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Abstract: A central question in geometric flow is to understand how the geometry and topology change after passing through singularities. I will explain how the local dynamics influence the shape of the flow near a singularity, and how the geometry and topology of the flow will change after passing through a singularity with generic dynamics. This talk is based on joint work with Zhihan Wang and Jinxin Xue

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Abstract: We show for a large class of entire functions, $f$, that after proper rescaling, on compact sets, the derivatives of $f$ converge to cosine, in particular their roots become evenly spaced. This proves a conjecture of Farmer and Rhoades [Trans. Amer. Math. Soc., 357(9):3789–3811, 2005] and Farmer [Adv. Math., 411:Paper No. 108781, 14, 2022] for our class of entire functions. A main ingredient of our proof is to show that high derivatives of high degree polynomials behave like Hermite polynomials, which we prove using the techniques from the newly developed field of finite free probability.

Abstract: The forward and inverse problems for radiative transfer are important in many applications, such as climate modeling, optical tomography, and remote sensing. However, these problems are notoriously challenging to compute due to their high dimensionality, significant memory requirements, and the computational expense associated with solving the inverse problem iteratively. To address these challenges, we present recent progress on two approaches. The first approach is hp-adaptive mesh refinement, which has proved effective in efficiently representing solutions where they are smooth with high-order approximations, while also providing the flexibility to resolve local features through adaptive refinements. For the forward problem, we demonstrate that exponential convergence with respect to degrees of freedom (DOFs) can be achieved even when the solution exhibits sharp gradients. For the inverse problem, we introduce a goal-oriented hp-adaptive mesh refinement method that blends the two optimization processes—one for inversion and one for mesh adaptivity—thereby reducing computational cost and memory requirements. The second approach, termed element learning, aims to accelerate finite element-type methods through machine learning. This approach retains the desirable features of finite element methods while substantially reducing training costs. It draws on principles from hybridizable discontinuous Galerkin (HDG) methods, replacing HDG's local solvers with machine learning models. Numerical tests for both approaches are presented to demonstrate their computational efficiency in addressing the forward and inverse computations of radiative transfer.

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Abstract: An important theme in contemporary condensed matter physics is “topological phases of matter”. This refers to exotic materials which exhibit a number of striking phenomena. E.g., they have a quantized macroscopic observable which is stable under large classes of perturbations and in their bulk they are insulators though they exhibit robust edge currents along their boundaries. To mathematically explain this, we define an appropriate topological space of quantum mechanical Hamiltonians which describe the motion of (single) electrons in an insulator, and calculate its path-connected components. Hamiltonians in the same path-component are said to be topologically “equivalent”. The presentation will be based on joint pre-prints together with Jui-Hui Chung.