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).

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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.

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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.

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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.

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Date/Time : 14 Sep 2023 15:30
Venue : S17 #05-11 (SR5)
Speaker : Daniel Huson, University of Tubingen

This talk has four parts. In the first part – topology -, we recall the classification of two-dimensional manifolds and orbifolds, and introduce their Conway-Macbeath notation. In the second part – geometry -, we discuss two-dimensional periodic tilings and their classification. In the third part – combinatorics -, we introduce Delaney-Dress symbols and indicate how they can be used to systematically enumerate periodic tilings. Finally, in the fourth part – algorithms and software -, we present Tegula

(https://ddec1-0-en-ctp.trendmicro.com:443/wis/clicktime/v1/query?url=http%3a%2f%2ftegula.husonlab.org&umid=7c8bc7cd-aafa-4ab0-9fe7-ed56d83a50f7&auth=8d3ccd473d52f326e51c0f75cb32c9541898e5d5-113b232b1017a668b66ec7edf1b8b0f54e074fb4),

a new computer program that can be used the explore the “galaxy” of all 2.4 billion periodic tilings of the euclidean plane, sphere and hyperbolic plane, of low Dress-complexity. This is joint work with Olaf Delgado and Rüdiger Zeller.

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Date/Time : 07 Sep 2023 15:30
Venue : S17 #05-11 (SR5)
Speaker : Wei Darren, National University of Singapore

Two flows are said to be Kakutani equivalent if one is isomorphic to the other after time change, or equivalently if there are Poincare sections for the flows so that the respective induced maps are isomorphic to each other. Ratner showed that if G=SL(2,R) and \Gamma is a lattice in G, and if U is a one parameter unipotent subgroup in G then the U action on G equipped with Haar measure is loosely Bernoulli, i.e. Kakutani equivalent to a circle rotation. Thus any two such systems SL(2,R)/\Gamma are Kakutani equivalent to each other. On the other hand, Ratner showed that if G’=SL(2,R) x SL(2,R) and \Gamma’ is a reducible lattice, and U’ is the diagonally embedded one parameter unipotent subgroup in G’, then (G’/\Gamma’, U’, m) is not loosely Bernoulli. We show that in fact in this case and many other situations one cannot have Kakutani equivalence between such systems unless they are actually isomorphic. This is a joint work with Elon Lindenstrauss.

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Date/Time : 17 Aug 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Yap Jit Wu, Harvard University

Let K be a number field, S a finite set of places containing all Archimedean places and f a rational map defined over K of degree d >_ 2. Given a non-preperiodic point a and non-exceptional b, both in P^1(K), Silverman has shown that there are only finitely many points among {a,f(a),f^2(a),…} that are S-integral relative to b. More recently, Hsia and Silverman showed an upper bound of the form O(4^|S|) on the number of such points as S varies. In this talk we will improve this upper bound to |S|^(1+o(1)) and show it is possible to get O(|S|) for certain rational maps.

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Date/Time : 17 Aug 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Jia Jia, Tsinghua University

TopoloAn automorphism of a compact complex space is called wild if there is no non-trivial proper invariant analytic subset. In this talk, I will show that a compact complex surface admitting a wild automorphism is either a complex torus or an Inoue surface of certain type, and this wild automorphism has zero entropy. This is based on a joint work with Long Wang.gy and geometry

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Date/Time : 16 Jun 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Zhu Feng, University of Wisconsin

Putting a hyperbolic metric on a complete finite-type surface gives us a linear representation (the holonomy representation) with many nice geometric and dynamical properties: for instance it is discrete and faithful, and in fact stably quasi-isometrically embedded. This picture can be generalised in (at least) two ways. First, the notion of geometric finiteness generalises this picture in the context of rank-one Lie groups such as PSL(2,R) or PSL(2,C). Second, Anosov representations generalise this picture to higher-rank Lie groups such as PSL(d,K) for d>2.

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