Performance analysis in Geometry and Topology focuses on evaluating the effectiveness of methods and algorithms used to study geometric structures, topological spaces, and their applications in fields such as computer science, mathematics, physics, and engineering. Both branches deal with the properties and relationships of objects, but while geometry focuses on spatial properties and measurements, topology studies the qualitative properties that remain invariant under continuous deformations.

1. Geometric Modeling and Algorithms

Geometric Algorithms

• Computational Geometry: Performance analysis of algorithms used for tasks such as convex hulls, Voronoi diagrams, Delaunay triangulation, and polygon intersection. Key performance indicators (KPIs) include algorithmic efficiency (e.g., time complexity), accuracy of geometric representation, and scalability for large datasets.
• Surface Reconstruction: Evaluating algorithms used in 3D surface reconstruction (e.g., Marching Cubes, Poisson Surface Reconstruction). Performance metrics include mesh accuracy, reconstruction speed, and handling of noisy data.
• Point Cloud Processing: Performance analysis of point cloud algorithms for 3D model generation and registration. Key metrics include registration accuracy, convergence rate, and robustness to noise.

Geometric Data Structures

• Bounding Volume Hierarchies (BVH): Performance analysis of BVH used in ray tracing and collision detection. Key metrics include tree construction time, query performance, and memory consumption.
• Quadtrees and Octrees: Performance of spatial partitioning structures such as quadtrees and octrees used for efficient spatial indexing and querying. Metrics include search time, update time, and space complexity.

2. Topological Data Analysis (TDA)

Persistent Homology

• Topological Features of Data: Performance analysis of persistent homology algorithms used to capture the topological features (e.g., connected components, loops, and voids) of data in various dimensions. Metrics include complexity of computation, simplicial complex size, and topological feature persistence.
• Multi-dimensional Data Analysis: Evaluating the application of persistent homology in multi-dimensional datasets such as those arising in biology, neuroscience, or sensor networks. Performance metrics include feature extraction accuracy, scalability, and computational efficiency.

Mapper Algorithms

• Mapper Algorithm for Topological Simplification: Performance of mapper algorithms used to analyze high-dimensional data and create simplified topological models. Key performance indicators include accuracy of topological simplification, runtime complexity, and scalability to large datasets.
• Topological Signature: Evaluating how mapper algorithms represent the topological features of datasets through graph-based representations. Metrics include persistence of structures, visualization quality, and detailed analysis.

3. Differential Geometry and Geometric Analysis

Surface Curvature Analysis

• Curvature Estimation: Performance analysis of algorithms used to compute Gaussian curvature, mean curvature, and principal curvatures of surfaces in 3D geometry. Metrics include accuracy, processing speed, and robustness to noise.
• Geometric Flow Methods: Evaluating the performance of geometric flow methods such as mean curvature flow or Ricci flow used to smooth or deform surfaces. Performance metrics include convergence rate, stability, and computational complexity.

Geometric Variational Problems

• Optimization in Geometry: Performance analysis of optimization algorithms (e.g., gradient descent, simulated annealing) applied to problems in shape matching, surface fitting, and registration. Key metrics include optimization accuracy, convergence speed, and energy functional minimization.

4. Topological Spaces and Continuity

Topological Transformations

• Homeomorphism Detection: Performance of algorithms designed to detect homeomorphisms between topological spaces, crucial in applications like data analysis and image processing. Performance metrics include detection accuracy, computation time, and scalability.
• Topology Optimization: Performance of topology optimization algorithms in fields like structural design and materials science, where the goal is to find optimal topological configurations that satisfy certain constraints. Metrics include convergence rate, solution accuracy, and computational cost.

Continuous Functions and Mapping

• Topological Properties of Continuous Maps: Assessing the effectiveness of methods for analyzing continuity and differentiability of continuous maps between topological spaces. Key performance indicators include computational accuracy, verification efficiency, and application to practical problems.

5. Geometric and Topological Modeling in Applications

Computer Graphics and Animation

• Geometric Representation in CG: Performance of algorithms used to represent geometrical shapes (e.g., Bezier surfaces, NURBS, B-splines) in computer graphics. Performance metrics include rendering speed, geometrical accuracy, and visual realism.
• Deformable Models in Animation: Performance analysis of deformable models (e.g., elastic and plastic deformations) used in character animation and simulation. Metrics include realism of deformation, processing time, and stability of the simulation.

Robotics and Path Planning

• Geometric Path Planning: Evaluating algorithms used for robot motion planning in geometric environments (e.g., A*, RRT). Key performance indicators include pathfinding efficiency, collision avoidance, and computational complexity.
• Geometric Constraints in Manipulation: Performance of constraint-based optimization techniques used to model and solve robotic manipulation problems under geometric constraints. Metrics include task success rate, computational time, and constraint satisfaction.

6. Geometric and Topological Visualization

Shape and Surface Visualization

• Mesh Visualization: Performance of algorithms used to visualize 3D meshes in computational geometry and finite element analysis. Key performance indicators include visualization speed, accuracy of mesh representation, and interactive rendering capabilities.
• Topological Visualization: Performance analysis of methods used to visualize topological spaces, such as persistent diagrams and barcodes in topological data analysis (TDA). Metrics include visual clarity, persistence accuracy, and computational efficiency.

High-Dimensional Data Visualization

• Dimension Reduction: Performance of dimensionality reduction techniques (e.g., t-SNE, UMAP) used to map high-dimensional data into lower-dimensional spaces for visualization. Key performance metrics include data representation accuracy, visualization interpretability, and scalability to large datasets.

S.no	Title	Subject Area	Print ISSN
1.	Annales de l’Institut Fourier	Geometry and Topology	3730956
2.	Journal of Algebraic Geometry	Geometry and Topology	10563911
3.	Journal of Differential Geometry	Geometry and Topology	0022040X
4.	Journal of Geometric Analysis	Geometry and Topology	10506926
5.	Journal of Geometry and Physics	Geometry and Topology	3930440
6.	Journal of Graph Theory	Geometry and Topology	3649024
7.	Linear Algebra and Its Applications	Geometry and Topology	243795
8.	Advances in Geometry	Geometry and Topology	1615715X
9.	Annals of Global Analysis and Geometry	Geometry and Topology	0232704X
10.	Balkan Journal of Geometry and its Applications	Geometry and Topology	12242780
11.	Beitrage zur Algebra und Geometrie	Geometry and Topology	1384821
12.	Graphical Models	Geometry and Topology	15240703
13.	Order	Geometry and Topology	1678094
14.	Communications in Analysis and Geometry	Geometry and Topology	10198385
15.	Differential Geometry and its Applications	Geometry and Topology	9262245
16.	Discrete and Computational Geometry	Geometry and Topology	1795376
17.	Electronic Journal of Combinatorics	Geometry and Topology	10778926
18.	Geometriae Dedicata	Geometry and Topology	465755
19.	Geometric and Functional Analysis	Geometry and Topology	1016443X
20.	Geometry and Topology	Geometry and Topology	14653060
21.	Topology and its Applications	Geometry and Topology	1668641
22.	Transformation Groups	Geometry and Topology	10834362
23.	Journal of Graph Algorithms and Applications	Geometry and Topology	15261719
24.	Computational Geometry: Theory and Applications	Geometry and Topology	9257721
25.	Fractals	Geometry and Topology	0218348X
26.	International Journal of Computational Geometry and Applications	Geometry and Topology	2181959
27.	Journal of Mathematical Imaging and Vision	Geometry and Topology	9249907
28.	Soft Computing	Geometry and Topology	14327643
29.	Mathematical Physics Analysis and Geometry	Geometry and Topology	13850172
30.	Conformal Geometry and Dynamics	Geometry and Topology	10884173
31.	Journal of Geometry	Geometry and Topology	472468
32.	Fixed Point Theory and Applications	Geometry and Topology	16871820
33.	Journal of Fixed Point Theory and Applications	Geometry and Topology	16617738
34.	Applied General Topology	Geometry and Topology	15769402
35.	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)	Geometry and Topology	18150659
36.	Rendiconti del Seminario Matematico dell ‘Universita’ di Padova/Mathematical Journal of the University of Padova	Geometry and Topology	418994
37.	Journal for Geometry and Graphics	Geometry and Topology	14338157
38.	Groups, Geometry, and Dynamics	Geometry and Topology	16617207
39.	Journal of Noncommutative Geometry	Geometry and Topology	16616952
40.	Set-Valued and Variational Analysis	Geometry and Topology	18770533
41.	Journal of Symplectic Geometry	Geometry and Topology	15275256
42.	Hiroshima Mathematical Journal	Geometry and Topology	182079
43.	Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales – Serie A: Matematicas	Geometry and Topology	15787303
44.	Hacettepe Journal of Mathematics and Statistics	Geometry and Topology	13035010
45.	Funkcialaj Ekvacioj	Geometry and Topology	5328721
46.	Algebraic and Geometric Topology	Geometry and Topology	14722747
47.	Mathematical Communications	Geometry and Topology	13310623
48.	Ars Mathematica Contemporanea	Geometry and Topology	18553966
49.	Fundamental and Applied Mathematics	Geometry and Topology	15605159
50.	Journal of Topology	Geometry and Topology	17538416
51.	Journal of Topology and Analysis	Geometry and Topology	17935253
52.	Journal of Singularities	Geometry and Topology	19492006
53.	Journal of Homotopy and Related Structures	Geometry and Topology	21938407
54.	Journal of Geometric Mechanics	Geometry and Topology	19414889
55.	Journal of Geometry and Symmetry in Physics	Geometry and Topology	13125192
56.	Real Analysis Exchange	Geometry and Topology	1471937
57.	Journal of Mathematical Physics, Analysis, Geometry	Geometry and Topology	18129471
58.	Mathematical Reports	Geometry and Topology	15823067
59.	Journal of Nonlinear and Convex Analysis	Geometry and Topology	13454773
60.	Quantum Topology	Geometry and Topology	1663487X
61.	Mathematics and Visualization	Geometry and Topology	16123786
62.	Analysis and Geometry in Metric Spaces	Geometry and Topology
63.	Journal of Spectral Theory	Geometry and Topology	1664039X
64.	Special Matrices	Geometry and Topology
65.	Progress in Mathematics	Geometry and Topology	7431643
66.	Journal of Nonlinear Functional Analysis	Geometry and Topology
67.	Annales de l’Institut Henri Poincare (D) Combinatorics, Physics and their Interactions	Geometry and Topology	23085827
68.	Axioms	Geometry and Topology
69.	Series on Knots and Everything	Geometry and Topology	2199769
70.	Complex Manifolds	Geometry and Topology
71.	Methods of Functional Analysis and Topology	Geometry and Topology	10293531
72.	Universitatis Iagellonicae Acta Mathematica	Geometry and Topology	8600120
73.	Rendiconti di Matematica e delle Sue Applicazioni	Geometry and Topology	11207183
74.	Algebraic Geometry	Geometry and Topology	23131691
75.	ACM Transactions on Spatial Algorithms and Systems	Geometry and Topology	23740353
76.	Discrete Analysis	Geometry and Topology
77.	Geometry, Integrability and Quantization	Geometry and Topology	13143247
78.	Forum of Mathematics, Sigma	Geometry and Topology
79.	Forum of Mathematics, Pi	Geometry and Topology
80.	Journal of Computational Geometry	Geometry and Topology	1920180X
81.	Epijournal de Geometrie Algebrique	Geometry and Topology
82.	Journal of Fractal Geometry	Geometry and Topology	23081309
83.	Innovations in Incidence Geometry	Geometry and Topology	26407337
84.	Proceedings of the International Geometry Center	Geometry and Topology	20729812
85.	Journal of Algebraic Systems	Geometry and Topology
86.	European Journal of Pure and Applied Mathematics	Geometry and Topology
87.	Advances in Dynamical Systems and Applications	Geometry and Topology	9735321
88.	SIAM Journal on Applied Algebra and Geometry	Geometry and Topology
89.	Cubo	Geometry and Topology	7167776
90.	Annals of PDE	Geometry and Topology	25245317
91.	Annales Mathematiques Blaise Pascal	Geometry and Topology	12591734
92.	Results in Nonlinear Analysis	Geometry and Topology
93.	Annals of K-Theory	Geometry and Topology	23791683
94.	Journal of Analysis	Geometry and Topology	9713611