Date/Time : 25 Jan 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Sungwoon Kim, Jeju National University

Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper, we relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions. This is a joint work with Jaejeong Lee and Misha Kapovich.

Add to calendar :

Date/Time : 24 Jan 2024 15:30
Venue : S17 #04-06 (SR1)
Speaker : Bong Lian, Brandeis University

We will consider a class of (typically) singular Calabi-Yau varieties given by cyclic branched covers of a fixed semi Fano manifold. The first prototype example goes back to Euler, Gauss and Legendre, who considered 2-fold covers of $\P^1$ branched over 4 points. Two-fold covers of $\P^2$ branched over 6 lines have been studied more recently by many authors, including Matsumoto, Sasaki, Yoshida and others, mainly from the viewpoint of their moduli spaces and their comparisons. I will outline a higher dimensional generalization from the viewpoint of mirror symmetry, and discuss the Riemann-Hilbert problem for periods of these singular varieties. The new insight here is the idea of `abelian gauge fixing’ and `fractional complete intersections’ that leads to a new interpretation of those classical results. The idea further points to a construction of large class of Calabi-Yau mirror pairs.

The lecture is based on joint work with S. Hosono, T.-J. Lee, M. Romo, L. Santilli, H. Takagi, S.-T. Yau.

Add to calendar :

Date/Time : 18 Jan 2024 15:30
Venue : S17 #04-06 (SR1)
Speaker : Biao Ma, Technion - Israel Institute of Technology

Let Mod(S) be the mapping class group of a connected, closed, oriented surface S of genus g ≥ 2. Based on ideas of Margulis, Athreya-Bufetov-Eskin-Mirzakhani gave an asymptotic formula for orbit growth of Mod(S) on the Teichmuller space with respect to the Teichmuller metric. In this talk, I will explain how we can get asymptotic formulas for orbit growth of certain subgroups of Mod(S) based on Roblin’s ideas. More precisely, let G be a non-elementary subgroup of Mod(S) so that the associated Bowen-Margulis measure is finite. We will discuss an asymptotic growth formula for G with respect to the Teichmuller metric. This talk is based on a joint work with Ilya Gekhtman.

Add to calendar :

Date/Time : 17 Jan 2024 16:00
Venue : S17 #06-11 (SR6)
Speaker : Tan Sheng Li, East China Normal University

Foliations are generalizations of fibrations on an algebraic surface. Some invariants of fibrations are generalized to foliations, e.g., Kodaira dimensions, geometric genus, Chern numbers, volume and slope. Fibrations with minimal slope must be hyperelliptic, which are double covers of ruled surfaces. We have no slope inequality for foliations, so it is interesting to construct some examples with small slopes. A nature way is to consider double covers of some transcendental foliations, we need to generalize the theory of double covers of fibrations to foliations. We will present an example whose slope is 12/7. This is a joint work with Jun LU and Shi XU.

Add to calendar :

Date/Time : 05 Dec 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Lu Qi, Princeton University

In this talk, I will discuss some convexity properties of multiplicities of filtrations on a local ring. In particular, the multiplicity function is convex along geodesics. As a major application, this gives a new proof of a theorem due to Xu and Zhuang on the uniqueness of normalized volume minimizers. In order to characterize strict convexity, we introduce the notion of saturation of a filtration, which turns out to be useful in other settings. For example, it allows us to generalizes a theorem by Rees on characterization of when two filtrations have equal multiplicities. It also allows us to introduce a metric on the space of filtrations. This talk is based on joint work with Harold Blum and Yuchen Liu and some ongoing work.

Add to calendar :

Date/Time : 09 Nov 2023 16:00
Venue : S17 #06-11 (SR6)
Speaker : Andreas Wieser, Hebrew University of Jerusalem

In the 80’s, Livingston asked whether there exist Seifert surfaces F,F’ for the same knot so that, after pushing their interiors into the four-ball, the surfaces are not topologically ambient isotopic. The question has recently been resolved by Hayden, Miller, Kim, Park, and Sundberg whose example satisfies that the interiors of the surfaces are disjoint. In joint work with Menny Aka, Peter Feller, and Alison B. Miller we classify all examples that can be recognized algebraically. This is based on an exact number-theoretic characterization of pairs of Seifert forms that can arise from Seifert surfaces with disjoint interiors. Here, the characterization is in terms of the Gauss composition of binary quadratic forms. In the second half of the talk we will also discuss asymptotic independence questions related to the above setting (based on work with Aka and Einsiedler) as well as open questions. No preliminaries are assumed. The first half of the talk will be focused on geometry and the second will be number-theoretical in nature (bordering on homogeneous dynamics).

Add to calendar :

Date/Time : 12 Oct 2023 15:30
Venue : S17 #06-11 (SR6)
Speaker : Luo Yujie, National University of Singapore

A curve on a surface is called a negative curve if its normal bundle has negative degree. We study negative curves on P^2 blowing up some general points, and we deduce some bounds and inequalities on the numerical invariants of those curves. This is an ongoing project with Runcao Lyu.

Add to calendar :

Date/Time : 04 Oct 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Wilderich Tuschmann, Karlsruhe Institute of Technology

Consider a smooth manifold with a Riemannian metric satisfying some sort of geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasakian, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also study this question for the corresponding moduli spaces of metrics, i.e., quotients of the former by the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure ‘how many’ different metrics and geometries the given manifold actually does exhibit.

In my talk, I will provide a short introduction to the subject with a focus on lower curvature bounds and then discuss non-traditional approaches to these objects via Gromov-Hausdorff convergence techniques which, in particular, also allow for the study of natural compactifications. Moreover, I will describe the moduli spaces of flat metrics on closed manifolds in a new way via symmetric spaces and, if time permits, also discuss some potential applications arising from this.

Add to calendar :

Date/Time : 21 Sep 2023 15:30
Venue : S17 #04-06 (SR1)
Speaker : Tan Ser Peow, National University of Singapore

The congruence subgroup Gamma_0(N) is the subgroup of the modular group PSL(2,Z) where the lower left-hand entry is congruent to 0 mod N. By a theorem of Kurosh, since PSL(2,Z) is freely generated by an element of order 2 and an element of order 3, any finite index subgroup of PSL(2,Z) can be freely generated by elements of order 2,3 and infinity, including Gamma_0(N).

Kulkarni used an algebraic-geometric method to give an algorithm for finding these generators, and he observed that when N is a prime less than 100, these generators can be chosen so that the lower left-hand entry is either N or 0. He conjectured that this is true for all primes. We call such a generating set an optimal generating system and we will discuss in this talk how to prove Kulkarni’s conjecture, with extensions to cases where N is the square of a prime. This is joint work with Nhat Minh Doan, Sanghyun Kim and Mong Lung Lang.

Add to calendar :