From the title article by Á. Lozano-Robledo and H.B. Daniels(AMS Notices, 2017) to the current "The Ubiquity of Elliptic Curves" by E.H. Goins (AMS Notices, 2019), what is the reason for 'The unreasonable effectiveness of elliptic curves'? This talk will illustrate the role of elliptic curves, and more generally Algebraic Geometry, in Mathematical Physics, Information Theory and Cryptography, and Machine Learning, presenting some of the speaker's results and open questions for future research.

Abstract: Inspired by the recent developments in differential geometry and calculus of variations, there have been several approaches to identifying a suitable notion of local (Ricci) curvature on non-smooth spaces, such as graphs and Markov chains. I will describe some of these approaches and review a few recent developments in this topic on discrete curvature. Some of the consequences include a tight Cheeger inequality in abelian Cayley graphs, and diameter bounds on the spectral gap of the graph Laplacian. Several open questions remain.

Abstract: The notion of asymptotically hyperbolic manifolds arises largely from two different contexts: as a compactifiable manifold by conformal deformation (like the Poincarè ball), or as an initial data set in general relativity (like a hyperboloid in the Minkowski space). This talk will mainly focus on the latter case. Beginning with the Einstein Field Equations, I am going to introduce an initial data set, Einstein constraint equations, and how the concept of asymptotically hyperbolic manifolds arises naturally. From there, I will talk about the positive mass theorem for asymptotically hyperbolic manifolds, and relevant result of the joint work with Huang and Martin, which concerns about the rigidity property of the mass.

I will also briefly mention the context of conformally compactifiable manifolds and some interesting results.

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Given a non-zero square integrable function $$g$$ and $$\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2$$ let \( \mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N.\) The Heil-Ramanathan-Topiwala (HRT) Conjecture is the question of whether $$\mathcal{G}(g, \Lambda)$$ is linearly independent. For the last two decades, very little progress has been made in settling the conjecture. In the first part of the talk, I will give an overview of the state of the conjecture. I will then describe some recent attempts in settling the conjecture for some special classes of functions.

Abstract: We will consider some mathematical aspects of a quantum state transfer. In particular, we will review a few recent papers on the matter and discuss the role of orthogonal polynomials and Jacobi matrices in the theory of spin transfer.

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Learning seminar in geometric flow