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We are still following the Uhlenbeck’s paper ‘Removable singularities in Yang-Mills fields’ and Donaldson’s survey of Uhlenbeck’s work. We will describe the construction of exponential gauges and estimates on the growth of the curvature, sketching thus the proof of the main theorem. Note that a proof of the Lie Group fact used during the talk can be found in Milnor's Morse Theory Book, also des...

We turn back to Yang-Mills equations. We start with recalling the concept of Coloumb gauge and work towards the theorem of Uhlenbeck about removable singularities. We will be mainly following Uhlenbeck’s paper ‘Removable singularities in Yang-Mills fields’ from 1982.

Following Schoen-Uhlenbeck's paper on energy minimizing maps, Devesh Rajpal uses tangent maps to prove that the singular set of an energy minimizer has Hausdorff dimension at most n-3, improving on the previously obtained n-2 bound.

In this talk Devesh gives a fundamental regularity theorem, but not the optimal version (stay tuned!).

In this talk Brad guides us through the small energy estimate. Enjoy!

Brad tells us about how to remove to remove singularities in harmonic maps, completing the proof of the Sacks-Uhlenbeck compactness theorem for harmonic maps.

Vishnu Mangalath takes us through the proof of Uhlenbeck's compactness result. In particular, we find some convenient gauges in which to study the curvature of a connection.

Max Orchard discusses connections on principal G-bundles, curvature, and concludes with the statement and variational interpretation of the Yang-Mills equations.

Vishnu Mangalath describes the structure of the Yang-Mills equations. In particular the Yang-Mills equations are not elliptic, but in Coloumb gauge they are ellipic.

In this talk Chris gives the detail on a fundamental important example of Yang-Mills, the Maxwell Equations.

Max Orchard introduces some basic concepts needed for the study of Yang-Mills theory. In particular, Max describes fibre and principal G-Bundles along with the notion of Ehresmann connection.

Stepan Hudecek, Rhuadi Burke, and Bai-Ling Wang introduce us to some aspects of Yang-Mills and 4-manifold topology. 00:00 Stepan Hudecek - Yang-Mills equations 30:13 Rhuaidi Burke - Homotopy classification of 4-manifolds via the intersection form 57:21 Bai-Ling Wang - Self dual/Anti self dual connections and problems outline

Stepan Hudecek (UQ) wraps up this year's seminar series with a discussional summary of some of the problems that arise in MCF of surfaces that are absent in higher dimensions.

Vishnu Mangalath continues discussing important gradient estimates in mean curvature flow.

Estimates for MCF in R^3

MCF with surgery part 2

MCF with surgery

Necks in MCF

Gradient Estimates in MCF II

Gradient Estimates in MCF

Cylindrical Estimates in MCF

Asymptotic Convexity II

Asymptotic Convexity in MCF (Part 1)

Convexity and Huisken's Convergence Theorem

The Second Fundamental Form at Singularities of MCF

Introduction to MCF Part 2

Introduction to Mean Curvature Flow

Kyle Broder: Curvature and Moduli Some Intimations and Propaganda

Peter Olanipekun: The Willmore energy in Four Dimensions