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.
- 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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Oct 29, 2020 (Thur) 4:00PM Zoom link |
Speaker: Yi Liu (Beijing International Center for Mathematical Research) Title: Volume of Seifert representations for graph manifolds and finite covers Abstract: Seifert volume is a topological invariant for closed orientable 3-manifolds. It is introduced by Brooks and Goldman as a generalization of the simplicial volume. In this talk, I will discuss some recent progress on this invariant for graph manifolds, and in particular, an effective formula that allows one to compute the volume of any representation into the motion group of the Seifert geometry. This is joint work with Pierre Derbez and Shicheng Wang. |
2020-21 Fall schedule
Date/Time/Venue | Talks |
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Aug 13, 2020 (Thur) 11:00AM Zoom link |
Speaker: Jonathan Luk (Stanford University) Title: Singularities in general relativity Abstract: General relativity is a theory of spacetime geometry and gravity in which the evolution of the spacetime is governed by the celebrated Einstein equations. It is well-known that solutions to the Einstein equations could develop singularities, for instance inside black holes. In this talk, I will survey some recent mathematical results regarding the nature of singularities in general relativity. |
Sep 3, 2020 (Thur) 10:00AM Zoom link |
Speaker: Pengzi Miao (University of Miami) Title: Mass in relativity via cubic polyhedra Abstract: Recently Stern has discovered a formula that relates scalar curvature to the level sets of harmonic maps. Prompted by Stern's formula, we find that the mass of an asymptotically flat 3-manifold has a geometric interpretation if evaluated along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov's scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning angle of slicing curves, the formula realizes the mass as integration of the angle defect detected by the boundary term in the Gauss-Bonnet theorem. |
Sep 10, 2020 (Thur) 11:00AM Zoom link |
Speaker: Huanchen Bao (National University of Singapore) Title: Flag manifolds over semifields Abstract: The study of totally positive matrices, i.e., matrices with positive minors, dates back to 1930s. The theory was generalised by Lustig to arbitrary split reductive groups using canonical bases, and has significant impacts on the theory of cluster algebras, higher teichmuller theory, etc.. In this talk, we survey basics of total positivity and explain its generalization to general semifields. This is based on joint work with Xuhua He. |
Sep 17, 2020 (Thur) 10:00AM Zoom link |
Speaker: Otis Chodosh (Stanford University) Title: Mean curvature flow of generic initial data Abstract: Mean curvature flow is the analogue of the heat equation in extrinsic differential geometry. Because mean curvature flow is nonlinear, there are necessarily singularities. In general, the singular behavior of the flow could be extremely complicated and is not well understood. I will discuss recent work with K. Choi, C. Mantoulidis, and F. Schulze concerning the mean curvature flow of a “generic” initial surface. In particular, we show that certain singularities do not arise in the case of a generic initial surface. |
Sep 24, 2020 (Thur) 9:00AM Zoom link |
Speaker: Wen-Ching Winnie Li (Pennsylvania State University) Title: The distribution of primes and zeta functions Abstract: The distribution of prime numbers has been one of the central topics in number theory. It has a deep connection with the zeros of the Riemann zeta function. The concept of "primes" also arises in other context. For example, in a compact Riemann surface, as introduced by Selberg, primitive closed geodesic cycles play the role of primes, while in a finite quotient of a finite-dimensional building, for each positive dimension, there are primes of similar nature. In this survey talk we shall discuss the distributions of such primes and their connections with the analytic behavior of the associated zeta and L-functions. |
Oct 8, 2020 (Thur) 11:00AM Zoom link |
Speaker: Aliakbar Daemi (Washington University in St. Louis) Title: Unitary Representations of 3-manifold Groups and the Atiyah-Floer Conjecture Abstract: A useful tool to study a 3-manifold is the space of the representations of its fundamental group, a.k.a. the 3-manifold group, into a Lie group. Any 3-manifold can be decomposed as the union of two handlebodies. Thus, representations of the 3-manifold group into a Lie group can be obtained by intersecting representation varieties of the two handlebodies. Casson utilized this observation to define his celebrated invariant. Later Taubes introduced an alternative approach to define Casson invariant using more geometric objects. By building on Taubes' work, Floer refined Casson invariant into a graded vector space whose Euler characteristic is twice the Casson invariant. The Atiyah-Floer conjecture states that Casson's original approach can be also used to define a graded vector space and the resulting invariant of 3-manifolds is isomorphic to Floer's theory. In this talk, after giving some background, I will give an exposition of what is known about the Atiyah-Floer conjecture and discuss some recent progress, which is based on a joint work with Kenji Fukaya and Maksim Lipyanskyi. I will only assume a basic background in algebraic topology and geometry. |
Oct 15, 2020 (Thur) 10:00AM Zoom link |
Speaker: Xin Zhou (Cornell University) Title: Generic scarring for minimal hypersurfaces Abstract: In classical spectral theory, Equidistribution and Scarring concern the distribution of normalized energy measures for Laplacian eigenfunctions on closed manifolds. The Quantum Ergodicity asserts that in negative curvature a density one subsequence of Laplacian eigenfunctions has their normalized energy measures equidistributing, while Scarring means that some particular subsequence of normalized energy measures concentrate on proper subsets. In this talk, we will present a scarring phenomenon for minimal hypersurfaces for a generic set of smooth metrics. In particular, for generic metrics, to each stable hypersurface, there exists a sequence of minimal hypersurfaces, with area and Morse index both diverging to infinity, that accumulate along the stable hypersurface in a quantitative way. This is a joint work with Antoine Song. |
Oct 22, 2020 (Thur) 10:00AM Zoom link |
Speaker: Artan Sheshmani (Harvard University CMSA) Title: Atiyah class and sheaf counting on local Calabi-Yau 4-folds Abstract: We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3. |
Oct 29, 2020 (Thur) 4:00PM Zoom link |
Speaker: Yi Liu (Beijing International Center for Mathematical Research) Title: Volume of Seifert representations for graph manifolds and finite covers Abstract: Seifert volume is a topological invariant for closed orientable 3-manifolds. It is introduced by Brooks and Goldman as a generalization of the simplicial volume. In this talk, I will discuss some recent progress on this invariant for graph manifolds, and in particular, an effective formula that allows one to compute the volume of any representation into the motion group of the Seifert geometry. This is joint work with Pierre Derbez and Shicheng Wang. |
Nov 5, 2020 (Thur) 4:00PM Zoom link |
Speaker: Raphael Zentner (Universität Regensburg) Title: TBA Abstract: TBA |
Nov 12, 2020 (Thur) 4:00PM Zoom link |
Speaker: Hans-Joachim Hein (University of Münster) Title: TBA Abstract: TBA |
Nov 19, 2020 (Thur) 10:00AM Zoom link |
Speaker: Valentino Tosatti (Northwestern University/McGill University) Title: TBA Abstract: TBA |
Nov 26, 2020 (Thur) 11:00AM Zoom link |
Speaker: Lu Wang (Caltech) Title: TBA Abstract: TBA |
Dec 3, 2020 (Thur) 4:00PM Zoom link |
Speaker: Yifeng Liu (Yale University) Title: TBA Abstract: TBA |
Dec 10, 2020 (Thur) 8:00PM Zoom link |
Speaker: Clifford Taubes (Harvard University) Title: TBA Abstract: TBA |