Date/Time : 13 Mar 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Yasushi Yamashita, Chuo University

The Riley slice of Schottky space is a family of Kleinian groups generated by two parabolic elements. It was studied in detail by Keen and Series. In this talk, we consider families of Kleinian groups generated by two elliptic elements, and show some computer generated pictures related to the groups.

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Date/Time : 13 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Lin Bo, National University of Singapore

The committor function is a central object for quantifying the transitions between metastable states of dynamical systems. Recently, a number of computational methods based on deep neural networks have been developed for computing the high-dimensional committor function. The success of the methods relies on sampling adequate data for the transition, which still is a challenging task for complex systems at low temperatures. This talk will present a deep learning method with two novel adaptive sampling schemes. In the two schemes, the data are generated actively from a modified potential where the bias potential is constructed from the learned committor function. We theoretically show that the data in the two schemes are uniformly distributed along the transition tube. The efficiency of the method is illustrated on high-dimensional systems including the alanine dipeptide and a dimer system immersed in the water box.

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Date/Time : 13 Mar 2024 16:00
Venue : S16 #03-09
Speaker : Hampus Nyberg, KTH Stockholm

Deep zero problems consider holomorphic functions satisfying local properties at a small number of points. We take a look at one such problem about uniqueness in the Bargmann-Fock space and show that it reduces down to a question about products such as $\frac{\sin{1}}{\sin{2}}\frac{\sin{3}}{\sin{4}}\frac{\sin{5}}{\sin{6}}…$.

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Date/Time : 07 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Ziyang Gao, Leibniz University Hannover

The goal of this talk is to propose a possible framework to study quadratic relations among holomorphic periods of CM abelian varieties. We define, for each Shimura variety and a CM point, a bi-$\overline{\mathbb{Q}}$-structure on the tangent space. Then we explain that in the case of the Siegel moduli variety, the numbers comparing the two $\overline{\mathbb{Q}}$-structures are precisely the products of the holomorphic periods of the CM abelian varieties parametrized by the CM point (up to $2\pi i$). Next we propose a hyperbolic analytic subspace conjecture, which is the analogue of Wüstholz’s analytic subgroup theorem in this context, and explain why it implies the desired consequence on the quadratic relations among these holomorphic CM periods. This is joint work with Emmanuel Ullmo and (partially with) Andrei Yafaev.

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Date/Time : 06 Mar 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Akiko Takeda, University of Tokyo

Random projection techniques based on the Johnson-Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems while approximately preserving their optimal values, which leads to smaller-scale optimization problems. In this talk, we show a few applications of random matrix techniques for constructing random subspace algorithms that iteratively solve smaller-scale subproblems and discuss their convergence speed. This talk is based on joint works with Ryota Nozawa, and Pierre-Louis Poirion.

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Date/Time : 06 Mar 2024 15:40
Venue : S17 #05-12 (SR4)
Speaker : Pierre-Louis Poirion, University of Tokyo

In this talk, we present a randomized subspace regularized Newton method for a non-convex function. We show that our method has global convergence under appropriate assumptions, and its convergence rate is the same as that of the full regularized Newton method. Furthermore, we can obtain a local linear convergence rate, under some additional assumptions, and prove that this rate is the best we can hope, under local strong convexity sumptions, when using random subspace. Finally, we will prove that under rank deficiency conditions, one can obtain super-linear convergence in the subspace setting.

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Date/Time : 06 Mar 2024 15:30
Venue : S17 #04-05 (SR2)
Speaker : Viet-Anh Nguyen, Universite de Lille

Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $\Sigma=(\Sigma_1,\ldots,\Sigma_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $\tau=(\tau_1,\ldots,\tau_\ell)$ is an $\ell$-tuple of positive real numbers, and let $H^0_0(X,L^p)$ be the space of holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $\tau_jp$ along $\Sigma_j$, $1\leq j\leq\ell$. If $Y\subset X$ is an irreducible analytic subset of dimension $m$, we consider the space $H^0_0 (X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to global holomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet $(L,\Sigma,\tau)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we give a general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When $L$ is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certain equilibrium current on $Y$. We apply this to the study of the equidistribution of zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as $p\to\infty$.

This is a joint-work with Dan Coman and George Marinescu.

Keywords:
(Partial) Bergman kernel function, big line bundle, singular Hermitian metric, holomorphic section, big cohomology class.

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Date/Time : 06 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Li Yifei, National University of Singapore

Designing a numerical scheme that can preserve the geometric structure for anisotropic surface diffusion with an arbitrary anisotropic surface energy is a long-standing problem. In this talk, we propose and analyze a structure-preserving parametric finite element methods (SP-PFEM) for the evolution of a closed curve in 2D, which preserve two geometric structures – area conservation and energy dissipation – at the full-discretized level, The SP-PFEM innovates with a novel surface energy matrix and the Cahn-Hoffman ξ-vector, leading to a new geometric identity for dealing with the weighted mean curvature. This new identity allows our SP-PFEM to be easily extended to various geometric flows with anisotropic effects. Extensive numerical results demonstrate its efficiency, stability, and success in other geometric flows.

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Date/Time : 06 Mar 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Rupert Hoelzl, Universitaet der Bundeswehr, Munich

A left-computable number x is called regainingly approximable if there is a computable increasing sequence (x_n)_{n in N} of rational numbers converging to x such that x-x_n < 2^-n for infinitely many n in N; and it is called nearly computable if there is such an (x_n)_n such that for every computable increasing function s:N -> N the sequence (x_{s(n+1)}-x_{s(n)}){n in N} converges computably to 0.

In this talk we study the relationship between both concepts by constructing on the one hand a non-computable number that is both regainingly approximable and nearly computable, and on the other hand a left-computable number that is nearly computable but not regainingly approximable; it then easily follows that the two notions are incomparable with non-trivial intersection. With this relationship clarified, we then hold the keys to answering an open question of Merkle and Titov: they studied speedable numbers, that is, left-computable numbers whose approximations can be sped up in a certain sense, and asked whether, among the left-computable numbers, being Martin-Loef random is equivalent to being non-speedable. As we show that the concepts of speedable and regainingly approximable numbers are equivalent within the nearly computable numbers, our second construction provides a negative answer.

This is joint work with Philip Janicki.

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