Differential Topology, 2027 (tentative)

A twelve-week differential topology course from LebSci, February 6–April 24, 2027, covering smooth manifolds, regular values, transversality, and degree.

Status

Tentative proposal: Saturday online sessions in English at 18:00 (Asia/Beirut), lasting 90 minutes. Dates, times, format, and syllabus remain provisional; nothing is confirmed. Applications are not open.

Suggested background

Multivariable calculus
Linear algebra
Point-set topology (open sets, continuity, compactness, connectedness)
A first course in real analysis
Familiarity with proof-based mathematics

Topics Charts, atlases, and smooth manifolds Smooth maps and diffeomorphisms Examples: spheres, tori, and open subsets of Euclidean space

Topics Tangent vectors as velocities of curves Derivatives of smooth maps and the chain rule Tangent bundles and vector fields: definitions and examples

Topics Inverse function theorem and local immersions An embedding is an immersion that is a homeomorphism onto its image Proper injective immersions are embeddings; embeddings need not be proper

Topics Submersions and regular values Preimage theorem for manifolds without boundary: dimension and tangent space Examples: spheres and regular level sets

Topics Critical points, critical values, and measure zero in charts Sard’s theorem for smooth maps: statement and consequences Examples of regular values and critical values

Topics Examples of embeddings into Euclidean space Weak Whitney embedding theorem: every smooth n-manifold embeds in R^(2n+1), statement only Tubular neighborhoods: geometric picture for compact embedded manifolds without boundary

Topics Half-space charts and the boundary manifold Preimage theorem for maps to a manifold without boundary: regularity for both the map and its boundary restriction Compact one-manifolds are finite disjoint unions of circles and closed intervals

Topics Transversality: tangent images and submanifold tangent spaces span the target tangent space Transverse preimage theorem for manifolds without boundary Parametric transversality for smooth families transverse to a submanifold: statement and examples

Topics Counting transverse intersections modulo 2 for a compact source without boundary and a closed submanifold of complementary dimension Homotopy invariance via compact one-manifolds with boundary Mod 2 degree for maps between compact connected manifolds without boundary of equal dimension

Topics Orientations of vector spaces and manifolds Degree for maps between compact connected oriented manifolds without boundary of equal dimension Computing degree from signed preimages of a regular value

Topics Vector fields and indices of isolated zeros Poincaré-Hopf theorem for compact manifolds without boundary and vector fields with isolated zeros: statement only Euler characteristic and vector field examples on spheres and tori

Topics Review of regular values, transversality, and degree No continuous retraction from the closed n-ball onto its boundary sphere for n >= 1 Brouwer fixed point theorem for continuous self-maps of the closed n-ball: statement and connection to non-retraction

Contacts

About

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Suggested background

Schedule

February 2027

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March 2027

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April 2027

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About

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Suggested background

Schedule

February 2027

Sat6Session 1Saturday, February 6, 20276:00 – 7:30 PMSmooth Manifolds (tentative)

Topics

Sat13Session 2Saturday, February 13, 20276:00 – 7:30 PMTangent Spaces and Derivatives (tentative)

Sat20Session 3Saturday, February 20, 20276:00 – 7:30 PMThe Inverse Function Theorem and Immersions (tentative)

Sat27Session 4Saturday, February 27, 20276:00 – 7:30 PMSubmersions and Regular Values (tentative)

March 2027

Sat6Session 5Saturday, March 6, 20276:00 – 7:30 PMSard’s Theorem (tentative)

Sat13Session 6Saturday, March 13, 20276:00 – 7:30 PMEmbeddings and Whitney’s Theorem (tentative)

Sat20Session 7Saturday, March 20, 20276:00 – 7:30 PMManifolds with Boundary (tentative)

Sat27Session 8Saturday, March 27, 20276:00 – 7:30 PMTransversality (tentative)

April 2027

Sat3Session 9Saturday, April 3, 20276:00 – 7:30 PMIntersection Theory Mod 2 (tentative)

Sat10Session 10Saturday, April 10, 20276:00 – 7:30 PMOrientation and the Degree (tentative)

Sat17Session 11Saturday, April 17, 20276:00 – 7:30 PMVector Fields and the Poincaré-Hopf Theorem (tentative)

Sat24Session 12Saturday, April 24, 20276:00 – 7:30 PMReview and Basic Applications (tentative)