Date/Time : 03 Nov 2021 16:00
Venue : via Zoom
Speaker : Zhong Guolei, National University of Singapore

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract:  We consider a projective klt pair (X,D) of arbitrary dimension with the anti-log canonical divisor -(K+D) being strictly nef. In this talk, I will first show that such X is rationally connected. In particular, when X is a threefold, we show that -(K+D) is indeed ample, which gives a positive answer to the singular version of Campana-Peternell’s conjecture (for the threefold case). Finally, we extend our first result to the (relative) fibration case and study the geometric structure of this fibration. This is a joint work with Jie Liu, Wenhao Ou, Juanyong Wang and Xiaokui Yang.

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Date/Time : 27 Oct 2021 16:00
Venue : via Zoom
Speaker : Beatrice Pozzetti, University of Heidelberg

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract:  Surprisingly there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups only consisting of injective representations with discrete image. Guichard and Wienhard introduced the notion of $\Theta$ positive representations as a conjectural framework to explain this phenomena. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of $\Theta$ positive representations in PO(p,q). As an application we deduce that they indeed form connected components of character varieties.

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Date/Time : 13 Oct 2021 16:00
Venue : via Zoom
Speaker : Jonas Beyrer, University of Bonn

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Meeting ID: 875 1147 6637
Passcode: 468399

Abstract:  The study of higher rank Teichmüller spaces has attracted a lot of attention in the last 20 years. Those spaces are connected components of the character variety Hom(H,G)/G that consist entirely of discrete and faithful representations; here G is a Lie group and H the fundamental group of a closed surface. Those components generalize classical Teichmüller space, the case G=PSL(2,R), which also identifies with the moduli space of hyperbolic structures on the surface. Representations in higher rank Teichmüller spaces, so called „positive representations“, are not only discrete and faithful, but have many other striking properties and similarities to hyperbolizations (e.g. constant entropy, collar lemmas, etc). However surprisingly there are many representations that are not in higher rank Teichmüller spaces and still share many of this striking properties: representations that satisfy a „partial positivity“. In this talk on joint work with Beatrice Pozzetti I want to discuss partially positive representations and their main difference to representations in higher rank Teichmüller spaces: degenerations.

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Date/Time : 06 Oct 2021 16:00
Venue : via Zoom
Speaker : Purvi Gupta, Indian Institute of Science, Bangalore

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Meeting ID: 875 1147 6637
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Abstract: A compact subset of $\mathbb{C^n}$ is said to be polynomially convex if it is defined by a family of polynomial inequalities. Polynomial convexity grants certain approximation-theoretic properties to the underlying set. When the set is a real submanifold of $\mathbb{C^n}$, its convexity properties are partly influenced by its topology, and the local and global structure of its CR (complex-real) singularities. The minimum complex dimension into which all compact real manifolds of a fixed dimension admit smooth polynomially convex embeddings is not known (although there are some obvious bounds). In this talk, we will discuss some recent improvements on the previously known bounds, especially focusing on the odd-dimensional case, where the embeddings cannot be produced by classical (local) perturbation techniques. This is joint work with R. Shafikov.

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Date/Time : 30 Sep 2021 10:00
Venue : via Zoom
Speaker : Brian Collier, University of California Riverside

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Meeting ID: 875 1147 6637
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Abstract:  In this talk, we will define a special class of nilpotent elements of complex semisimple Lie algebra and describe how a Slodowy slice construction defines connected components of moduli spaces of G-Higgs bundles on a compact Riemann surface for specific real groups G. Through the nonabelian Hodge correspondence, these components define connected components of the G-character variety with many nice features. We will then describe how these nice features, together with recent work of Guichard-Labourie-Wienhard on positive representations imply that such components are higher Teichmuller spaces, that is, they consist entirely of discrete and faithful representations. In the remaining time, we will describe what remains to be proven to finish the classification of higher Teichmuller spaces.

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Date/Time : 08 Sep 2021 16:00
Venue : via Zoom
Speaker : Duc-Manh Nguyen, University of Bordeaux

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Meeting ID: 875 1147 6637
Passcode: 468399

Since the work of Eskin-Okounkov in 2001, it has been known that in any stratum of translation surfaces, the number of square-tiled surafces constructed from at most n squares grows like $c\pi^{2g}n^d$, where $d$ is the (complex) dimension of the stratum, $g$ is the genus of the surfaces, and $c$ is a rational number. Similar phenomena also occur in strata of quadratic differentials. Counting square-tiled surfaces in a given stratum is more or less the same as counting quadrangulations of a topological surface, with some constraints on the singularities and the holonomy of the associated flat metric. More recently, Engel showed that the asymptotics of the numbers of quadrangulations and triangulations with at most $n$ tiles are of the form $\alpha n^d$, where $\alpha$ is a constant in $Q[\pi]$ or $Q[\sqrt{3}\pi]$. In this talk, we will explain how the asymptotics above is related to the volume of some moduli spaces. Moreover, in some situations they can be computed from the self-intersection number of a divisor in some complex projective varieties. This implies in particular that the constant $\alpha$ belongs actually to either $Q\cdot\pi^d$, or  $Q\cdot(\sqrt{3}\pi)^d$ in those cases. This is joint work with Vincent Koziarz.

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Date/Time : 01 Sep 2021 16:00
Venue : via Zoom
Speaker : Elisha Falbel, Mathematics Institute of Jussieu–Paris Rive Gauche (IMJ-PRG)

Join Zoom Meeting https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09 Meeting ID: 875 1147 6637 Passcode: 468399 A path structure on a three-manifold is a contact structure (with fixed contact form) with a choice of two directions on each contact plane. It appears, for instance, in the theory of Anosov flows. I will explain how to classify such structures on a compact manifold with a large automorphism group extending a classification result of Ghys for Anosov flows. This is joint work with M. Mion-Mouton and J. M. Veloso.

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Date/Time : 17 Feb 2021 10:00
Venue : via Zoom
Speaker : Luo Yusheng, University of Michigan

Join Zoom Meeting https://nus-sg.zoom.us/j/83866564646?pwd=UnBpaEZqTjc1aXlZeGJZaEZCS2dqdz09 Meeting ID: 838 6656 4646 Password: 332346 Abstract:  The Sullivan dictionary provides a conceptual framework for understanding the connections between dynamics of rational maps and Kleinian groups. In this talk, we will explain how an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps can be established. As an application, we show how this dynamical correspondence allows us to solve mating problems. We further establish a connection on the parameter/deformation space by proving an analogue of Thurston’s compactness theorem for acylindrical hyperbolic 3-manifolds in the rational map setting. If time permits, we will also talk about some open questions and further studies.

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Date/Time : 10 Feb 2021 17:00
Venue : via Zoom
Speaker : Thomas Peternell, University of Bayreuth

Join Zoom Meeting https://nus-sg.zoom.us/j/84061304664?pwd=NEkvd0V6djhFK3dtelBPbG41V3BTZz09 Meeting ID: 840 6130 4664 Password: 903906 Abstract:  I will discuss a Chern class criterion to when a semistable sheaf on a mildly singular variety is projectively flat and apply this to the uniformization of varieties with nef anticanonical bundles („ singular semipositive Ricci“).

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