Graduate Geometric Analysis Reading Seminar

The graduate reading seminar in differential geometry / geometric analysis meets Tuesdays 4-6pm in Van Vleck B123. Students will give literature talks over the semester with faculty participation. To join the mailing list, please send an email to: math-geom-reading+subscribe@g-groups.wisc.edu.

The topic for Fall 2025 is Introduction to gauge theory. The basic reference will be these notes. Here is the tentative schedule:

Date	Speaker	Title	References
9/9		Connections, gauge transformations, curvature	Math 865 notes, Donaldson-Kronheimer, "The Geometry of Four-Manifolds" Ch. 2, Kobayashi-Nomizu "Foundations of Differential geometry," ...
9/16		Chern-Weil Theory
9/23		Definition of YM functional, first variation, Maxwell-Dirac equations
9/30		The Hodge Theorem	Jost, "Riemannian Geometry and Geometric Analysis" Ch. 2, Griffiths-Harris, "Principles of Algebraic Geometry" Ch. 0
10/7	Anuk	Yang-Mills in 2D and 4D	Atiyah, "Geometry of Yang-Mills fields" Ch. 2, Atiyah-Hitchin-Singer, "Self-duality in four-dimensional Riemannian geometry"
10/14		Uhlenbeck's gauge-fixing theorem I	Uhlenbeck, "Connections with L^p bounds on curvature," Donaldson-Kronheimer Ch. 2
10/21		Uhlenbeck's gauge-fixing theorem II
10/28	Alex	Uhlenbeck compactness	Donaldson-Kronheimer Ch. 4
11/4		Holomorphic bundles, Chern connection, the integrability theorem	Griffiths-Harris Ch. 0, Donaldson-Kronheimer Ch. 2&6
11/11		Mumford-Takemoto stability, Narasimhan-Seshadri Theorem
11/18		Donaldson's proof of Narasimhan-Seshadri	Donaldson, "A new proof of the theorem of Narasimhan and Seshadri"
12/2		Atiyah-Bott I: Equivariant cohomology, topology of the gauge group	Atiyah-Bott, "The Yang-Mills equations over Riemann surfaces"
12/9		Atiyah-Bott II: Second variation, perfectness of YM, idea of proof

Past topics:

Spring '25: Quantitative differentiation in Riemannian geometry and elliptic/parabolic PDEs

Date	Speaker	Title	Reference
1/28	Ruobing Zhang	Introduction to cone structures and monotonicity
2/4	Ruobing Zhang	Quantitative linear approximations of differentiable functions on $\mathbb{R}$: models and future plans
2/11	Ruobing Zhang	A comparative review of quantitative stratifications of the singular sets in various contexts
2/18	Zihan Zhang	Monotonicity of Almgren's frequency and applications to the nodal set estimates
2/25	Ziji Ma	Schauder estimates by scaling I	Leon Simon's paper
3/4	Ziji Ma	Schauder estimates by scaling II
3/11	Ruobing Zhang	Quantitative stratification and the critical/singular set of elliptic PDEs
3/18	Yue Su	Cheeger-Colding's segment inequality and Poincaré inequality
4/1	Anuk Dayaprema	Energy identity for stationary Yang-Mills I	Naber-Valtorta's paper
4/8	Anuk Dayaprema	Energy identity for stationary Yang-Mills II
4/15	Anuk Dayaprema	Energy identity for stationary Yang-Mills III
4/22		No seminar
4/29		Talk moved to next week due to Distinguished Lectures
5/6	Ruobing Zhang	Almost-volume-cone implies almost-metric-cone	Cheeger-Colding

Fall '24: Ricci flow

Date	Speaker	Title	Reference
9/10	Sigurd Angenent	Introduction to the Ricci flow
9/17	Alex Waldron	Rapid course in Riemannian geometry	Notes
9/24	Ruocheng Yang	Evolution equations under Ricci flow	Topping Ch. 2, Notes
10/1	Kaiyi Huang	The maximum principle	Topping Ch. 3, Notes
10/8	Anuk Dayaprema	Short-time existence for the Ricci flow	Topping Ch. 4-5
10/15	Yijie He	Ricci flow as a gradient flow	Topping Ch. 6
10/22	Ruobing Zhang	The compactness theorem for the Ricci flow	Topping Ch. 7
10/29	Alex Waldron	Curvature pinching and preserved curvature properties	Topping Ch. 9
11/05	Andoni Royo-Abrego (Tübingen)	Ricci flow and sphere theorems	Notes
11/12	Anuk Dayaprema	Perelman's W-functional	Topping Ch. 8

Spring '24: Heat-kernel approach to the Atiyah-Singer index theorem

Fall '23: G₂ geometry

Spring '23: Yau's proof of the Calabi conjecture

Fall '22: Spin geometry and the index theorem

Spring '22: Differential-geometric approach to GIT.