**Title:** Entropy, Stability and Yang-Mills Flow

**Speaker:** Casey Kelleher, UC Irvine

**Date:** Monday, December 15, 2014

**Time:** 11:30am – 12:30pm

**Place:** Lecture Hall 102, Simons Center

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**Abstract:** (Joint work with Jeffrey Streets). Following the work of Colding and Minicozzi in the case of mean curvature flow, I will define a notion of entropy for connections over Euclidean n-space which has shrinking Yang-Mills solitons as critical points. This entropy is defined implicitly, making it difficult to work with analytically. I will discuss a characterization of entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This characterization leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying “generic singularities” of Yang-Mills flow, and I will discuss the differences in the strategy in dimension n=4 versus larger n.

**Title:** Ricci flow of regions with curvature bounded below in dimension three

**Speaker:** Miles Simon, OVGU

**Date:** Friday, December 12, 2014

**Time:** 11:30am – 12:30pm

**Place:** Lecture Hall 102, Simons Center

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**Abstract:** We consider smooth complete solutions to Ricci flow with bounded curvature on manifolds without boundary in dimension three. Assuming a ball at time zero of radius one has curvature bounded from below by -1, then we prove estimates which show that compactly contained subregions of this ball will be smoothed out by the Ricci flow for a short but well defined time interval. The estimates we obtain depend on the initial volume of the ball and the distance from the compact region to the boundary of the initial ball. They do not depend on the upper bound of the curvature on the ball at time zero. Versions of these estimates for balls of radius r follow using scaling arguments.

**Title:** Minimal two-sphere with constant curvature in the Grassmannians

**Speaker:** Xiaowei Xu, USTC

**Date:** Thursday, December 11, 2014

**Time:** 12:30pm – 03:30pm

**Place:** Lecture Hall 102, Simons Center

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**Abstract:** In this talk, I wish to introduce some problems and our recent works on minimal two-sphere with constant curvature in the real and complex Grassmann manifold.

**Title:** Hypersurface flows and geometric inequalities in space forms

**Speaker:** Junfang Li, University of Alabama at Birmingham

**Date:** Wednesday, December 10, 2014

**Time:** 02:30pm – 03:30pm

**Place:** Lecture Hall 102, Simons Center

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**Abstract:** Consider the space forms Nn+1 with metric ds2=dρ2+φ2(ρ)dz2. We discuss two different types of normalized hypersurface curvature flows : mean curvature and inverse mean curvature types. These flow equations are closely related to isoperimetric type of inequalities. Some of the flow equations are new even in Euclidean space. These work are joint with Pengfei Guan.

**Title:** Deforming symplectomorphisms of Kahler-Einstein manifolds by the mean curvature flow

**Speaker:** Mu-Tao Wang, Columbia

**Date:** Wednesday, December 10, 2014

**Time:** 11:30am – 12:30pm

**Place:** Lecture Hall 102, Simons Center

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**Abstract:** The mean curvature flow is applied to study the symplectomorphism group of a Kahler-Einstein manifold. In particular, I will discuss a pinching theorem for symplectomorphisms of the complex projective space. It is shown that there is a dimension-dependent constant Lambda such that any Lambda pinched symplectomorphism is symplectically isotopic to a biholomorphic isometry of the complex projective space. The talk is based on joint work with Ivana Medos.

**Title:** A Lagrangian mean curvature type flow for holomorphic line bundles

**Speaker:** Adam Jacob, Harvard

**Date:** Friday, December 05, 2014

**Time:** 02:30pm – 03:30pm

**Place:** Lecture Hall 102, Simons Center

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**Abstract:** Let L be a holomorphic line bundle over a compact Kahler manifold X. Motivated by mirror symmetry, in this talk I will address the deformed Hermitian-Yang-Mills equation on L, which is the line bundle analogue of the special Lagrangian equation in the case that X is Calabi-Yau. I will show solutions are unique global minimizers of a positive functional. To address existence of solutions, I will introduce a line bundle analogue of the Lagrangian mean curvature flow, and prove convergence in certain cases. This is joint work with S.-T. Yau.

**Title:** The long-time behavior of the Calabi flow

**Speaker:** Kai Zheng, Gottfried Wilhelm Leibniz Universität

**Date:** Friday, November 21, 2014

**Time:** 11:30am – 12:30pm

**Place:** Seminar Room 313, Simons Center

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**Title:** Kahler-Ricci Flow with conic singularities

**Speaker:** Yuanqi Wang, UC Santa Barbara

**Date:** Wednesday, November 19, 2014

**Time:** 02:30pm – 03:30pm

**Place:** Seminar Room 313, Simons Center

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**Abstract:** Inspired by Donaldson’s program, we introduce the Kahler Ricci flows with conic singularities. The main part of this talk is to show that the conical Kahler Ricci flow exists for short time in a proper space. These existence results are hight related to the heat kernel. We will also discuss the long time existence and convergence of this flow to conic Ricci flat metrics.

**Title:** About Hermitian metrics whose scalar curvature of the Chern connection is constant

**Speaker:** Simone Calamai, Università di Firenze

**Date:** Monday, November 17, 2014

**Time:** 02:30pm – 03:30pm

**Place:** Seminar Room 313, Simons Center

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**Abstract:** Given a smooth compact complex manifold, we consider the problem on existence, in a fixed conformal class, of Hermitian metrics whose scalar curvature induced by the Chern connection is constant. In particular, we describe the interesting role in the picture played by the Gauduchon metrics (work in progress with Daniele Angella and Cristiano Spotti).

**Title:** Networks of Curves Evolving by Curvature in the Plane

**Program:** Geometric Flows

**Speaker:** Felix Shulze, University College London

**Date:** Wednesday, November 12, 2014

**Time:** 11:03am – 12:30pm

**Place:** Lecture Hall 102, Simons Center

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**Abstract:** The network flow is the evolution of a regular network of embedded curves under curve shortening flow in the plane, where it is allowed that at triple points three curves meet under a 120 degree condition. A network is called non-regular if at multiple points more than three embedded curves can meet, without any angle condition but with distinct unit tangents. Studying the singularity formation under the flow of regular networks one expects that at the first singular time a non-regular network forms. In this talk we will present recent work together with Tom Ilmanen and Andre Neves, showing that starting from any non-regular initial network there exists a flow of regular networks.