Introduction to Gauge theory
This course provides an introduction to Gauge theory in mathematics for graduate and advanced undergraduate students. We focus on Donaldson's theory on 4-manifolds, which involves the moduli space of anti-self-dual connections, and related gauge-theoretic invariants in four dimensions. The course begins with a review of connections and curvatures on vector bundles and principal G-bundles. Our aim is to understand the Uhlenbeck compactification of the moduli space, its role in formulating the Donaldson invariants, and the connection to the algebraic counterpart in the moduli problem of algebraic geometry. If time permits, we may discuss more advanced topics from the current research field.
Lecturer
Date
14th March ~ 30th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday | 13:30 - 16:55 | A14-203 | ZOOM 07 | 559 700 6085 | BIMSA |
Prerequisite
manifolds, vector bundles, principal bundles, and the characteristic classes of them such as the Pontryagin and Chern ones
Syllabus
1. Connections, curvatures, and gauge transformations
2. Yang-Mills connections and anti-self-dual instantons
3. Sobolev spaces, elliptic operators, and linearisation
4. Atiyah-Hitchin-Singer complex, Kuranishi model, and Freed-Uhlenbeck perturbation
5. Reducible connections, orientation issues of the moduli space
6. Uhlenbeck compactness, Taubes' gluing, and Donaldson's theorem
7. Uhlenbeck compactification and Gieseker compactifiation
8. Donaldson invariants, Witten's conjecture, and other related invariants
9. Advanced topics
2. Yang-Mills connections and anti-self-dual instantons
3. Sobolev spaces, elliptic operators, and linearisation
4. Atiyah-Hitchin-Singer complex, Kuranishi model, and Freed-Uhlenbeck perturbation
5. Reducible connections, orientation issues of the moduli space
6. Uhlenbeck compactness, Taubes' gluing, and Donaldson's theorem
7. Uhlenbeck compactification and Gieseker compactifiation
8. Donaldson invariants, Witten's conjecture, and other related invariants
9. Advanced topics
Reference
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, 1990.
D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Second edition, Mathematical Sciences Research Institute Publications, 1991.
R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer-Verlag, 1994.
R. Friedman and J. W. Morgan (eds.), Gauge theory and the topology of four-manifolds, IAS/Park City Mathematics Series, Volume 4, American Mathematical Society, 1998.
C. H. Taubes, Metrics, Connections and gluing theorems. CBMS Regional Conference Series in Mathematics, 89, American Mathematical Society, 1996.
C. H. Taubes, Differential Geometry. Bundles, connections, metrics and curvature, Oxford Graduate Texts in Mathematics, 23, Oxford University Press, 2011.
D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Second edition, Mathematical Sciences Research Institute Publications, 1991.
R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer-Verlag, 1994.
R. Friedman and J. W. Morgan (eds.), Gauge theory and the topology of four-manifolds, IAS/Park City Mathematics Series, Volume 4, American Mathematical Society, 1998.
C. H. Taubes, Metrics, Connections and gluing theorems. CBMS Regional Conference Series in Mathematics, 89, American Mathematical Society, 1996.
C. H. Taubes, Differential Geometry. Bundles, connections, metrics and curvature, Oxford Graduate Texts in Mathematics, 23, Oxford University Press, 2011.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
My research interests are primarily centred on Gauge theory within mathematics. Recently, my focus has been on semistable Higgs sheaves on complex projective surfaces and associated gauge-theoretic invariants, employing algebro-geometric methods. However, I also have a strong interest in working within the analytic category.