School of Science Department of Mathematics 49 Modeling the Statistical Structures of Brain-Wide Activity Using Recurrent Neural Circuits Supervisor: HU Yu / MATH Student: SUN Mengxi / DSCT Course: UROP 1100, Summer This UROP project implements an RNN using the PINing algorithm to reproduce results from the thesis. The RNN was first trained on synthetic ground truth activity generated with a known recurrent weight matrix Wtrue, already been learned to fit the experimental data. While the model achieved a high R² score and captured scale invariance in both the training and generalization phases, the correlation between the trained weight matrix Wtrain and Wtrue was low. This discrepancy is likely due to error accumulation from the ReLU activation function. The RNN also successfully fit experimental data, reproducing scale-invariance and realistic dynamics. However, whether the model captures more than pairwise correlations needs further analysis on Wtrain and its transpose. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: CHEN Tairun / MATH-PMA Course: UROP 1100, Fall UROP 2100, Spring For a simply-connected, connected, simple complex algebraic group of rank , with double Bruhat cell , the function ring ( , ) admits an cluster structure ℂ such that = ̅. Similar results also hold for Grassmannian. This report will focus on the work in [IOS]. Given a connected marked surface Σ with at least two marked points and no punctures, the cluster algebra ,Σ associated with the pair ( , Σ) coinsides with its upper cluster algebra ,Σ, as the function ring ( ,Σ× ) of the moduli space of decorated twisted -local systems on Σ, which is generated by matrix coefficients of Wilson lines. Application to finite type cases can recover results in [Muller] for G = SL2, and answer conjectures in [IY23] and [IY22] for = 3, 4, with a description involving skein algebras. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: CHENG Yat Long / QFIN Course: UROP 1100, Spring This report is for fulfilment of UROP1100 requirements at HKUST. The project is titled “cluster algebra” and supervised by Prof. IP, Ivan Chi Ho. This is a short expository article on cluster algebra, summarizing the resources the author read during the project. The resources include parts of lecture notes written by the advisor, covering basic motivations and definitions of cluster algebra, and introducing cluster algebra on surfaces, which involves the triangulations on surfaces with marked points. In a research paper, Musiker and Schiffler proposed a new formula for evaluating cluster variables in cluster algebras associated with unpunctured surfaces. The formula expresses cluster variables as a summation over perfect matchings from a graph constructed from the surface. In this article, we will provide a brief definition of cluster algebra, discuss the results in the paper, and give a worked-out example of the formula.
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