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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Date/Time : 03 May 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Marco Abate, University of Pisa

In this talk we shall describe the asymptotic behaviour of sequences of functions obtained by iteratively left-composing or right-composing holomorphic self-maps of a hyperbolic Riemann surface $X$. We shall consider in detail two cases: when the holomorphic self-maps to be composed have values in a Bloch subdomain of~$X$; and when the holomorphic self-maps to be composed are sufficiently close to a given self-map. This is a joint work with Argyrios Christodoulou and Ian Short.

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Date/Time : 19 Apr 2023 15:00
Venue : S17 #04-04 (SR3)
Speaker : Lucas Kaufmann, University of Orleans

Consider a (singular) holomorphic foliation F on a complex manifold. In the 1970s, Baum-Bott constructed residue classes associated with each singular component of F. These classes satisfy an index theorem, which computes characteristic classes of F.

A natural question since their work is how to compute or find natural representatives of such cohomology classes. Apart from the case where the foliation is of rank one and has isolated singularities, no general result is known. In this talk I will show that each Baum-Bott class can be naturally represented by a residue current supported by the singular component of F. This is a joint work with Richard Lärkäng and Elizabeth Wulcan.

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