**MATH-IMS Joint Colloquium Series (Pure Mathematics)**

**Organizers:**Yi-Jen Lee and Martin Li**Hosts:**Department of Mathematics & Institute of Mathematical Sciences, CUHK**Contact:**Please email to yjlee@math.cuhk.edu.hk or martinli@math.cuhk.edu.hk for any inquiries. The recording of the talks are available upon request.**Remark:**Due to current COVID-19 situation, we will be hosting the colloquium talks online via ZOOM until safe travels are permitted. You can access the talk via the ZOOM link below.

Upcoming talk

Date/Time/Venue | Talks |
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Dec 2, 2022 (Fri) 9:30AM Zoom link |
Speaker: Chongchun Zeng (Georgia Institute of Technology) Title: Capillary Gravity Water Waves Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping Abstract:
We consider the 2-dim capillary gravity water wave problem -- the free boundary problem of the Euler equation with gravity and surface tension -- of finite depth $x_2 \in (-h,0)$ linearized at a uniformly monotonic shear flow $U(x_2)$. Our main focus are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes (i.e. embedded eigenvalues), we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain stronger decay for the remainders. This is a joint work with Xiao Liu. |

2022-23 Fall schedule

Date/Time/Venue | Talks |
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Sep 30, 2022 (Fri) 9:30AM Zoom link |
Speaker: Yuri Trakhinin (Sobolev Institute of Mathematics) Title: Well-posedness for moving interfaces with surface tension in ideal compressible MHD Abstract:
We discuss recent results on the local well-posedness for an interface with surface tension that separates a perfectly conducting inviscid fluid from a vacuum. The fluid flow is governed by the equations of three-dimensional ideal compressible magnetohydrodynamics (MHD), while the vacuum magnetic and electric fields are supposed to satisfy the pre-Maxwell equations. The fluid and vacuum magnetic fields are tangential to the interface. This renders a nonlinear hyperbolic-elliptic coupled problem with a characteristic free boundary. We prove the local existence and uniqueness of solutions of this free boundary problem. Both the non-collinearity condition for the fluid and vacuum magnetic fields and the Rayleigh-Taylor sign condition required for the case of zero surface tension become unnecessary in our result, which verifies the stabilizing effect of surface tension on the evolution of moving vacuum interfaces in ideal compressible MHD. This is a joint work with Tao Wang. |

Oct 14, 2022 (Fri) 4:30PM Zoom link |
Speaker: Andrea Mondino (University of Oxford) Title: Smooth and non-smooth aspects of Ricci curvature lower bounds Abstract:
After recalling the basic notions coming from differential geometry, the talk will be focused on spaces satisfying Ricci curvature lower bounds. The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the 80s and was pushed by Cheeger and Colding in the 90s who investigated the fine structure of possibly non-smooth limit spaces. A completely new approach via optimal transportation was proposed by Sturm and Lott-Villani around fifteen years ago. Via such an approach one can give a precise definition of what means for a non-smooth space to have Ricci curvature bounded below. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seems to be new even for smooth Riemannian manifolds. The talk is meant to be an introduction to the topic, accessible to non-specialists and as self-contained as possible. |

Oct 21, 2022 (Fri) 11:00AM Zoom link |
Speaker: Yi Lai (Stanford University) Title: Steady Ricci solitons with positive curvature operators Abstract:
Steady Ricci solitons are fundamental objects in the study of Ricci flow, as they are self-similar solutions and often arise as singularity models. Classical examples of steady solitons are the 2D cigar soliton (a.k.a. Witten's black hole), and the 3D rotational symmetric Bryant soliton and its generalization to higher dimensions. Hamilton conjectured that there exists a family of steady Ricci solitons in 3D called flying wings, which are “between” the 3D Bryant soliton and the product of a line and the 2D cigar soliton. In this talk, I will first discuss the construction of new families of steady gradient solitons with positive curvature operators in any dimension greater than or equal to three. This leads to the resolution of Hamilton's conjecture. Then I will discuss the symmetry of 3D steady Ricci solitons and show that they are all O(2)-symmetry. |

Oct 28, 2022 (Fri) 9:30AM Zoom link |
Speaker: Michael Groechenig (University of Toronto) Title: Rigid local systems Abstract:
An irreducible representation of a group G is said to be rigid, if it cannot be continuously deformed to a non-isomorphic representation. If G happens to be the fundamental group of a complex projective manifold, rigid representations are expected to have fundamentally different properties. In this talk I will introduce the basic notions related to rigidity and subsequently turn to two conjectures by Simpson (motivicity and integrality). I will then report on joint work with Esnault which proves the integrality conjecture for cohomologically rigid local systems and comment on recent developments with regard to motivicity. |

Nov 4, 2022 (Fri) 4:30PM Zoom link |
Speaker: Anne-Marie Aubert (Sorbonne University) Title: Generalized affine Hecke algebras and the Langlands program Abstract:
The Langlands program is a broad tissue of conjectures that relates different areas of mathematics, including representation theory and number theory. Affine Hecke algebras are deformations of group algebras of Coxeter systems of affine type. We will introduce generalizations of them which occur naturally in both the representation theory of $p$-adic reductive groups and the arithmetic side of the local Langlands correspondence. After having provided examples of these generalized affine Hecke algebras, we will explain how they can be used to build the correspondence in a large amount of cases. |

Nov 11, 2022 (Fri) 4:30PM Zoom link |
Speaker: Felix Schulze (University of Warwick) Title: Singularities along the Lagrangian mean curvature flow of surfaces Abstract:
It is an open question to determine which Hamiltonian isotopy classes of Lagrangians in a Calabi-Yau manifold have a special Lagrangian representative. One approach is to follow the steepest descent of area, i.e. the mean curvature flow, which preserves the Lagrangian condition. But in general such a flow will develop singularities in finite time, and it has been open how to continue the flow past singularities. We will give an introduction to the problem and explain recent advances where we show that in the simplest possible situation, i.e. the Lagrangian mean curvature flow of surfaces, when the singularity is the special Lagrangian union of two transverse planes, then the flow forms a “neck pinch”, and can be continued past the singularity. In a different direction we show that ancient solutions of the flow, whose blow-down is given by two planes meeting along a line, must be translators. These are joint works with Jason Lotay and Gábor Székelyhidi. |

Nov 18, 2022 (Fri) 9:00AM Zoom link |
Speaker: Keith Promislow (Michigan State University) Title: Frustration in the Packing of Soft Materials Abstract:
Many processes in material science involve entropic contributions from packing – the constraints imposed by volume occupied by other material. Diblock polymers offer a rich environment to study the packing of soft materials as gradient flows of a system energy. Ideas from $\Gamma$ convergence provide powerful tools to extract simplified models in certain singular limits. We present examples of packing dichotomies in both continuous and discrete formulations and identify cases in which limiting problems may be more complex. We present a derivation of a random phase reduction of self-consistent mean field models, identify regimes in which they converge to functionalized Cahn-Hilliard energy, and provide a discrete system for the packing of soft balls that exhibits large-system frustration: the inability of gradient flows to obtain the global energy minimum, that significantly complicates the extraction of limiting processes. |

Nov 25, 2022 (Fri) 4:30PM Zoom link |
Speaker: Toshiyuki Kobayashi (University of Tokyo) Title: Basic Questions in Group-Theoretic Analysis on Manifolds Abstract:
Symmetry of geometry induces symmetry of function spaces through the regular representation. In turn, it provides a group theoretic approach to global analysis such as the classical Fourier series expansion or more generally the spherical harmonics expansions where the "symmetry" is abelian or compact groups. In this talk, we address some basic questions about the global analysis on manifolds X acted algebraically by highly non-commutative groups G such as SL(n,R) Problem A. Does the group G control "sufficiently" the space of function on X? Problem B. What can we say about "spectrum" for L2(X) We plan to discuss some recent progress with emphasis on "multiplicity" for Problem A and "decay of matrix coefficients" for Problem B. |

Dec 2, 2022 (Fri) 9:30AM Zoom link |
Speaker: Chongchun Zeng (Georgia Institute of Technology) Title: Capillary Gravity Water Waves Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping Abstract:
We consider the 2-dim capillary gravity water wave problem -- the free boundary problem of the Euler equation with gravity and surface tension -- of finite depth $x_2 \in (-h,0)$ linearized at a uniformly monotonic shear flow $U(x_2)$. Our main focus are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes (i.e. embedded eigenvalues), we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain stronger decay for the remainders. This is a joint work with Xiao Liu. |

**Past Colloquium**