Date/Time : 14 Feb 2024 16:30
Venue : S17 #04-06 (SR1)
Speaker : Liu Xin, Chinese Academy of Sciences

We propose constraint dissolving approaches for optimization problems over a class of Riemannian manifolds. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint dissolving function named CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and Łojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of the proposed approaches.

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Date/Time : 13 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Lei Zhen, Fudan University

More than 20 years ago, Soffer-Weinstein considered a class of nonlinear Klein-Gordon equations with nice potentials. For the first time they proved that spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity, via energy transfer from the discrete to continuum modes, under the condition that the discrete modes are close to the continuous spectral modes. Since then, a long-standing open question is to study the the corresponding problem with small eigenvalues, which was also raised in the paper of Soffer-Weinstein 1999. In this talk, we will report our recent result on this problem which settles the above open question. This is a joint work with my students Jie Liu and Zhaojie Yang.

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Date/Time : 08 Feb 2024 14:00
Venue : S17 #05-11 (SR5)
Speaker : Peter A Markowich, KAUST

We propose a mesoscopic modeling framework for optimal transportation networks with biological applications. The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in physical space. The energy expenditure of the network is given by a functional consisting of a pumping (kinetic) and metabolic power-law term, constrained by a Poisson equation accounting for local mass conservation. We establish convexity and lower semicontinuity of the functional on appropriate sets. We then derive its gradient flow with respect to the 2-Wasserstein topology on the space of probability measures, which leads to a transport equation, coupled to the Poisson equation. To lessen the mathematical complexity of the problem, we derive a reduced Wasserstein gradient flow, taken with respect to the tensor-valued conductivity variable only. We then construct equilibrium measures of the resulting PDE system. Finally, we derive the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE. We calculate its equilibrium states, represented by measures concentrated on a hypersurface in the phase space.

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Date/Time : 08 Feb 2024 15:00
Venue : S17-05-11 (SR 5)
Speaker : Tan Bui-Thanh, The University of Texas at Austin

Digital twins (DTs) are digital replicas of systems and processes. At the core of a DT is a  physical/mathematical model which captures the behavior of the real system across temporal and spatial scales. One of the key roles of DTs is enabling “what if” scenario testing of hypothetical simulations to understand the implications at any point throughout the life cycle of the process, to monitor the process, and to calibrate parameters to match the actual process. In this talk we will present various (faster than) real time Scientific Deep Learning (SciDL) approaches for forward, inverse, and UQ problems. Both theoretical and numerical results for various problems including transport, heat, Burgers, Euler, and Navier-Stokes equations will be presented.

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Date/Time : 08 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Rui Chen, IASM, Zhejiang University

Branching law is a classical topic in representation theory which concerns certain Hom functors. It is natural to consider Ext functors associated to those Hom functors. In this talk we will report some recent progress on the study of these Ext spaces, and show a conjecture of D. Prasad, which asserts that the Ext-analogue of GGP model vanishes for tempered representations. We will also discuss some applications.

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Date/Time : 07 Feb 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Alexander Rabinovich, Tel Aviv University

Church’s Problem asks for the construction of a procedure which, given a logical specification S(I,O) between input-strings I and output-strings O, determines whether there exists an operator F that implements the specification in the sense that S(I,F(I)) holds for all inputs I. Buechi and Landweber gave a procedure to solve Church’s problem for MSO specifications and operators computable by finite-state automata. We investigate a generalization of the Church synthesis problem to the continuous time of the non-negative reals. It turns out that in the continuous time there are phenomena which are very different from the canonical discrete time domain of the natural numbers.

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Date/Time : 07 Feb 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Marco Badran, ETH Zurich

Given a closed manifold $N$ and a compact, non-degenerate minimal submanifold $M$ of codimension 2 we construct a collection of solutions $U_\varepsilon$ to the magnetic Ginzburg-Landau equations such that $U_\varepsilon$ concentrates around $M$ as $\varepsilon\to 0$, establishing a codimension two analogous of the celebrated result by Pacard and Ritoré in the Allen-Cahn setting. The construction is carried out via an asymptotic gluing technique, yielding precise asymptotic expansion of the concentration profile.

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Date/Time : 07 Feb 2024 16:00
Venue : S17 #05-11 (SR5)
Speaker : Didac Martinez-Granado, University of Luxembourg and National University of Singapore

We define a notion of hyperplane for a hyperbolic group G and study G-invariant measures on the space of hyperplanes of its Gromov boundary, which we call “co-geodesic currents´´. Co-geodesic currents are induced by many classical objects such as geodesic currents for surface groups, certain cocompact actions of hyperbolic groups on CAT(0) cube complexes, quasi-fuchsian representations on a surface group, etc. Moreover, there is a natural intersection pairing between co-geodesic currents and the more classical geodesic currents, introduced by Bonahon, i.e. G-invariant measures on the square of the boundary of G. This pairing generalizes Bonahon’s intersection number when G is a surface group. Furthermore, every co-geodesic current \mu induces natural dual pseudo-metric space, in the sense that the intersection of \mu with the current determined by the conjugacy class of an element g in G recovers the stable length of g in the pseudo-metric space. This is joint work in progress with Eduardo Reyes.

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Date/Time : 06 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Dong Bin, Peking University

This presentation delves into the recent advancements in applying artificial intelligence (AI) to mathematical exploration. Initially, we will outline the current state of integrating AI methodologies, including machine learning, deep learning, and neural networks, into mathematical research, thereby fostering enhanced approaches and innovative concepts. Subsequently, we will highlight some preliminary results from ongoing research in this intersecting domain. Concluding the discussion, we will anticipate the future trajectory of research at the intersection of AI and mathematics, focusing on the potential of AI techniques to tackle new challenges and uncover novel problems in mathematical studies.

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