Date/Time : 16 Sep 2020 14:00
Venue : via Zoom
Speaker : Jialun Li , University of Zürich, Switzerland

Join Zoom Meeting https://nus-sg.zoom.us/j/96068904418?pwd=YkY1YmhEaHlvMk9TbERJSEllMFJ5UT09 Meeting ID: 960 6890 4418 Password: 788160 Abstract: Let mu be a Borel probability measure on SL2(R) with a finite exponential moment, such that the support of mu generates a Zariski dense subgroup in SL2(R). We can define a unique probability measure on the circle, which is called the mu stationary measure or Furstenberg measure. We will prove, using Bourgain’s discretized sum-product estimate, that the Fourier coefficients of this measure go to zero with a polynomial speed. Starting from this result, we can obtain a spectral gap of the transfer operator, whose properties enable us to get an exponential error term in the renewal theorem in the context of random products of matrices.

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Date/Time : 09 Sep 2020 11:00
Venue : via Zoom
Speaker : Hsueh-Yung Lin, Kavli-IPMU, Japan

Join Zoom Meeting https://nus-sg.zoom.us/j/95748660096?pwd=M2Y1eVJvb01iTzQrZ29CNGZuREM4UT09 Meeting ID: 957 4866 0096 Password: 871058 Abstract: Let X be a compact Kähler manifold. The celebrated Kodaira embedding theorem asserts that if the Kähler cone of X contains a rational cohomology class, then X is projective. Instead of considering Kähler classes, we will study the algebraicity of X when X carries a 1-dimensional positive rational Hodge class. We will first discuss and formulate some precise problems (depending on the notion of positivity) then focus on the following results: 1) If the interior of the dual Kähler cone of X contains a rational class, then the Picard variety of X is projective. 2) If the interior of the dual pseudoeffective cone of a compact Kähler threefold X contains a rational class, then X is projective.

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Date/Time : 02 Sep 2020 10:00
Venue : via Zoom
Speaker : Hongtaek Jung , Institute for Basic Science - South Korea

Join Zoom Meeting https://nus-sg.zoom.us/j/94193089627?pwd=SGZ6cUtXNnp6ejFpZUpGWHFkalM5dz09 Meeting ID: 941 9308 9627 Password: 788615 Abstract: It is known that the PSL(n,R)- Hitchin components of closed surfaces of negative Euler characteristic admit global Darboux coordinates with respect to the Goldman symplectic form. I will discuss an analogous result for the “PSL(3,R)-Hitchin components”  or the deformation spaces C(O) of convex projective structures of closed orbifolds O of negative Euler characteristic.  For this, I will present how to construct the Goldman type symplectic form on C(O).  Then I will introduce two ingredients to construct the global Darboux coordinates: a decomposition theorem that splits C(O) into simpler symplectic manifolds and an action-angle principle. This is joint work with Suhyoung Choi.

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Date/Time : 26 Aug 2020 10:00
Venue : via Zoom
Speaker : Li Qiongling , Chern Institute of Mathematics, Nankai University

Join Zoom Meeting https://nus-sg.zoom.us/j/91329086826?pwd=RFNXQzROaDhicTdKQkxkeWxpL3lhUT09 Meeting ID: 913 2908 6826 Password: 648142 We start this talk with some basic linear algebras. Consider m(A)=[A,A*] on sl(n,C). Ness studied the (normalized) square of norm |m|^2 and showed some nice properties of |m|^2 which characterize the unitary orbits inside the nilpotent orbits of sl(n,C). We generalize Ness’ results to general orbits by modifying |m|^2, motivated by the curvature formula of symmetric spaces. As geometric applications, we show some domination results in n-Fuchsian fibers in the Hitchin fibration of the moduli space of Higgs bundles over a Riemann surface. And we also show some existence and uniqueness results of equivariant minimal/maximal surfaces in certain product spaces. This is joint work with Song Dai.    

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Date/Time : 19 Aug 2020 10:00
Venue : via Zoom
Speaker : Junho Peter Whang, MIT

Join Zoom Meeting https://nus-sg.zoom.us/j/94766702852?pwd=UG9SclBVcTVlVDFnSnRQcFJjaUw4QT09 Meeting ID: 947 6670 2852 Password: 569325 Abstract: Moduli spaces for special linear rank two local systems (with prescribed boundary traces) on a topological surface are basic objects in geometry. The study of integer points on these algebraic varieties can be traced back to 1880 work of Markoff, in the case where the surface is a one-holed torus. For general surfaces, we will show that the moduli spaces are log Calabi-Yau, and use mapping class group dynamics and differential geometric tools to establish a structure theorem for their integer points. In addition, we will also give an analysis of algebraic curves on the moduli spaces. Throughout the talk, emphasis will be put on the interplay between the dynamical, arithmetic, and topological aspects of surfaces and their local systems.  

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Date/Time : 29 Jan 2020 14:00
Venue : S17 #05-11 SR5
Speaker : Zhong Guolei , NUS

Date/Time:29 Jan 2020 14:00 Venue: S17 #05-11 SR5 Speaker: Zhong Guolei, NUS Int-Amplified endomorphisms of compact Kähler spaces Let X be a normal compact Kähler space of dimension n. A surjective endomorphism f of such X is int-amplified if f *ξ – ξ = η for some Kähler classes ξ and η. First, we show that this definition generalizes the notation in the projective setting. Second, we prove that for the cases of X being smooth, a surface or a threefold with mild singularities, if X admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a Q-torus. Finally, we consider a normal compact Kähler threefold Y with only terminal singularities and show that, replacing f by a positive power, we can run the minimal model program (MMP) f-equivariantly for such Y and reach either a Q-torus or a Fano (projective) variety of Picard number one. Add to calendar: 

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Date/Time : 22 Nov 2019 14:00
Venue : S17 #04-04 SR3
Speaker : Brice Loustau, TU Darmstat

Date/Time:22 Nov 2019 14:00 Venue: S17 #04-04 SR3 Speaker: Brice Loustau , TU Darmstat Computing harmonic maps between Riemannian manifolds The theory of harmonic maps between Riemannian manifolds goes back to the foundational work of Eells-Sampson and Hartman. Using the heat flow method, i.e. the gradient flow for the energy functional, they showed existence and uniqueness of harmonic maps between compact Riemannian manifolds when the target has negative sectional curvature. In order to compute such harmonic maps effectively, we discretize the theory by introducing meshes of the domain manifold, which consist of a geodesic triangulation with vertex and edge weights. By studying the convexity properties of the energy functional, we show that the discrete heat flow method converges to unique discrete harmonic maps. Moreover, under suitable assumptions, the discrete theory converges back to the smooth theory when taking finer and finer meshes. In particular, the smooth harmonic map can be obtained as the limit of discrete harmonic maps, equivalently as the double limit of the discrete heat flow when iterating both the time and space discretization steps. Time permits, we will also discuss center of mass methods to compute harmonic maps. We feature a concrete illustration of these methods withHarmony, a computer software with a graphical user interface that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps between hyperbolic disks. Add to calendar: 

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Date/Time : 06 Nov 2019 16:00
Venue : S17 #06-11
Speaker : Jonathon Teo , National University of Singapore

Date/Time:06 Nov 2019 16:00 Venue: S17 #06-11 Speaker: Jonathon Teo, National University of Singapore Branes in the moduli space of parabolic Higgs bundles In this talk, we will be looking at some constructions of families of branes in the moduli space of stable parabolic Higgs bundles. We will first be introducing the moduli space of rank 2 stable parabolic Higgs bundles over the complex projective curve, which is a hyperkahler manifold and contains the cotangent bundle of the moduli space of stable parabolic bundles. The family of branes are constructed from complex or Lagrangian submanifolds of the moduli space of stable parabolic bundles. Thus, a deformation of the submanifolds in the moduli space of stable parabolic bundles gives us a deformation of branes. We will also show that this deformation space is degenerate. Add to calendar: 

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Date/Time : 30 Oct 2019 16:00
Venue : S17 #06-11
Speaker : Xu Binbin , University of Luxembourg

Date/Time:30 Oct 2019 16:00 Venue: S17 #06-11 Speaker: Xu Binbin, University of Luxembourg Ambartzumian-Pleijel identity in hyperbolic geometry Given a convex compact domain in Euclidean plane with C1 boundy, a measure on the space of all straight lines invariant under the isometry group of Euclidean plane induces a distribution on the positive real line, by considering the length of the intersection of straight lines with this domain. Ambartzumian-Pleijel identity is about computing the integral of any real valued C1 function on the positive real line with respect to this distribution. In this talk, we’ll introduce the counterpart of this identity in hyperbolic geometry. Then we will discuss some of its applications. In particular, it can be used to prove Croftin formula and the isoperimetric inequality in hyperbolic geometry. I’ll also discuss some of its generalizations in other context. Add to calendar: 

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