Slide 1: Introduction to Algebraic Topology
Algebraic topology is a branch of mathematics that uses algebraic structures to study topological spaces. It provides powerful tools for analyzing the shape and structure of geometric objects.
import networkx as nx
import matplotlib.pyplot as plt
# Create a simple graph
G = nx.Graph()
G.add_edges_from([(1, 2), (2, 3), (3, 1)])
# Draw the graph
nx.draw(G, with_labels=True)
plt.title("A Simple Graph Representation")
plt.show()Slide 2: Simplicial Complexes
Simplicial complexes are fundamental objects in algebraic topology, representing higher-dimensional generalizations of graphs.
from scipy.spatial import Delaunay
import numpy as np
# Generate random points in 2D
points = np.random.rand(20, 2)
# Create Delaunay triangulation
tri = Delaunay(points)
# Plot the triangulation
plt.triplot(points[:, 0], points[:, 1], tri.simplices)
plt.plot(points[:, 0], points[:, 1], 'o')
plt.title("2D Simplicial Complex")
plt.show()Slide 3: Homology Groups
Homology groups are algebraic structures that capture the essence of holes in topological spaces. They provide a way to count and classify different types of holes.
import gudhi
# Create a simplicial complex
simplex_tree = gudhi.SimplexTree()
simplex_tree.insert([1, 2, 3])
simplex_tree.insert([2, 3, 4])
simplex_tree.insert([3, 4, 5])
# Compute homology
homology = simplex_tree.persistence()
print("Homology groups:")
for interval in homology:
dim, (birth, death) = interval
print(f"Dimension {dim}: birth = {birth}, death = {death}")Slide 4: Fundamental Group
The fundamental group is a topological invariant that captures information about loops in a space.
import sympy as sp
def fundamental_group_presentation(generators, relations):
G = sp.generators(' '.join(generators))
return sp.presentation_free(G, relations)
# Example: Fundamental group of a torus
torus_group = fundamental_group_presentation(['a', 'b'], ['a*b*a^-1*b^-1'])
print("Fundamental group of a torus:", torus_group)Slide 5: Euler Characteristic
The Euler characteristic is a topological invariant that relates the number of vertices, edges, and faces in a polyhedron.
def euler_characteristic(vertices, edges, faces):
return vertices - edges + faces
# Example: Cube
cube_vertices = 8
cube_edges = 12
cube_faces = 6
print("Euler characteristic of a cube:",
euler_characteristic(cube_vertices, cube_edges, cube_faces))Slide 6: Betti Numbers
Betti numbers are topological invariants that count the number of holes in each dimension of a space.
import numpy as np
from scipy.spatial import Delaunay
from persim import plot_diagrams
import ripser
# Generate a point cloud in the shape of a circle
t = np.linspace(0, 2*np.pi, 100)
circle = np.column_stack((np.cos(t), np.sin(t)))
# Compute persistent homology
diagrams = ripser.ripser(circle)['dgms']
# Plot persistence diagrams
plot_diagrams(diagrams, show=True)Slide 7: Homotopy Equivalence
Homotopy equivalence is a relation between topological spaces that preserves their essential topological properties.
import networkx as nx
import matplotlib.pyplot as plt
def contract_edge(G, u, v):
G.add_edges_from((u, w) for w in G.neighbors(v) if w != u)
G.remove_node(v)
# Create two homotopy equivalent graphs
G1 = nx.cycle_graph(4)
G2 = nx.cycle_graph(4)
contract_edge(G2, 0, 1)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))
nx.draw(G1, with_labels=True, ax=ax1)
ax1.set_title("Original Graph")
nx.draw(G2, with_labels=True, ax=ax2)
ax2.set_title("Homotopy Equivalent Graph")
plt.show()Slide 8: Cohomology
Cohomology is the dual notion to homology, providing additional algebraic tools for studying topological spaces.
import numpy as np
from scipy.linalg import null_space
def compute_cohomology(boundary_matrix):
kernel = null_space(boundary_matrix.T)
image = np.column_stack([boundary_matrix[:, i]
for i in range(boundary_matrix.shape[1])])
betti_number = kernel.shape[1] - np.linalg.matrix_rank(image)
return betti_number
# Example: Compute 1-dimensional cohomology of a triangle
boundary_matrix = np.array([
[-1, 1, 0],
[-1, 0, 1],
[0, -1, 1]
])
print("1-dimensional cohomology:", compute_cohomology(boundary_matrix))Slide 9: Persistent Homology
Persistent homology is a method for computing topological features of a space at different spatial resolutions.
import numpy as np
import matplotlib.pyplot as plt
from ripser import ripser
from persim import plot_diagrams
# Generate a noisy circle
t = np.linspace(0, 2*np.pi, 100)
circle = np.column_stack((np.cos(t), np.sin(t)))
noisy_circle = circle + 0.1 * np.random.randn(*circle.shape)
# Compute persistent homology
diagrams = ripser(noisy_circle)['dgms']
# Plot the persistence diagram
plot_diagrams(diagrams, show=True)Slide 10: Simplicial Homology Computation
Simplicial homology is a concrete way to compute homology groups for simplicial complexes.
import numpy as np
from scipy.linalg import null_space
def simplicial_homology(boundary_matrix):
kernel = null_space(boundary_matrix)
image = np.column_stack([boundary_matrix[:, i]
for i in range(boundary_matrix.shape[1])])
betti_number = kernel.shape[1] - np.linalg.matrix_rank(image)
return betti_number
# Example: Compute 1-dimensional homology of a triangle
boundary_matrix = np.array([
[-1, -1, 0],
[1, 0, -1],
[0, 1, 1]
])
print("1-dimensional homology:", simplicial_homology(boundary_matrix))Slide 11: Topological Data Analysis
Topological Data Analysis (TDA) applies techniques from algebraic topology to analyze and extract insights from complex datasets.
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from ripser import ripser
from persim import plot_diagrams
# Generate a noisy dataset
X, _ = datasets.make_circles(n_samples=300, noise=0.05, factor=0.5)
# Compute persistent homology
diagrams = ripser(X)['dgms']
# Plot the dataset and persistence diagram
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))
ax1.scatter(X[:, 0], X[:, 1])
ax1.set_title("Noisy Circles Dataset")
plot_diagrams(diagrams, show=False, ax=ax2)
ax2.set_title("Persistence Diagram")
plt.show()Slide 12: Mapping Cylinder
The mapping cylinder is a construction in algebraic topology that helps visualize continuous maps between topological spaces.
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
def mapping_cylinder(t, theta):
x = (1 - t) * np.cos(theta)
y = (1 - t) * np.sin(theta)
z = t
return x, y, z
t = np.linspace(0, 1, 100)
theta = np.linspace(0, 2*np.pi, 100)
T, Theta = np.meshgrid(t, theta)
X, Y, Z = mapping_cylinder(T, Theta)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X, Y, Z, cmap='viridis')
ax.set_title("Mapping Cylinder")
plt.show()Slide 13: Real-Life Example: Protein Structure Analysis
Algebraic topology can be applied to analyze the structure of proteins, helping to understand their function and interactions.
from Bio.PDB import PDBParser
import numpy as np
from ripser import ripser
from persim import plot_diagrams
# Load a protein structure (you need to download a PDB file)
parser = PDBParser()
structure = parser.get_structure("protein", "path/to/protein.pdb")
# Extract alpha carbon coordinates
coords = np.array([atom.coord for atom in structure.get_atoms() if atom.name == 'CA'])
# Compute persistent homology
diagrams = ripser(coords)['dgms']
# Plot persistence diagram
plot_diagrams(diagrams, show=True)Slide 14: Real-Life Example: Network Analysis
Algebraic topology techniques can be used to analyze complex networks, revealing hidden structures and patterns.
import networkx as nx
import numpy as np
from ripser import ripser
from persim import plot_diagrams
# Generate a random graph
G = nx.erdos_renyi_graph(100, 0.1)
# Compute the adjacency matrix
adj_matrix = nx.adjacency_matrix(G).todense()
# Compute persistent homology
diagrams = ripser(adj_matrix, distance_matrix=True)['dgms']
# Plot persistence diagram
plot_diagrams(diagrams, show=True)Slide 15: Additional Resources
For further exploration of Algebraic Topology:
- "Computational Topology: An Introduction" by Herbert Edelsbrunner and John Harer ArXiv: https://arxiv.org/abs/cs/0503002
- "Algebraic Topology" by Allen Hatcher Available online: https://pi.math.cornell.edu/~hatcher/AT/ATpage.html
- "Topological Data Analysis" by Gunnar Carlsson ArXiv: https://arxiv.org/abs/0907.2721
These resources provide in-depth coverage of the topics discussed in this presentation and offer advanced concepts for further study.