School of Science Department of Mathematics 39 Department of Mathematics Low-Rank and Sparsity Reconstruction in Data Science Supervisor: CAI Jianfeng / MATH Co-supervisor: YE Guibo / MATH Student: LEI, Yicong / DSCT Course: UROP 1100, Summer In various scientific and engineering applications, one frequently encounters the challenge of addressing an ill-posed inverse problem, where the number of available measurements is less than the dimension of the model to be estimated. However, in many practical scenarios of interest, structural constraints on models mean that they possess only a few degrees of freedom relative to their ambient dimension. This paper is generally a literature review about two papers: Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information and The Convex Geometry of Linear Inverse Problems. In short, this paper summarizes some concepts and methods which may be used in low-rank and sparsity reconstruction. Chromatic Polynomials of Graphs and Signed Graphs Supervisor: CHEN Beifang / MATH Student: CHEN Tairun / MATH-PMA Course: UROP 1100, Summer Started with Whitney, the combinatorial theory of matroids axiomatically abstracts and generalizes the notions, linear independence in vector spaces and connectedness in graphs. It turns out to be a unified way to study linear algebra, graph theory, and combinatorics via finite geometry. This report summarizes my study of the first few sections of the great paper "Hodge theory for combinatorial geometries". We shall concern various combinatorial properties of convex polytopes and focus on some interesting examples. We’ll see fancy geometric realizations of some abstract combinatorial objects. The author expects a further study after formally learning Algebraic Geometry and Algebraic Topology. Chromatic Polynomials of Graphs and Signed Graphs Supervisor: CHEN Beifang / MATH Student: DINH Vu Tung Lam / MATH-PMA Course: UROP 1100, Fall It is known the recurrence formula of chromatic polynomial for a simple graph, which plays a significant role in understanding the structure of any directed graph if the graph remains undetermined. One of the important results in this topic is Whitney’s Broken Circuit Theorem (1932). Corresponding to the chromatic polynomial topic is the unimodal sequence, which gets a more concise property on the coefficients of the polynomial in any graph. The purpose of the problem in the future is to extend the coloring and chromatic polynomial of graphs to signed graphs, and to study properties of its coefficients. The problem is related to several branches of mathematics such as combinatorics, topology and geometry.
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