# Difference between revisions of "Geometry and Topology Seminar"

(→Fall 2020) |
(→Fall Abstracts) |
||

(13 intermediate revisions by the same user not shown) | |||

Line 30: | Line 30: | ||

|Nov. 6 | |Nov. 6 | ||

|Jiyuan Han (Purdue) | |Jiyuan Han (Purdue) | ||

− | | | + | | On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations |

|(Chen) | |(Chen) | ||

+ | |- | ||

+ | |Nov. 13 | ||

+ | |Ilyas Khan (Madison) | ||

+ | |Rigidity Properties of Mean Curvature Flow Translators with Curvature Conditions | ||

+ | |(Local) | ||

|- | |- | ||

|Nov. 20 | |Nov. 20 | ||

|Max Hallgren (Cornell) | |Max Hallgren (Cornell) | ||

− | | | + | |The Entropy of Ricci Flows with a Type-I Scalar Curvature Bound |

|(Huang) | |(Huang) | ||

|- | |- | ||

Line 48: | Line 53: | ||

===Yu Li=== | ===Yu Li=== | ||

Ancient solutions model the singularity formation of the Ricci flow. In two and three dimensions, we currently have complete classifications for κ-noncollapsed ancient solutions, while the higher dimensional problem remains open. This talk will survey some recent developments of κ-noncollapsed ancient solutions with nonnegative curvature in higher dimensions. | Ancient solutions model the singularity formation of the Ricci flow. In two and three dimensions, we currently have complete classifications for κ-noncollapsed ancient solutions, while the higher dimensional problem remains open. This talk will survey some recent developments of κ-noncollapsed ancient solutions with nonnegative curvature in higher dimensions. | ||

+ | |||

+ | ===Yi Lai=== | ||

+ | We found a family of $\mathbb{Z}_2\times O(2)$-symmetric 3d steady gradient Ricci solitons. We show that these solitons are all flying wings. This confirms a conjecture of Hamilton. | ||

+ | |||

+ | ===Jiyuan Han=== | ||

+ | Let (X,D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1,1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampere equations that correspond to generalized and twisted Kahler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show | ||

+ | that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kahler-Ricci/Mabuchi solitons or Kahler-Einstein metrics. This is a joint work with Chi Li. | ||

+ | |||

+ | ===Ilyas Khan=== | ||

+ | In this talk we discuss some uniqueness results for mean curvature flow translators. Under certain curvature conditions, we classify the blow-down limits of translating solutions of the mean curvature flow and employ recent techniques from the theory of ancient MCF solutions to show the uniqueness of translators with these blow-down limits. | ||

+ | |||

+ | ===Max Hallgren=== | ||

+ | In this talk, we study the singularities of closed Ricci flow solutions which satisfy a Type-I scalar curvature assumption. Bamler's structure theory of Ricci flows with bounded scalar curvature shows that singularities are modeled on shrinking Ricci solitons with singularities of codimension 4. We extend the analysis by characterizing the singular set of the flow in terms of a Gaussian density functional, and also establish entropy uniqueness of dilation limits at a fixed point, generalizing results previously known assuming a Type-I bound on the full curvature tensor. We also show that in dimension 4, the singular Ricci soliton is smooth away from finitely many points, which are conical smooth orbifold singularities. | ||

== Archive of past Geometry seminars == | == Archive of past Geometry seminars == |

## Latest revision as of 23:58, 13 November 2020

The Geometry and Topology seminar meets in room **901 of Van Vleck Hall** on **Fridays** from **1:20pm - 2:10pm**. For more information, contact Shaosai Huang.

In the fall of 2020, we will hold **online meetings** on
Zoom platform
(available every **Friday 1:00pm - 2:30pm**).

## Contents

## Fall 2020

date | speaker | title | host(s) |
---|---|---|---|

Oct. 23 | Yu Li (Stony Brook) | On the ancient solutions to the Ricci flow | (Huang) |

Oct. 30 | Yi Lai (Berkeley) | A family of 3d steady gradient solitons that are flying wings | (Huang) |

Nov. 6 | Jiyuan Han (Purdue) | On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations | (Chen) |

Nov. 13 | Ilyas Khan (Madison) | Rigidity Properties of Mean Curvature Flow Translators with Curvature Conditions | (Local) |

Nov. 20 | Max Hallgren (Cornell) | The Entropy of Ricci Flows with a Type-I Scalar Curvature Bound | (Huang) |

Dec. 4 | Yang Li (IAS) | TBA | (Chen) |

## Fall Abstracts

### Yu Li

Ancient solutions model the singularity formation of the Ricci flow. In two and three dimensions, we currently have complete classifications for κ-noncollapsed ancient solutions, while the higher dimensional problem remains open. This talk will survey some recent developments of κ-noncollapsed ancient solutions with nonnegative curvature in higher dimensions.

### Yi Lai

We found a family of $\mathbb{Z}_2\times O(2)$-symmetric 3d steady gradient Ricci solitons. We show that these solitons are all flying wings. This confirms a conjecture of Hamilton.

### Jiyuan Han

Let (X,D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1,1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampere equations that correspond to generalized and twisted Kahler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kahler-Ricci/Mabuchi solitons or Kahler-Einstein metrics. This is a joint work with Chi Li.

### Ilyas Khan

In this talk we discuss some uniqueness results for mean curvature flow translators. Under certain curvature conditions, we classify the blow-down limits of translating solutions of the mean curvature flow and employ recent techniques from the theory of ancient MCF solutions to show the uniqueness of translators with these blow-down limits.

### Max Hallgren

In this talk, we study the singularities of closed Ricci flow solutions which satisfy a Type-I scalar curvature assumption. Bamler's structure theory of Ricci flows with bounded scalar curvature shows that singularities are modeled on shrinking Ricci solitons with singularities of codimension 4. We extend the analysis by characterizing the singular set of the flow in terms of a Gaussian density functional, and also establish entropy uniqueness of dilation limits at a fixed point, generalizing results previously known assuming a Type-I bound on the full curvature tensor. We also show that in dimension 4, the singular Ricci soliton is smooth away from finitely many points, which are conical smooth orbifold singularities.

## Archive of past Geometry seminars

2019-2020 Geometry_and_Topology_Seminar_2019-2020

2018-2019 Geometry_and_Topology_Seminar_2018-2019

2017-2018 Geometry_and_Topology_Seminar_2017-2018

2016-2017 Geometry_and_Topology_Seminar_2016-2017

2015-2016: Geometry_and_Topology_Seminar_2015-2016

2014-2015: Geometry_and_Topology_Seminar_2014-2015

2013-2014: Geometry_and_Topology_Seminar_2013-2014

2012-2013: Geometry_and_Topology_Seminar_2012-2013

2011-2012: Geometry_and_Topology_Seminar_2011-2012

2010: Fall-2010-Geometry-Topology

Dynamics_Seminar_2020-2021