Date/Time : 10 Feb 2021 14:00
Venue : via Zoom
Speaker : Allan Francis Merino, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/88282572726?pwd=a1QrdVFHSUt0WGxDZER4Uy9za3FmQT09 Meeting ID: 882 8257 2726 Password: 353941 Abstract:  For every irreducible reductive dual pair (G, G’) in Sp(W), R. Howe proved the existence of an isomorphism between the spaces R(G) and R(G’), where R(G) is the set of infinitesimal equivalence classes of irreducible admissible representations of \tilde{G} (preimage of G in the metaplectic group) which can be realized as a quotient of the metaplectic representation. All the representations appearing in the correspondence have a distribution character, and characters are analytic objects completely identifying the irreducible representations. In particular, one natural question is to understand the transfer of characters in the theta correspondence (or Howe’s duality). In 2000, T. Przebinda introduced the Cauchy-Harish-Chandra integral and conjectured that the transfer of characters should be obtained via this map. This conjecture is known to be true if G is compact and was proven by Przebinda for unitary representations in the stable range case. In general, it has been established that the Cauchy-Harish-Chandra integral sends the \tilde{G}-invariant eigendistributions on \tilde{G} into the \tilde{G’}-invariant eigendistributions on \tilde{G’} as long as rk(G) \leq rk(G’). In my talk, after recalling carefully the construction of the Cauchy-Harish-Chandra integral and stating Przebinda’s conjecture, I am going to explain, by using results of A. Paul, how to prove this conjecture for the pair (G,G’) = (U(p, q), U(r, s)), with p+q = r+s, starting with a discrete series representation of \tilde{G}. At the end of my talk, I will discuss some auxiliary results I got in my paper and an ongoing project on transfer of characters for (G, G’), with rk(G) \leq rk(G’), starting from a discrete series representation.

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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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Date/Time : 10 Feb 2021 16:00
Venue : via Zoom
Speaker : Vincent Tan , National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82921718900?pwd=VHVEeWszTUZWT0o5KzQrWXVPbXRxZz09&from=addon Meeting ID: 829 2171 8900 Password: 968735 Abstract: We consider the classical problem of learning tree-structured graphical models but with the twist that the observations are corrupted in independent noise. For the case in which the noise is identically distributed, we derive the exact asymptotics via the use of probabilistic tools from the theory of strong large deviations. Our results strictly improve those of Bresler and Karzand (2020) and Nikolakakis et al. (2019) and demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds. When the noise is non-identically distributed, Katiyar et al. (2020) showed that although the exact tree structure cannot be recovered, one can recover a “partial” tree structure; that is, one that belongs to the equivalence class containing the true tree. We propose Symmetrized Geometric Averaging (SGA), a statistically robust algorithm for partial tree recovery. We provide error exponent analyses and extensive numerical results on a variety of trees to show that the sample complexity of SGA is significantly better than the algorithm of Katiyar et al. (2020). SGA can be readily extended to Gaussian models and is shown via numerical experiments to be similarly superior.

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Date/Time : 05 Feb 2021 09:00
Venue : via Zoom
Speaker : Chenyang Xu, Princeton University

Join Zoom Meeting Meeting ID: 884 1179 8539 Password: 528772 Abstract: From many view points, all varieties are built from three basic types, depending on the sign of their first Chern class. Those with positive first Chern class are historically named after Italian Mathematician G. Fano. For a long time, it is mysterious to people on how to parametrize Fano varieties, due to some fundamental pathological behaviors for families of such varieties. However, in the last decade, it was realized an extra notion called K-stability, which was first introduced by complex geometers to characterize the existence of KE metric, indeed gives a right framework to construct a ‘good’ parametrizing space, called the K-moduli space. In my talk, I will discuss this with an aim make some main ideas accessible to general audiences.

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Date/Time : 04 Feb 2021 16:00
Venue : via Zoom
Speaker : Zhou Xiang, City University of Hong Kong

Join Zoom Meeting https://nus-sg.zoom.us/j/81177164345?pwd=Z1BiMWx1dWI1bU1xdlN5WFl3b3ZJdz09 Meeting ID: 811 7716 4345 Password: 311396 Abstract:  Optimal control problem is typically cast as a nonlinear Hamiltonian-Jacobi-Bellman PDE problem which the value function satisfies. In policy iteration, the nonlinear HJB equation is split into alternative iteration between a linear HJB equation with a given control (policy evaluation) and an point-wise optimization for control variable (policy improvement), but the latter is in fact determined only by the gradient of value function. In this talk, under smoothness assumption, we derive the PDE system for the (vector-valued) gradient of the value function (value-gradient function), which is closed and linear and enjoys a nice component-decoupling property. This PDE system of value-gradient can be solved by the method of characteristics as the linear HJB equation: one curve of characteristics will produce the data for both value and value-gradient. Supplemented by this additional value-gradient data, the value function is then computed by minimizing the sum of two mean square errors between the data and the parametric function approximations. We show by a few numerical examples the improvement of both robustness and accuracy when such value-gradient is taken into account. This is joint work with A. Bensoussan and P. Yam and JY Han.

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Date/Time : 03 Feb 2021 14:00
Venue : via Zoom
Speaker : Allan Francis Merino, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/86961364011?pwd=SU9Xa05IUW9kL1NVY21peGVUUTNHUT09 Meeting ID: 869 6136 4011 Password: 065002 Abstract: For every irreducible reductive dual pair (G, G’) in Sp(W), R. Howe proved the existence of an isomorphism between the spaces R(G) and R(G’), where R(G) is the set of infinitesimal equivalence classes of irreducible admissible representations of \tilde{G} (preimage of G in the metaplectic group) which can be realized as a quotient of the metaplectic representation. All the representations appearing in the correspondence have a distribution character, and characters are analytic objects completely identifying the irreducible representations. In particular, one natural question is to understand the transfer of characters in the theta correspondence (or Howe’s duality). In 2000, T. Przebinda introduced the Cauchy-Harish-Chandra integral and conjectured that the transfer of characters should be obtained via this map. This conjecture is known to be true if G is compact and was proven by Przebinda for unitary representations in the stable range case. In general, it has been established that the Cauchy-Harish-Chandra integral sends the \tilde{G}-invariant eigendistributions on \tilde{G} into the \tilde{G’}-invariant eigendistributions on \tilde{G’} as long as rk(G) \leq rk(G’). In my talk, after recalling carefully the construction of the Cauchy-Harish-Chandra integral and stating Przebinda’s conjecture, I am going to explain, by using results of A. Paul, how to prove this conjecture for the pair (G,G’) = (U(p, q), U(r, s)), with p+q = r+s, starting with a discrete series representation of \tilde{G}. At the end of my talk, I will discuss some auxiliary results I got in my paper and an ongoing project on transfer of characters for (G, G’), with rk(G) \leq rk(G’), starting from a discrete series representation.

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Date/Time : 03 Feb 2021 17:00
Venue : via Zoom
Speaker : Wong Tin Lok, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/96860201432?pwd=cVdwZmd2clVFaEhaTmJjaGdXMFdmdz09 Meeting ID: 968 6020 1432 Password: Is P=NP? Abstract: Arithmetic generally does not admit any non-trivial quantifier elimination. I will talk about one exception, where the negation of an induction axiom is included in the theory. Here the Weak Koenig’s Lemma from reverse mathematics arises as a model completion. This work is joint with Marta Fiori-Carones, Leszek Aleksander Kolodziejczyk and Keita Yokoyama.

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Date/Time : 03 Feb 2021 10:00
Venue : via Zoom
Speaker : Andrew Zimmer, University of Winsconsin-Madison

Join Zoom Meeting https://nus-sg.zoom.us/j/87303863968?pwd=VFhyUkU5V05jM3FxVnhFQTk0UXZkQT09 Meeting ID: 873 0386 3968 Password: 687478 Abstract: A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. When the group is word hyperbolic, these representations are Anosov representations (which have been extensively studied over the past 15 years). The non-word hyperbolic is less understood and in this talk I will describe some recent results about the general structure of the groups that can admit convex co-compact representations. Most of what I will discuss is joint work with Mitul Islam.

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Date/Time : 27 Jan 2021 16:00
Venue : via Zoom
Speaker : Subhroshekhar Ghosh, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/86838713064?pwd=SG5JNGpXN2owVnd5R1lJa0g5SEtFQT09 Meeting ID: 868 3871 3064 Password: 519115 Abstract: The classical paradigm of randomness in the sciences in general, and data science in particular, is the model of independent and identically distributed (i.i.d.) random variables. In fact, going beyond i.i.d. is often considered a difficulty and a challenge to be overcome. In this talk, we will explore a different perspective, wherein stochastic systems with constraints in fact help to understand fundamental problems in the math of data. Our constrained systems are well-motivated from statistical physics, including models like the random critical points and determinantal probability measures. These will be used to shed important light on natural questions of relevance in understanding data, including the problems of constrained likelihood maximization and dimension reduction. En route, we will explore connections to spiked random matrix models in the PCA setting and novel asymptotics for the log likelihood in the context of the classical Wilk’s theorem. Based on multiple joint works with Philippe Rigollet (MIT) and Sanjay Chaudhuri (NUS). No special background assumptions will be made.

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