension and other geometric characteristics of measures and sets.

   Quantization problems arise naturally on a wide variety of spaces, including Euclidean spaces, Riemannian manifolds, spherical surfaces, fractal sets, and other nonlinear metric spaces. Research in this area includes geometric, variational, and algorithmic methods for computing optimal quantization sets, as well as extensions of quantization theory to graphs, networks, and combinatorial metric structures.

   In geometric and analytic approaches to quantization in mathematical physics, the subject includes geometric quantization of symplectic manifolds and related structures, including the study of prequantum line bundles, polarizations, and the correspondence between classical observables and quantum operators. Closely related directions include deformation quantization through star products, Poisson brackets, and formal deformations of commutative algebras of observables.

   Further developments include stochastic quantization based on Langevin dynamics and path-integral formulations of quantum field theories. Additional related frameworks include phase-space quantization, Berezin–Toeplitz quantization methods, and operator-algebraic perspectives that link classical phase-space geometry with noncommutative quantum structures.

2. Fractal Geometry

   Self-similar, self-affine, and random fractal sets; Moran constructions, generalized Moran structures, and iterated function systems (IFS) with or without overlap; fractal functions, fractal interpolation schemes, and IFS-generated curves and surfaces; classical and generalized dimensions (Hausdorff, packing, box-counting, Assouad, and lower dimensions); multifractal analysis and spectrum theory for measures and dynamical quantities; fractal boundaries, attractors, invariant sets, Julia sets, and limit sets; fractal geometry on manifolds, spheres, and curved surfaces; and applications to dynamical systems, geometric structures, optimization, and measure-theoretic modeling.

3. Topology

   Topological spaces and structures arising in quantization, geometry, and dynamical systems; general topology of metric and topological spaces, including compactness, connectedness, continuity, convergence, and completeness; low-dimensional and geometric topology, including knot theory, link theory, and embeddings of curves and graphs in manifolds, with connections to dynamical systems, geometry, and mathematical physics; topological manifolds and spaces appearing in geometric, fractal, and dynamical settings; topological and symbolic dynamics, shift spaces, subshifts of finite type, and coding of dynamical systems; invariant sets, attractors, limit sets, orbit closures, recurrence, and minimality; topological entropy, conjugacy, semi-conjugacy, and structural stability; Cantor-type sets and totally disconnected spaces; topological dimension theory and its relationships with Hausdorff, packing, box-counting, and other fractal dimensions; topology of self-similar, self-affine, and dynamically defined sets; graph, network, and combinatorial topology; topological constraints and stability phenomena in constrained and conditional quantization; and connections with shape theory, inverse limits, persistent homology, and topological data analysis, with applications to variational problems, optimal transport, and quantization on nonlinear and singular spaces.

4. Dynamical Systems

   Uniformly, partially, and non-uniformly hyperbolic systems; symbolic dynamics, Markov partitions, and subshifts; thermodynamic formalism including pressure, equilibrium states, Gibbs measures, and large deviation principles; invariant sets, attractors, limit sets, Lyapunov exponents, entropy, and stability theory; chaotic dynamics, mixing, robustness, and quantum and semiclassical aspects of dynamical systems (quantum chaos); and random, parametric, and skew-product systems.

   The area also includes infinite-dimensional dynamical systems generated by partial differential equations, including fluid-dynamical flows, with focus on invariant structures, attractors, mixing, and ergodic properties.

   The scope further encompasses conformal and complex dynamics, including rational, meromorphic, and entire maps; Julia and Fatou sets; rational and transcendental semigroups; conformal graph-directed systems; dimension theory for conformal and holomorphic iterated function systems; and the thermodynamic and fractal properties of complex and random holomorphic dynamical systems.

5. Ergodic Theory

   Ergodic measures, invariant transformations, and ergodic decompositions; unique ergodicity, rigidity phenomena, and classification of invariant measures; mixing, weak mixing, strong mixing, and decay of correlations; metric and topological entropy, recurrence theory, and orbit distributions; applications to quantization, information theory, and geometric measure structures; and connections with number theory, Diophantine approximation, and dynamical systems on homogeneous and arithmetic spaces.

6. Operator Theory

   Linear and nonlinear operators on Banach and Hilbert spaces; bounded, compact, and self-adjoint operators; spectral theory and spectral geometry, including eigenvalue problems, spectral asymptotics, spectral measures, resonances, and inverse spectral questions; functional calculus, semigroups of operators, and generator theory; C*-algebras, von Neumann algebras, and noncommutative geometric methods in dynamics and geometry; transfer operators, Perron–Frobenius and Koopman operators arising in ergodic theory and dynamical systems; operator-theoretic methods in quantization, including spectral decompositions, compactness and stability properties, and operator-valued metrics; Laplacians, Schrödinger-type operators, and differential operators on manifolds, graphs, quantum graphs, and fractal sets; and applications to geometric analysis, harmonic analysis, complex dynamics, and mathematical physics.

7. Fixed Point Theory

   Classical and modern fixed point theorems, including Banach, Schauder, Brouwer, Kakutani, Browder, and Krasnosel’skii-type results; fixed points of linear and nonlinear operators, multivalued maps, and nonexpansive mappings on Banach, Hilbert, and general metric spaces; contraction principles and generalized fixed point frameworks in metric and topological settings; invariant points and equilibrium states in dynamical systems, iterated function systems, and random and parametric processes; fixed points of transfer operators, Perron–Frobenius operators, and Markov operators arising in ergodic theory and thermodynamic formalism; applications to fractal geometry, self-similarity, and attractor theory; fixed point methods in variational problems, evolution equations, and partial differential equations; and connections with quantization, optimal transport, geometric optimization, and stability theory.

8. Harmonic Analysis

   Harmonic analysis studies the representation of functions, operators, and signals through oscillatory components, Fourier-analytic methods, and spectral decompositions. It plays a central role in modern analysis and interacts deeply with geometry, operator theory, probability, dynamics, partial differential equations, and data science.

   Harmonic Analysis on Fractals and Metric Spaces

   Analysis on non-smooth and irregular structures such as fractals, graphs, networks, and general metric-measure spaces; Fourier and wavelet analysis on self-similar and self-affine sets; frames, sampling theory, and multiresolution and multiscale constructions; spectral properties of Laplacians and related operators; heat kernel methods and diffusion processes; and applications to signal analysis, image processing, and data representation on complex geometric structures.

   Harmonic Analysis and Convex Geometry

   Fourier-analytic methods in convex geometry, including geometric and functional inequalities, Brunn–Minkowski theory, restriction phenomena with geometric content, and connections with geometric measure theory, probability, and high-dimensional convexity.

   Harmonic Analysis and Operator Theory

   Operator-theoretic methods in harmonic analysis, including singular integrals, pseudodifferential operators, Calderón–Zygmund theory, multiplier operators, functional calculi, and spectral techniques, with applications to partial differential equations, dynamics, quantization, and mathematical physics.

   Harmonic Analysis on Manifolds and Lie Groups

   Harmonic analysis on manifolds, Lie groups, and homogeneous spaces, including representation theory, non-Euclidean Fourier analysis, invariant differential operators, heat kernels, and eigenfunction expansions, with strong links to geometry, topology, and mathematical physics.

   Spectral Theory, Frames, and Sampling

   Spectral analysis of operators and spaces; eigenvalue problems and spectral decompositions; frame theory, sampling and reconstruction, wavelets, multiresolution analysis, and stability questions in Euclidean and non-Euclidean settings.

   Applications to Dynamics, PDEs, and Data Science

   Applications of harmonic-analytic techniques to dynamical systems, ergodic theory, and partial differential equations, as well as modern applications to signal processing, image analysis, learning theory, and data representation on complex geometric and multiscale structures.

9. Functional Analysis (Foundations and Infinite-Dimensional Analysis)

   Banach and Hilbert space theory; locally convex spaces; weak, strong, and weak-* topologies; compactness, reflexivity, and duality methods; infinite-dimensional analysis; variational and convex-analytic frameworks; functional-analytic foundations underlying quantization, ergodic theory, and thermodynamic formalism; operator semigroups and evolution equations from a functional-analytic perspective; and interactions with infinite-dimensional dynamical systems, diffusion processes, and analysis on metric-measure and fractal spaces.

10. Geometric Analysis

    Geometric analysis on smooth manifolds, including Riemannian and metric geometry; curvature-dependent geometric PDEs and variational problems; classical and sharp geometric inequalities with applications to quantization and optimization; Laplace–Beltrami operators, semiclassical and microlocal analysis, eigenvalue problems, and geometric spectral theory; structure and analysis of measure and metric-measure spaces; and spherical geometry involving small circles, spherical polygons, geodesic configurations, and geometric optimization on curved surfaces.

11. Geometric Measure Theory

    Rectifiable and unrectifiable sets, geometric densities, and Hausdorff measures; projection theorems, slicing techniques, and dimension theory; measure-theoretic and geometric foundations of fractals, invariant sets, and dynamical systems; and applications to quantization, geometric approximation, and optimization on irregular, singular, or fractal geometric spaces.

12. Optimal Transport

    Wasserstein distances (Wp), optimal transport maps, and Kantorovich formulations; transport-based approaches to quantization, clustering, and geometric approximation; geometric, analytic, and dynamical applications of optimal transport; gradient flows and evolution equations in Wasserstein spaces; transport and functional inequalities, including Talagrand, HWI, and logarithmic Sobolev inequalities; curvature-dimension conditions and displacement convexity; and metric-measure structures and geometric analysis arising from optimal transport theory.

13. Variational Methods

    Variational principles underlying quantization, geometric partitioning, and energy-based optimization; minimization of geometric and analytic energies on Euclidean spaces, manifolds, and fractal domains; classical and modern tools from the calculus of variations, including Euler–Lagrange equations, variational inequalities, and geometric functionals; nonlocal variational models, peridynamics, and nonlocal continuum mechanics; discrete-to-continuum limits, Γ-convergence, and asymptotic analysis for local and nonlocal energies; and deep connections with elliptic and geometric partial differential equations, optimal transport, and geometric optimization on curved, singular, and irregular spaces.

14. Geometric Optimization

    Centroidal Voronoi tessellations (CVTs) and optimal tessellations; energy-minimizing and equilibrium configurations on Euclidean domains, manifolds, and curved surfaces; sphere packing, covering, and best-packing problems; geometric coding theory, spherical designs, and optimal spherical codes; optimal partitions and quantization on geometric and metric-measure spaces; and applications to geometric approximation and optimization on curved and non-Euclidean surfaces.

15. Probability Theory and Stochastic Processes

    Stochastic processes and random walks on manifolds, graphs, and fractal spaces; Markov chains, Markov processes, and random dynamical systems; Brownian motion and diffusions on irregular geometries; stochastic differential equations and stochastic flows; probabilistic potential theory and martingale methods in dimension theory; and random constructions in fractals, invariant measures, and quantization.

16. Computational Mathematics and Numerical Methods

    Numerical algorithms for optimal quantization, clustering, and geometric approximation; simulation and computation of dynamical systems, fractal structures, and geometric flows; numerical schemes for partial differential equations and variational problems on manifolds, metric spaces, and irregular domains; numerical analysis and simulation of nonlocal and peridynamic models, including multiscale, adaptive, and nonlocal discretization techniques; high-performance and large-scale computing for geometric, dynamical, and stochastic models; and discrete and computational approaches to spectral problems, energy minimization, inverse problems, and optimal transport.

17. Machine Learning, AI Geometry, and Data Science (Mathematical Foundations)

    Vector quantizers, prototype-based models, and clustering algorithms; geometric and manifold-based learning; Wasserstein and transport-based methods in data science; fractal and multiscale features in learning and signal analysis; mathematical foundations of geometric deep learning and representation learning; and rigorous links between quantization, statistical learning theory, and modern data-driven methods.

18. Mathematical Physics and Discrete Energy Models

    Minimal-energy particle systems and Coulomb/Riesz-type interactions; discrete energies and equilibrium configurations on manifolds, spheres, graphs, and fractal sets; quantum and semiclassical models with geometric, spectral, or fractal structure; Schrödinger operators, quantum graphs, and spectral geometry; Gibbs states, free energy minimization, phase transitions, and quantum ergodicity; and statistical-mechanical frameworks involving potentials, equilibrium distributions, and thermodynamic limits with geometric or dynamical features.

19. Information Geometry and Statistical Mechanics

    Divergences and information-theoretic distances (including α-divergences, Rényi and Tsallis entropies, and Bregman-type divergences); statistical manifolds, Fisher information metrics, and information-geometric flows; entropy-based variational principles and free energy functionals; gradient flows and evolution equations in spaces of probability measures; and connections with optimal transport, thermodynamic formalism, and equilibrium states in dynamical systems and quantization.

20. Applications and Interdisciplinary Directions

    Applications in data compression, signal processing, and information theory; machine-learning methods including vector quantizers, clustering algorithms, and prototype-based models; numerical approximation and discretization techniques on manifolds and geometric domains; geometric modeling, mesh generation, and computer graphics; mathematical physics involving minimal-energy particle systems, Coulomb/Riesz interactions, and discrete energies; and statistical-mechanical frameworks involving free energy, potentials, and equilibrium distributions.

21. Additional and Emerging Research Directions

    1. Nonlocal Models and Peridynamics.
       Nonlocal diffusion and interaction models; nonlocal variational formulations and peridynamics; discrete-to-continuum limits and convergence analysis; numerical and analytical methods for nonlocal operators; and applications to materials science, continuum mechanics, fracture modeling, and multiscale geometric structures.
    2. Dimension Theory and Multiscale Geometry.
       Dimension theory across geometry, dynamics, probability, and fractals; dimension spectra for measures and invariant sets; multiscale geometric analysis; and rigidity and stability phenomena.
    3. Metric Geometry and Analysis on Metric Spaces.
       Gromov–Hausdorff convergence; analysis on metric-measure spaces; curvature notions in nonsmooth settings; and applications to quantization and optimal transport.
    4. Random and Stochastic Geometry.
       Random geometric structures; Poisson point processes; random graphs and tessellations; stochastic geometry on manifolds and fractals; and probabilistic quantization methods.
    5. Discrete Geometry and Discrete Dynamical Systems.
       Combinatorial geometry; graph dynamics; cellular automata; and energy minimization and quantization on discrete structures.
    6. Inverse Problems and Geometric Reconstruction.
       Inverse problems on geometric and fractal domains; reconstruction from quantized or noisy data; stability and identifiability; and applications to imaging and data analysis.
    7. Partial Differential Equations on Irregular Domains.
       PDEs on fractals and singular spaces; heat, wave, and diffusion equations; spectral and variational aspects; and interactions with operator theory and harmonic analysis.
    8. Mathematical Foundations of Scientific Computing.
       Convergence and stability of numerical schemes; quantization-based discretization; and theoretical aspects of computation in geometry and dynamics.
    9. Mathematical Biology, Networks, and Complex Systems.
       Dynamical systems and networks in biological and complex systems; geometric and probabilistic modeling; quantization and clustering methods; and large-scale interacting systems.
    10. Spectral, Semiclassical, and Scaling Phenomena.
        Spectral asymptotics; semiclassical limits; renormalization and scaling methods; multifractal spectra of measures and eigenfunctions; spectral optimization and shape optimization; and interfaces between spectra, geometry, and dynamics.

The above list is not exhaustive; ISQGD warmly welcomes contributions at the interfaces of these areas, as well as new and emerging directions aligned with its mission.