Date/Time : 01 Dec 2022 16:00
Venue : Zoom
Speaker : Daren Wei, Hebrew University of Jerusalem

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https://nus-sg.zoom.us/j/82637054452?pwd=ZkpEVllwN3g1YUNYRHIwMXF3TmUzdz09

Meeting ID: 826 3705 4452
Passcode: 332211

Unipotent flows have some striking rigidity properties: for instance, every orbit is recurrent, and every orbit closure as well as every invariant measure is homogeneous. In particular, the isomorphism rigidity theorem tells us that a measurable isomorphism between two flows in the class of unipotent flows on quotients of semisimple groups implies an algebraic isomorphism between their corresponding groups and lattices. Moreover, such systems always have zero entropy and thus they cannot be distinguished by classical entropy invariants.

We extend the isomorphism rigidity for unipotent flows to its time changes. More precisely, one parameter unipotent flows on quotients of semisimple groups fall into two categories: 1. unipotent flows that are time changes of linear irrational flows on T^2 and hence are all time changes of each other; 2. The measurable isomorphism between their time changes implies the much stronger (algebraic) equivalence as above. We also show that the complexity of time changes of unipotent flows can be described explicitly in terms of the corresponding adjoint action, and associated Jordan block‐like structures. Moreover, this is also true for the complexity of high rank abelian unipotent actions.

This is based on joint works with Adam Kanigowski, Philipp Kunde, Elon Lindenstrauss and Kurt Vinhage.

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Date/Time : 30 Nov 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ji Hui, National University of Singapore

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https://nus-sg.zoom.us/j/85864142504?pwd=aUUyaFRETE5hYit6WlYyMGhhdDZYUT09

Meeting ID: 858 6414 2504
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Imaging has been vital in advancing science, medicine, engineering, and industrial applications. Today’s imaging systems increasingly rely on computing power and algorithms to indirectly form images (visual interpretation) from only partial and noisy measurements. Most image reconstruction tasks are about solving a challenging linear or non-linear ill-posed inverse problem. For the past few decades, we have witnessed phenomenal developments in mathematical theories, models, and algorithms for solving inverse imaging problems. This talk will start by reviewing our works on prevailing model-driven mathematical approaches in computational imaging with manually-crafted model/transforms, including wavelets and sparsity-based regularization. Then we will describe how to advance these studies into the era of data-driven approaches that directly learn intrinsic structures from image data. Finally, we will discuss the connection between these data-driven models/algorithms and deep convolutional neural networks, which led to further advances in self-supervised deep learning for computational imaging.

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Date/Time : 25 Nov 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Slawomir Solecki, Cornell University

One of the areas of interest of Descriptive Set Theory is dynamics of Polish groups, that is, groups carrying a group topology that is separable and completely metrizable. Such groups, like for example, the unitary group of the infinite dimensional Hilbert space or the homeomorphism group of the unit interval, are not, in general, locally compact. Therefore, in studying their dynamics, classical methods relying on Haar measure are not available. They can often be replaced by descriptive set theoretic tools.

I will describe the descriptive set theoretic background and indicate how this point of view lead to a recent answer to an old question in Ergodic Theory. The question lies within a long-established theme, going back to the work of Halmos and Rokhlin, of investigating generic measure preserving transformations.

To be more specific: The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation has emerged. This picture included substantial evidence that pointed to these groups being all topologically isomorphic to a single group, namely, $L^0$—the group of all Lebesgue measurable functions from $[0,1]$ to the circle. Glasner and Weiss asked if this was the case. I will sketch a negative answer to this question that rests on an analysis of unitary representations of the non-locally compact Polish group $L^0$.

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Date/Time : 25 Nov 2022 13:30
Venue : S17 #04-06 (SR1)
Speaker : Gennady Samorodnitsky, Cornell University

This is a quick introduction to simplicial complexes, both deterministic and random, simplicial homology and connections to graphs.

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Date/Time : 23 Nov 2022 15:30
Venue : S17 #04-06 (SR1)
Speaker : Gennady Samorodnitsky, Cornell University

This is a quick introduction to simplicial complexes, both deterministic and random, simplicial homology and connections to graphs.

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Date/Time : 22 Nov 2022 10:00
Venue : S17 #04-06 (SR1)
Speaker : Carsten Chong, Columbia University

In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behaviour of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter H<0.5. In this paper, we derive a consistent and asymptotically mixed normal estimator of H based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

This talk is based on joined work with Marc Hoffmann (Paris Dauphine), Yanghui Liu (Baruch College), Mathieu Rosenbaum and Grégoire Szymanski (both Ecole Polytechnique).

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Date/Time : 18 Nov 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Hoang-Hung Vo, Saigon University

In this talk, I will present some recent advances for the qualitative properties of the principal eigenvalue of nonlocal operators, which arise from many natural phenomena, especially in ecology and population dynamics. I shall first introduce several notions of generalized principal eigenvalue in the sense of Berestycki-Nirenberg-Varadhan and then employ them to investigate the qualitative properties of the principal eigenvalue the asymptotic behavior with respect to the dispersal range and dispersal rate. Different limits are obtained and linked to different phenomena in ecology. We also obtain the long time dynamics of nonlocal Fisher-KPP equation assuming that the environment is unfavorable far away. By taking into account on the seasonal effect on the population, the qualitative properties of the principal eigenvalue of time-periodic nonlocal operators are also discussed, new limiting properties of the principal eigenvalue and long time dynamics nonlocal Fisher-KPP equation with time periodic coefficients are further investigated.

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Date/Time : 17 Nov 2022 15:30
Venue : S17 #04-04 (SR3)
Speaker : Renaud Leplaideur, University of New Caledonia

This is a joined work with A. Baraviera from UFRGS (Brazil). We introduce a notion of substitution on the colored binary trees.

In the first step I will present the motivtions for this work. Strangely (perhaps), they come from questions in ergodic theory and thermodynamic formalism.

In the second step I will present how we define these substreetutions, and then present one example.

The main result is that the associated “attractor” is minimal and non-periodic. I will also present many open questions about these objects.

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Date/Time : 16 Nov 2022 16:00
Venue : S15 #03-15 (Seminar Room L3)
Speaker : Cristian Calude, University of Auckland

Abstract: We construct a class of 3-dimensional photonic quantum random number generators and prove that each generates maximally unpredictable digits via measurements that are robust to errors. In particular, every sequence generated is strongly incomputable; hence its quality is provable better than that of every pseudo-random sequence. We also briefly contrast 2-dimensional and 3-dimensional quantum random number generators, discuss photonic implementations and show the superiority of the latter ones. These results suggest that incomputability in physics is real and practically useful.

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