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#Betti Numbers

Showing: 22 courses
University of Chicago Department of Mathematics

The Weil Conjectures and A1-homotopy Theory

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Dive into the Weil Conjectures and A1-homotopy theory, exploring connections between algebraic topology and finite field equations in this mathematical colloquium.
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1 Lesons
1 hour 7 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Topological Data Analysis of Collective Motion

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Explore topological data analysis for collective motion in nature, examining bird flocks, fish schools, and insect swarms through innovative mathematical models and simulations.
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1 Lesons
56 minutes
On-Demand
Free-Video
Institute for Advanced Study

The External Activity Complex of a Pair of Matroids

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Delve into advanced matroid theory, exploring external activity complexes and their relationship to Schubert varieties, with applications to tautological classes and Speyer's matroid invariant.
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1 Lesons
1 hour 8 minutes
On-Demand
Free-Video
IMSA

Nonrational Toric Geometry III - Quasifolds, Foliations, Combinatorics and One-parameter Families

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Explores nonrational toric geometry through quasifolds, foliations, and combinatorics, extending classical toric geometry to new contexts and examining one-parameter families of toric quasifolds.
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23 Lesons
53 minutes
On-Demand
Free-Video
IMSA

Nonrational Toric Geometry II - Quasifolds, Foliations, Combinatorics and One-parameter Families

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Explore complex toric quasifolds, their structure, and relation to rational toric varieties. Learn about extending constructions to nonsimple convex polytopes in symplectic and complex settings.
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12 Lesons
55 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Statistical Invariance of Betti Numbers in the Thermodynamic Regime

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Explore statistical invariance of Betti numbers in random topology, examining conditions for topological inference through parametric statistics lens. Investigates asymptotic behavior and group invariance concepts.
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10 Lesons
29 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

A Limit Theorem for Betti Numbers of Random Simplicial Complexes

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Explore asymptotic behavior of Betti numbers in homogeneous and spatially independent random simplicial complexes, extending previous work on Linial-Meshulam complexes using local weak convergence.
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7 Lesons
24 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Configuration Spaces of Hard Squares

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Exploring topology of configuration spaces with hard squares, focusing on Betti numbers' growth rate in confined regions. New insights into complex geometric arrangements.
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1 Lesons
59 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Torsion Homology Growth and Related Topics

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Explore torsion homology growth in compact CW complexes, examining Betti number growth in finite covers and connections to amenable category and minimal volume entropy.
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1 Lesons
52 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Discrete Morse Theory Meets Multi-Parameter Persistence

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Exploring connections between discrete Morse theory and multiparameter persistence in topological data analysis, focusing on critical cells, fibered rank invariant, and computational efficiency.
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1 Lesons
52 minutes
On-Demand
Free-Video
IMSA

Fake Projective Planes: Properties and Recent Progress

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Explore fake projective planes: complex surfaces with unique properties, their construction via uniformization, and recent advancements in understanding their algebraic and geometric characteristics.
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1 Lesons
42 minutes
On-Demand
Free-Video
ICTP Mathematics

What Are the Betti Numbers of a Manifold? - Don Zagier - 2017

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Explore the concept of Betti numbers in manifolds with renowned mathematician Don Zagier, delving into their significance and applications in topology and geometry.
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1 Lesons
1 hour 19 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Strange Random Topology of the Circle

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Explore the intriguing behavior of "topological noise" in random Cech complexes constructed from the circle, revealing unexpected homotopy equivalences and higher Betti numbers in specific filtration radii intervals.
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1 Lesons
39 minutes
On-Demand
Free-Video
Insights into Mathematics

Delta Complexes, Betti Numbers, and Torsion in Algebraic Topology

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Explore homology groups of torus and projective plane using delta-complexes. Learn about Betti numbers and torsion in algebraic topology, with practical computations and insights into fundamental concepts.
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1 Lesons
48 minutes
On-Demand
Free-Video
International Centre for Theoretical Sciences

Lower Dimensional Betti Numbers of Homogeneous Spaces of Lie Groups

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Exploration of lower dimensional Betti numbers in homogeneous spaces of Lie groups, discussing mathematical techniques and applications in algebraic topology and differential geometry.
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1 Lesons
26 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

The Asymptotics of Betti Numbers in Preferential Attachment Clique Complexes

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Explore the higher-order connectivity of preferential attachment networks through Betti numbers, examining growth rates and asymptotic behavior in this advanced mathematical analysis.
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1 Lesons
46 minutes
On-Demand
Free-Video
Centre International de Rencontres Mathématiques

Random Algebraic Geometry - Lecture 1

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Explore random algebraic geometry concepts in this advanced mathematics lecture, delving into complex theoretical foundations and applications.
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1 Lesons
57 minutes
On-Demand
Free-Video
Centre International de Rencontres Mathématiques

Random Algebraic Geometry - Lecture 4

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Explore advanced concepts in random algebraic geometry, focusing on key theories and applications in this fourth lecture of the series.
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1 Lesons
57 minutes
On-Demand
Free-Video
Centre International de Rencontres Mathématiques

Random Algebraic Geometry - Lecture 2

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Explore advanced concepts in random algebraic geometry, focusing on theoretical foundations and practical applications in mathematical research.
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1 Lesons
1 hour 1 minute
On-Demand
Free-Video
Centre International de Rencontres Mathématiques

Random Algebraic Geometry - Lecture 3

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Explore advanced concepts in random algebraic geometry, focusing on cutting-edge research and theoretical developments in this mathematical field.
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1 Lesons
52 minutes
On-Demand
Free-Video
Centre de recherches mathématiques - CRM

Subgroups of Hyperbolic Groups, Finiteness Properties and Complex Hyperbolic Lattices

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Explore finiteness properties of subgroups in hyperbolic groups and complex hyperbolic lattices, focusing on cocompact arithmetic lattices with positive first Betti number.
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1 Lesons
42 minutes
On-Demand
Free-Video
Applied Algebraic Topology Network

Spectral Relaxations of the Persistent Rank Invariant - Applied Algebraic Topology Network

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Spectral relaxations of persistent rank invariant: A framework for continuous relaxations, offering stability and differentiability while preserving key properties and enabling efficient GPU-parallelized computation.
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1 Lesons
53 minutes
On-Demand
Free-Video