Mini Project Topics

Topic 1

Title:

Exhaustive Enumeration of Simplicial Mappings Between Small Complexes

Abstract:

Develop an algorithm to enumerate all possible simplicial mappings between two small, randomly generated simplicial complexes. Analyze the properties of these mappings and classify them based on their topological characteristics.

Technical description:

This project requires understanding the definition and properties of simplicial mappings. You'll need to:

• Generate pairs of small simplicial complexes
• Implement an algorithm to enumerate all possible simplicial mappings between them
• Classify these mappings based on properties such as injectivity, surjectivity, and preservation of homology
• Visualize the complexes and selected mappings

The project will involve combinatorial algorithms, group theory for the automorphism groups of simplicial complexes, and computational topology. SageMath's SimplicialComplex class will be useful, but you may need to implement custom functions for generating and analyzing the simplicial mappings.

References:

• Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. Section 2.1 on simplicial approximation.
• Dłotko, P. Applied and Computational Topology Tutorial, section on simplicial maps.
• Munkres, J. R. (1984). Elements of Algebraic Topology. Addison-Wesley. Chapter 1.

Topic 2

Title:

Constructing Simplicial Complexes with Prescribed Homology

Abstract:

Given a sequence of abelian groups, with the first group being free abelian, construct a simplicial complex of dimension at most 4 whose homology groups match the given sequence. This project explores the realization problem in algebraic topology.

Technical description:

This project involves:

• Understanding the constraints on homology groups of simplicial complexes
• Implementing algorithms to construct complexes with prescribed homology
• Verifying the homology of the constructed complexes
• Optimizing the construction to minimize the number of simplices used

The mathematics involved includes the theory of CW complexes, obstruction theory, and the realization problem in algebraic topology. You'll use SageMath's SimplicialComplex class and its homology calculations, and may need to implement additional algorithms based on theoretical results.

References:

• Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. Chapter 4.
• Munkres, J. R. (1984). Elements of Algebraic Topology. Addison-Wesley. Chapters 4 and 5.
• Carlsson, G. (2009). “Topology and Data”. Bulletin of the American Mathematical Society, 46(2), 255–308.

Topic 3

Title:

Killing Homology by Adding Simplices

Abstract:

Develop a program that modifies a given simplicial complex by adding new simplices in order to kill a selected homology group H_n(K), while preserving all lower-dimensional homology groups H_k(K) for k < n. The project explores how attaching higher-dimensional simplices changes the topology of a complex.

Technical description:

This project requires understanding how homology classes can be eliminated by making cycles into boundaries. You will need to:

• Read or generate a finite simplicial complex
• Compute its homology groups
• Identify generators of a chosen homology group H_n(K)
• Find cycle representatives of these generators
• Add suitable (n+1)-simplices, possibly using new vertices, whose boundaries kill the selected n-cycles
• Verify that H_n(K) becomes trivial
• Check that all lower-dimensional homology groups H_k(K), for k < n, remain unchanged
• Output and visualize the modified complex

The project connects computational topology with the classical construction of CW complexes with prescribed homology. SageMath's SimplicialComplex class may be useful for homology computations, but implementing the actual modification procedure will likely require custom chain-level algorithms.

References:

• Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. Chapter 2.
• Munkres, J. R. (1984). Elements of Algebraic Topology. Addison-Wesley. Chapters 1 and 4.
• Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society.

Topic 4

Title:

Killing Homology by Removing Simplices

Abstract:

Develop a program that modifies a simplicial complex by removing carefully chosen simplices in order to kill a selected homology group H_n(K). The goal is to eliminate n-dimensional cycles while preserving all homology groups H_k(K) for k < n-1, allowing H_n-1(K) to change.

Technical description:

This project studies how removing simplices changes the homology of a simplicial complex. Unlike adding simplices, this procedure can create new lower-dimensional cycles, so the removal strategy must be chosen carefully. You will need to:

• Read or generate a finite simplicial complex
• Compute its homology groups before modification
• Identify n-dimensional homology generators and representative cycles
• Select n-simplices whose removal destroys these cycles
• Ensure that the resulting object is still a valid simplicial complex
• Verify that the new complex has trivial H_n
• Check that H_k is unchanged for all k < n-1
• Analyze how H_n-1 changes after the removal
• Compare the homology groups before and after modification

This project involves chain complexes, boundary maps, cycle and boundary groups, and algorithmic manipulation of simplicial complexes. SageMath can be used for homology computations, while the removal procedure may require implementing custom routines for maintaining the simplicial-complex structure.

References:

• Hatcher, A. (2002). Algebraic Topology. Cambridge University Press. Chapters 2 and 4.
• Munkres, J. R. (1984). Elements of Algebraic Topology. Addison-Wesley. Chapters 1 and 4.
• Edelsbrunner, H., & Harer, J. (2010). Computational Topology: An Introduction. American Mathematical Society.