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 : 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 : 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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Date/Time : 27 Jan 2021 17:00
Venue : via Zoom
Speaker : Simone Diverio, Roma Sapienza

Join Zoom Meeting https://nus-sg.zoom.us/j/82968914141?pwd=ckgvWDNRZE5nblJKNS9KUGZidkw1UT09 Meeting ID: 829 6891 4141 Password: 841138 Abstract: In the last few years there has been a renewed interest around an old conjecture by Griffiths characterizing which should be the positive characteristic forms for any given Griffiths positive holomorphic Hermitian vector bundle. According to this conjecture, they should be precisely the characteristic forms belonging to the positive cone spanned by the Schur forms. After recalling the various notions of positivity for holomorphic Hermitian vector bundles, and how they are (or should be) related, we shall explain a recent result obtained in collaboration with my PhD student F. Fagioli, which gives a partial confirmation of the above conjecture. Such a result is obtained as a consequence of a pointwise, differential-geometric Gysin formula for the push-forward of the curvature of the tautological line bundles over flag bundles.

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Date/Time : 22 Jan 2021 09:00
Venue : via Zoom
Speaker : Amol Aggarwal, Columbia University

Join Zoom Meeting Meeting ID: 896 0780 2118 Password: 815869 Abstract: Lévy matrices are symmetric matrices whose entries are random variables with infinite variance; they are governed by a parameter \alpha \in (0, 2) dictating the power law decay of their entries. For \alpha < 1, they are believed to serve as one of the few examples of a matrix model exhibiting a mobility edge, also called an Anderson transition, that separates chaotic (GOE) eigenvalue spacing statistics from ordered (Poisson) ones. In this talk we describe results concerning the statistics for the eigenvalue spacings and eigenvector entries of Lévy matrices. In particular, for \alpha \in (1, 2) their eigenvalue statistics asymptotically follow those of the GOE throughout the spectrum, and for \alpha < 1 the same statement holds around small eigenvalues. These describe joint works with Patrick Lopatto, Jake Marcinek, and Horng-Tzer Yau.

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Date/Time : 21 Jan 2021 09:00
Venue : via Zoom
Speaker : Chao Li, Columbia University

Join Zoom Meeting https://nus-sg.zoom.us/j/84339085667?pwd=ZDZvZkZPMEJiSW1HcDNNNnVIbHlSQT09 Meeting ID: 843 3908 5667 Password: 816448 Abstract: For certain automorphic representations π on unitary groups, we show that if L(s, π) vanishes to order one at the center s=1/2, then the associated π-localized Chow group of a unitary Shimura variety is nontrivial. This proves part of the Beilinson-Bloch conjecture for unitary Shimura varieties, which generalizes the BSD conjecture. Assuming the modularity of Kudla’s generating series of special cycles, we further prove a precise height formula for L'(1/2, π). This proves the conjectural arithmetic inner product formula, which generalizes the Gross-Zagier formula to Shimura varieties of higher dimension. We will motivate these conjectures and discuss some aspects of the proof. This is joint work with Yifeng Liu.

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Date/Time : 20 Jan 2021 16:00
Venue : via Zoom
Speaker : Marko Weber, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/82724882064?pwd=RFdxUWF4REdhblJKWi9vazJzUW9TQT09&from=addon Meeting ID: 827 2488 2064 Password: 167307 Abstract: We explicitly solve an equilibrium model in which several long-lived agents with heterogenous absolute risk-aversion and discount rates make consumption and investment decisions, trading an asset that pays a dividend whose growth rate randomly fluctuates over time. We analyze equilibrium consumption shares, stock price, risk-free rate, and optimal trading strategies. We introduce a novel technique to study the market equilibrium in an incomplete model.

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Date/Time : 20 Jan 2021 16:00
Venue : via Zoom
Speaker : Frédéric Naud, Institut de Mathématiques de Jussieu - Paris 6

Join Zoom Meeting https://nus-sg.zoom.us/j/85799518051?pwd=N0JDU2pGbXAwcThKZVE3UEh5MmJXZz09 Meeting ID: 857 9951 8051 Password: 336911 We will define a notion of random covers of hyperbolic surfaces (compact or non- compact) through random representations of their fundamental groups. We will then show how some tools from group combinatorics can be used to study, in a probabilistic sense, the “low spectrum” of these surfaces in the large euler characteristic limit. Joint work with Michael Magee and Doron Puder.

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Date/Time : 18 Jan 2021 09:00
Venue : via Zoom
Speaker : Sourav Sarkar, University of Toronto

Join Zoom Meeting https://nus-sg.zoom.us/j/85233351827?pwd=U2dESEhORVN4eHJwZlhNZEt0MEE3Zz09 Meeting ID: 852 3335 1827 Password: 149940 Abstract: Universality in disordered systems has always played a central role in the direction of research in Probability and Mathematical Physics, a classical example being the Gaussian universality class (the central limit theorem). In this talk, I will describe a different universality class for random growth models, called the KPZ universality class. Since Kardar, Parisi and Zhang introduced the KPZ equation in their seminal paper in 1986, the equation has made appearances everywhere from bacterial growth, fire front, coffee stain to the top edge of a randomized game of Tetris; and this field has become a subject of intense research interest in Mathematics and Physics for the last 15 to 20 years. The random growth processes that are expected to have the same scaling and asymptotic fluctuations as the KPZ equation and converge to the universal limiting object called the KPZ fixed point, are said to lie in the KPZ universality class, though this KPZ universality conjecture has been rigorously proved for only a handful of models till now. Here, I will talk about some universal geometric properties of the KPZ fixed point and show that the KPZ equation and exclusion processes converge to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ universality conjecture for these models, which were long-standing open problems in this field.

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