Date/Time : 11 Nov 2020 14:00
Venue : via Zoom
Speaker : Zhu Chengbo, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/89815068389?pwd=UXpnNXFNczNvTzhza3BqTVRHTllBQT09 Meeting ID: 898 1506 8389 Password: 543629 Abstract:  The idea that the study of finite dimensional representations of compact Lie groups amounts to the study of rational representations of reductive algebraic groups can be traced back to Hermann Weyl in the 1930’s. It was I.M. Gelfand who advocated that infinite-dimensional representations of non-compact Lie groups should be investigated by methods of functional analysis, and more specifically theory of distributions, in his ICM Plenary Lecture in 1954 (Some aspects of functional analysis and algebra) and in the six-volume work by I.M. Gelfand et al. on “Generalized Functions” (1960’s). Through two angles, I will explain why one needs to and how one applies, the theory of distributions. 1. Models of representations of the group SL(2,R); 2. Basic invariants of smooth representations: matrix coefficients and characters

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Date/Time : 11 Nov 2020 16:00
Venue : via Zoom
Speaker : Cai Zhenning, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/83949543700?pwd=MVBqSlBTaUtwRmxWRndta0NZZ2tFZz09 Meeting ID: 839 4954 3700 Password: 380740 Abstract: The complex Langevin (CL) method is a numerical approach to alleviate the numerical sign problem in the computation of path integrals in lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high-dimensional and oscillatory integrals with the form of an ensemble average. The method was developed in 1980s. However, after it was proposed, it had very few applications due to its subtle nature. It is often observed that the CL method converges but the limiting result is incorrect. Less than one decade ago, an improvement of the CL method, called gauge cooling, was proposed, after which the CL method acquired much more attention and was later successfully applied to a number of fields including finite density quantum chromodynamics, superstring theory, and the spin-orbit coupling. In this talk, I will explain the basic idea of the CL method, the reason of its failure, the mechanism and new development of gauge cooling, and the current limitation of the method.

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Date/Time : 11 Nov 2020 09:00
Venue : via Zoom
Speaker : Yu Hui, Columbia University

Join Zoom Meeting https://nus-sg.zoom.us/j/85408305509?pwd=OHpHQkdtUSt3YzdLN1BTVkxOeTRXQT09 Meeting ID: 854 0830 5509 Password: 843914 Abstract:  The N-membrane problem is the study of shapes of elastic membranes being pushed against each other. The main questions are the regularity of the functions modeling the membranes, and the regularity of the contact regions between consecutive membranes. These are classical questions in free boundary problems. However, very little is known when N is larger than 2. In this case, there are multiple free boundaries that cross each other, and most known techniques fail to apply. In this talk, we discuss, for general N, the optimal regularity of the solutions in arbitrary dimensions, and a classification of blow-up solutions in 2D. Then we focus on the regularity of the free boundaries when N=3. We also discuss how the techniques developed here can be applied to other problems involving multiple free boundaries. This talk is based on two recent joint works with Ovidiu Savin (Columbia University).

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Date/Time : 09 Nov 2020 09:00
Venue : via Zoom
Speaker : Yang Yunan, New York University

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https://nus-sg.zoom.us/j/83496316224?pwd=STdTeDhKQk9pTnlLVFREaWs5UzFkZz09 Meeting ID: 834 9631 6224 Password: 665418

Optimal transport has been one interesting topic of mathematical analysis since Monge (1781). The problem’s close connections with differential geometry and kinetic descriptions were discovered within the past century, and the seminal work of Kantorovich (1942) showed its power to solve real-world problems. Recently, we proposed the quadratic Wasserstein distance from optimal transport theory for inverse problems, tackling the classical least-squares method’s longstanding difficulties such as nonconvexity and noise sensitivity. The work was soon adopted in the oil industry. As we advance, we discover that the advantage of changing the data misfit is more general in a broader class of data-fitting problems by examing the preconditioning and “implicit” regularization effects of different mathematical metrics as the objective function in optimization, as the likelihood function in Bayesian inference, and as the measure of residual in numerical solution to PDEs.

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Date/Time : 06 Nov 2020 15:00
Venue : via Zoom
Speaker : Ya Deng, Institut des Hautes Études Scientifiques

Join Zoom Meeting Meeting ID: 922 6123 0313 Password: 505350 Abstract: The study of hyperbolicity is intimately related to fascinating questions of geometry (no entire curves) and number theory (finiteness of rational points). The aim of this talk is to present recent progress on the geometric side of the problem. A conjecture by Kobayashi in 1970 states that the complement of a general hypersurface in the complex projective space of high degree is Kobayashi hyperbolic. I will first report a joint work with Damian Brotbek in 2018, in which we confirmed Kobayashi’s conjecture with an effective degree bound. In the second part, I will report my recent work on the hyperbolicity for moduli spaces of polarized manifolds, and for period domains of variation of Hodge structures. This includes a solution of a conjecture by Viehweg-Zuo in 2003.

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Date/Time : 05 Nov 2020 10:00
Venue : via Zoom
Speaker : Rahul Dalal, University of California, Berkeley

Join Zoom Meeting https://nus-sg.zoom.us/j/94719446250?pwd=bzRVL1Z6blh1RUViMTUzVnFFblllUT09 Meeting ID: 947 1944 6250 Password: 022183 Abstract:  Since Automorphic representations for general groups are very difficult to study individually, they are often studied in families instead. The Arthur-Selberg trace formula lends itself very naturally to answering questions about averages of various parameters of the local components of automorphic representations in so-called harmonic families. In their 2016 work, Shin and Templier realized that, in the special case of representations with discrete series at infinity, the trace formula simplified dramatically enough to compute statistics with good error bounds. These bounds were good enough for applications: an averaged Sato-Tate law analogous to Sato-Tate for families of elliptic curves and computations of the specific random-matrix statistics followed by low-lying zeros of L-functions in the family. Shin-Templier’s work used the invariant trace formula which disallowed families that distinguish representations with infinite component in the same L-packet. However, which member of this L-packet a representation might correspond to determines some important characteristics–whether the representation is holomorphic or quaternionic for example. I will discuss how methods related to the stable trace formula can remove this restriction. The key idea is applying a certain “hyperendoscopy” formulation of stabilization used first by Ferrari, though many technical difficulties come up.

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Date/Time : 04 Nov 2020 17:00
Venue : via Zoom
Speaker : Andre Nies, University of Auckland

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract: The K-trivial sets are antirandom in the sense that the initial segment complexity in terms of prefix-free Kolmogorov complexity K grows as slowly as possible. Chaitin introduced this notion in about 1975, and showed that each K-trivial is Turing below the halting set. Shortly after, Solovay proved   that a K-trivial set can be noncomputable. In the past two decades, many alternative characterisations of this class have been found: properties such as  being low for K, low for Martin-Löf (ML) randomness, and a basis for  ML randomness, which state in one way or the other that the set is close to computable. Initially, the class of noncomputable K-trivials appeared to be  amorphous. More recently, evidence of an internal structure has been found. Most of these results can be phrased in the language of a mysterious  reducibility on the K-trivials which is weaker than Turing’s: A is ML-below B if each ML-random computing B also computes A. Bienvenu, Greenberg, Kucera, Nies and Turetsky (JEMS 2016) showed that there an ML-complete K-trivial set. Greenberg, Miller and Nies  (JML, 2019) established   a dense hierarchy of subclasses of the K-trivials based on fragments of Omega computing the set, and each such subclass is an initial segment for ML.  More recent results  generalise these approaches using so-called cost functions.  They also show that each K-trivial set  is ML-equivalent to a c.e. K-trivial. Alternative reducibilities on the K-trivials will be considered near the end of the talk. One of them is related to the extreme lowness notion of strong jump traceability, and appears to be orthogonal to ML-reducibility. Very recent work with Greenberg and Turetsky provides manyfold characterisations in terms of cost functions, and random oracles.

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Date/Time : 04 Nov 2020 09:00
Venue : via Zoom
Speaker : Luna Lomonaco, IMPA, Rio de Janeiro - Brazil

Join Zoom Meeting https://nus-sg.zoom.us/j/88295756539?pwd=RGVDZGNJUFJMdk1LbGJaQUFVVmY2dz09 Meeting ID: 882 9575 6539 Password: 608485 Abstract: Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere (implicit maps sending z to w). The iteration of such multi-valued map generates a dynamical system on the Riemann sphere (dynamical system which generalise rational maps and finitely generated Kleinian groups). We consider a specific 1-(complex)parameter family of (2:2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every a in the connectedness locus M_{\Gamma}, this family is a mating between the modular group and rational maps in the family Per_1(1); we develop for this family a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials; and we show that M_{\Gamma} is homeomorphic to the parabolic Mandelbrot set M_1. This is joint work with S. Bullett.

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Date/Time : 04 Nov 2020 16:00
Venue : via Zoom
Speaker : Katharina Schratz, Sorbonne Université

Join Zoom Meeting  https://nus-sg.zoom.us/j/88145005740?pwd=RjVlcmtKRU5iSmV0KysrY0FNYjdZdz09 Meeting ID: 881 4500 5740  Password: 233926  Abstract: In recent years, a large toolbox of numerical schemes for dispersive equations has been established, based on different discretisation techniques such as discretising the variation of constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low-regularity and/or with high oscillations. Classical schemes fail indeed to capture the oscillatory nature of the solution, a fact that leads to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of these new schemes is to tackle and deeply embed the underlying structure of resonances into the numerical discretisation. As in the continuous case, these resonances are central to structure preservation and provide the new schemes with strong geometric properties at low regularity.

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