For this session, Tacos talks will be held live on Zoom!<\/p>\n\n\n\n
The speakers for the ninth session, Hermitian Yang-Mills Connections<\/em> are:<\/p>\n\n\n\n K\u00e4hler-Yang-Mills equations and gravitating vortices<\/strong><\/p>\n\n\n\n I will start introducing the K\u00e4hler\u2013Yang\u2013Mills equations on a holomorphic vector bundle over a compact complex manifold. These equations, inspired by the Hitchin\u2013Kobayashi correspondence for bundles and the Yau\u2013Tian\u2013Donaldson conjecture for constant scalar curvature K\u00e4hler metrics, intertwine the curvature of a Hermitian\u2013Yang\u2013Mills connection on the bundle and the scalar curvature of a K\u00e4hler metric on the manifold. After this, I will consider special symmetric solutions on a compact Riemann surface known as gravitating vortices. <\/p>\n\n\n\n Hermitian-Einstein metrics in singular settings<\/strong><\/p>\n\n\n\n We will discuss a few results concerning the existence of HE metrics for stable sheaves on K\u00e4hler spaces with log-terminal singularities, and their relevance for the Bogomolov-Gieseker inequality.<\/p>\n\n\n\n Local Wall-Crossing and HYM Connections<\/strong><\/p>\n\n\n\n We will review some recent results on the behaviour of Hermitian Yang-Mills connections with respect to variations of the polarisation. In particular, we will focus on their convergence when the polarisation reaches the boundary of the stable locus. The methods presented are versatile and can be used for similar perturbation problems for geometric PDEs with moment map pictures.<\/p>\n\n\n\n First Introductory Lecture<\/strong><\/p>\n\n\n\n In the first lecture, I will survey the history and development of solutions to the Hermitian-Einstein equations. On the algebro-geometric side, this goes back to a theorem of Weil in the 1930s on the existence of holomorphic connections on curves. This motivated the work of Narasimhan-Seshadri, and then in turn higher dimensional versions of stability of sheaves. A second motivation comes from gauge theory and solutions to the anti-self-duality equations in Yang-Mills theory. Finally, I will briefly discuss some of the key applications and generalizations of the Hermitian-Einstein equations, such as Miyaoka’s version of the Bogomolov-Gieseker inequality.<\/p>\n\n\n\n Second Introductory Lecture<\/strong><\/p>\n\n\n\n In the second lecture, I will discuss some of the important techniques that go into the proof of the Donaldson-Uhlenbeck-Yau theorem, such as Uhlenbeck’s weak compactness theorem, weakly holomorphic subbundles, and the Donaldson functional. I will then mention more recent results, such as the solution to a conjecture of Bando-Siu, and the extension of Hermitian-Einstein equations to singular varieties.<\/p>\n\n\n\n Algebraic and analytic compactifications of moduli spaces of sheaves<\/strong><\/p>\n\n\n\n I will discuss the relationship between projective compactifications of moduli spaces of sheaves on higher dimensional projective manifolds, and the Uhlenbeck-Tian compactifications of solutions to the Hermitian-Einstein equations. This generalizes older work of Jun Li in the case of surfaces. This is joint work with Greb, Sibley, and Toma, and separately with Xuemiao Chen.<\/p>\n\n\n\n <\/p>\n\n\n\n The conference will be held on May 28-30, 2024 with two 45-minute talks per day. Richard A. Wentworth will give two introductory talks on the topic, then, we will have four research-oriented talks, one from each of the speakers. The tentative schedule can be found below (with all times listed in US Eastern time):<\/p>\n\n\n\n May 28<\/p>\n\n\n\n May 29<\/p>\n\n\n\n May 30<\/p>\n\n\n\n The conference will be held online on Zoom:<\/p>\n\n\n\n https:\/\/vanderbilt.zoom.us\/j\/91283512875?pwd=ZEgrTHg1Q25ZbUtvNHhMNHNJYmNKUT09<\/a><\/p>\n\n\n\n Meeting ID: 912 8351 2875 If you\u2019d like to receive updates from us, write an email to gtacos20@gmail.com<\/a> and we will add you to our mailing list.<\/p>\n","protected":false},"excerpt":{"rendered":" For this session, Tacos talks will be held live on Zoom! The speakers for the ninth session, Hermitian Yang-Mills Connections are: Oscar Garc\u00eda-Prada (ICMAT) K\u00e4hler-Yang-Mills equations and gravitating vortices I will start introducing the K\u00e4hler\u2013Yang\u2013Mills equations on a holomorphic vector bundle over a compact complex manifold. These equations, inspired by the Hitchin\u2013Kobayashi correspondence for bundles and the … <\/p>\n
Passcode: 914422<\/p>\n\n\n\n