Date/Time : 25 Jan 2022 16:00
Venue : via Zoom
Speaker : Feng Zhu, Technion - Israel Institute of Technology

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https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: Putting a hyperbolic metric on a closed surface gives us a linear representation (the holonomy representation) with many nice geometric and dynamical properties. Anosov representations generalise this picture, or more generally the theory of convex cocompact subgroups of rank-one Lie groups such as PSL(2,R) or PSL(2,C), to higher-rank Lie groups such as PSL(d,K) for d>2. They give us an open subset of the space of representations consisting of quasi-isometric embeddings of word-hyperbolic groups.

There are many equivalent characterisations of Anosov representations; in particular, Labourie’s original definition uses a flow coming from the hyperbolic group to obtain additional control over the image subgroup. In this talk, I will introduce a relative notion of Anosov representations, similarly formulated in terms of flows, which allows for (certain) parabolic elements / controlled breakdowns of hyperbolicity.

This generalises work of Canary–Zhang–Zimmer and is joint work in progress with Andrew Zimmer.

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Date/Time : 25 Jan 2022 15:00
Venue : via Zoom
Speaker : Haiping Yang, Imperial College London

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https://nus-sg.zoom.us/j/82929931113?pwd=dFlvb3U4SU9ta05jdCtPdWQrOVFodz09

Meeting ID: 829 2993 1113
Passcode: 528816

Abstract: The category of D-mods was introduced in the 1970s and has been a huge area of interest. While it is interesting on its own right, it also has many applications in representation theory and geometry. In particular, holonomicity can be used to show finiteness of certain invariants. For example, finiteness of Skein modules of 3-manifolds [GJS] and finiteness of Poisson (co)homology of certain Poisson algebras [ES, PS].

In the first part of this talk, we are going to introduce the category of D-modules on a (not necessarily smooth) variety. In particular, we will see the derived category is compactly generated, and therefore construct a derived equivalence to the category of dg-modules over an explicit dg-ring. When the variety has only cuspidal singularities, we show that the equivalence is the derived version of the abelian equivalence described by Ben-Zvi and Nevins.

In the second part of the talk, we are going to explore the holonomicity idea and define Hochschild-de Rham homology. This version of Hochschild homology behaves better for a symplectic singularity. In particular, we are going to see a version of Nest-Tsygen theorem for symplectic singularities.

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Date/Time : 24 Jan 2022 10:00
Venue : via Zoom
Speaker : Zhiwen Zhang, The University of Hong Kong

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https://nus-sg.zoom.us/j/83299433896?pwd=UnpRbldHK2pKZXN2QVl2bFcxUXJwQT09

Meeting ID: 832 9943 3896
Passcode: 783017

Abstract: We study the propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficient on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting method for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical method. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space.

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Date/Time : 19 Jan 2022 16:00
Venue : via Zoom
Speaker : Frédéric Rousset, Université Paris-Sud

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https://nus-sg.zoom.us/j/89641856972?pwd=dmVFYXo1bXQ5UGxXZCtHYzdWbTh3UT09

Meeting ID: 896 4185 6972
Passcode: 203052

Abstract: We shall study the well-posedness of singular kinetic equations of Vlasov type arising in plasma physics (quasineutral limit) or fluid mechanics (variational solutons of the incompressible Euler equation). Singular means that the force field is given by the gradient of a macroscopic quantity (the density for example) without smoothing. A stability condition on the profile of the initial data is necessary for well-posedness in Sobolev spaces. (joint work with D. Han-Kwan (Ecole Polytechnique))

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Date/Time : 19 Jan 2022 16:00
Venue : via Zoom
Speaker : Ye Jinhe, Institut de Mathematiques de Jussieu - Paris Rive Gauche

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

Abstract: Consider the class of fields with Char(K)=0 and x^4+y^4=1 has only 4 solutions in K, we show that this class has a model companion, which we denote by curve-excluding fields. Curve-excluding fields provides (counter)examples to various questions. Model theoretically, they are model complete. Field theoretically, they are not large and unbounded. Time permitting, we will discuss other aspects such as decidability of such fields.

Joint work with Will Johnson and Erik Walsberg.

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Date/Time : 19 Jan 2022 16:30
Venue : via Zoom
Speaker : Christian Webb, University of Helsinki

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https: https://nus-sg.zoom.us/j/81980150378?pwd=UDdlbHhNSDdEQmFLMUp6bHRSK2dFUT09

Meeting ID: 819 8015 0378
Passcode: 600818

Abstract: Logarithmically correlated fields are stochastic processes whose covariance has a logarithmic singularity on the diagonal. In the past few decades, it has been discovered that such objects play an important role in various probabilistic models ranging in areas ranging from number theory and random matrix theory to quantum field theory and random PDEs. In this talk, I will describe what these objects are and what kind of role they have played in various recent research questions.

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Date/Time : 18 Jan 2022 10:00
Venue : via Zoom
Speaker : Ara Basmajian, City University of New York

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https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract:  There are various contexts for which it is natural to study spaces that look the same from any point of the space (that is, “homogeneous spaces”). For example, the simply connected Riemann surfaces, namely the Riemann sphere, complex plane, and unit disc are conformally homogeneous Riemann surfaces. In fact they are the only ones. On the other hand, given any surface it is not difficult to cook up a diffeomorphism between any two points of the surface. Hence one needs a notion that is not as strong as conformality and not as weak as differentiability. The key observation is that while smooth maps can distort infinitesimal circles to ellipses with unbounded eccentricity (the ratio of the major to the minor axis can be arbitrarily large), conformal maps do not distort infinitesimal circles at all. This leads to the notion of a homeomorphism being K-quasiconformal (has eccentricity bounded by K). Conformal homeomorphisms are 1-quasiconformal.

A Riemann Surface X is said to be K-quasiconformally homogeneous if for any two points x and y on it, there exists a K-quasiconformal self-mapping taking x to y. If such a K exists we say that X is a QCH Riemann surface. After introducing the basics, the focus of this talk will be on Riemann surface structures that are QCH, and their connections to the topology and hyperbolic geometry of the surface.

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Date/Time : 18 Jan 2022 09:00
Venue : via Zoom
Speaker : Gaston Naredo, University of Sydney

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https://nus-sg.zoom.us/j/81663160111?pwd=WnVRdlh0SVV4Z1M1L24rT3F2OXArUT09

Meeting ID: 816 6316 0111
Passcode: 390667

Abstract: In this talk, I will explore the top-heaviness and unimodality behaviors of certain sequences. I will explain (following the work of Björner and Ekedahl) why Deligne’s hard Lefschetz theorem implies the top-heaviness theorem for the Betti numbers of a proper variety X having an algebraic cell decomposition.

The unimodality conjecture is false even for “type A” Schubert varieties. However, I believe the conjecture is true for dominant alcoves in affine Weyl groups. I will show examples of this in ranks 2, 3, and 4. I will show some geometric visualisations in ranks 2 and 3 and explain the research directions I am taking to address this problem.

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Date/Time : 14 Jan 2022 09:00
Venue : via Zoom
Speaker : Alexander Kiselev, Duke University

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https://nus-sg.zoom.us/j/81290641127?pwd=eUowVGZYN3REWndWVHRKK2piQ1pUUT09

Meeting ID: 812 9064 1127
Passcode: 612753

Abstract: The Euler equation describing motion of ideal fluid goes back to 1755. The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its potential connection with the arguably greatest unsolved problem of classical physics, turbulence. Results on small scale and singularity formation in two dimensions and for a number of related models will also be presented.

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