MATH-IMS Joint Colloquium Series (Pure Mathematics)
2020-21 Term 1

Date/Time/Venue Talks
Aug 13, 2020 (Thur)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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Speaker: Raphael Zentner (Universität Regensburg)
Title: Irreducible SU(2)-representations of toroidal integral homology 3-spheres

Abstract: Somewhat an analogue of the L-space conjecture for Heegaard Floer homology is the following question: Does instanton Floer homology detect the 3-sphere among integral homology 3-spheres? The generators of the underlying instanton chain complex is given by irreducible SU(2)-representations of the fundamental group. Hence a weaker question than the above is whether all integral homology 3-spheres admit irreducible SU(2)-representations. It is conjectured that this holds, since it has been proved for large classes by various authors. We show that toroidal integral homology 3-spheres have irreducible SU(2)-representations. This is joint work with Tye Lidman and Juanita Pinzon Caicedo.
Nov 12, 2020 (Thur)
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Speaker: Hans-Joachim Hein (University of Münster)
Title: Smooth asymptotics for collapsing Calabi-Yau metrics

Abstract: Yau's solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion of the degenerating metrics in a natural class of examples. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
Nov 19, 2020 (Thur)
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Speaker: Valentino Tosatti (Northwestern University/McGill University)
Title: Geometry and dynamics on K3 surfaces

Abstract: K3 surfaces are a class of compact complex manifolds that enjoys many special properties and play an important role in several areas of mathematics. In this colloquium I will discuss a new interplay between complex geometry and analysis on K3 surfaces equipped with their Calabi-Yau metrics, and dynamics of holomorphic diffeomorphisms of these surfaces, that Simion Filip and I have been investigating recently.
Nov 26, 2020 (Thur)
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Speaker: Lu Wang (Caltech)
Title: Nonuniqueness in Mean Curvature Flow

Abstract: Mean curvature flow is the gradient flow of area functional that decreases the area in the steepest way. In general the flow will develop singularities in finite time. It is known that there may not be a unique way to continue the flow through singularities. In this talk, we will discuss some global features of the space of mean curvature flows that emerge from cone-like singularities. This is joint with Jacob Bernstein.
Dec 3, 2020 (Thur)
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Speaker: Yifeng Liu (Yale University)
Title: Cycles on arithmetic varieties

Abstract: In this talk, we study algebraic cycles on algebraic varieties over number fields. We introduce a series of conjectures and tools in such study proposed by Beilinson and Bloch. Then we use modular curves and more general Shimura varieties as playground to test these observations. At the end, we will introduce a recent result for unitary Shimura varieties obtained by Chao Li and myself.
Dec 10, 2020 (Thur)
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Speaker: Clifford Taubes (Harvard University)
Title: Studying Z/2 eigenfunctions and eigenvalues of the Laplacian on the round 2-sphere

Abstract: I once thought this: Certainly by the year 2020, the Laplacian on the round 2-sphere would have nothing new to offer. As it turns out, I was wrong. I will describe a sequence of eigenvector/eigenvalue problems on the 2-sphere that Yingying Wu and I are studying that take every opportunity to do the unexpected.

© Martin Li, Department of Mathematics, The Chinese University of Hong Kong