Hang Lung Mathematics Awards 2023

3. Diocesan Boys’ School On the Upper-bound of Anchored Packing Problem The Lower-Left Anchored Rectangle Packing (LLARP) problem is well-studied in optimisation. Given a finite set of points P in the unit square [0, 1]2, where (0, 0) ∈ P, we seek a set of rectangles S that has maximal area coverage, each of which has a lower-left corner anchored at a point pi ∈ P. S satisfies three constraints: the rectangles are pairwise non-overlapping, completely lie within [0, 1]2 and the sides are parallel to the coordinate axes. This paper looks into a variant of LLARP, concerning convex N-gons, anchored at their lower-left vertex and with the two edges adjacent parallel to the coordinate axes such that a right angle is formed. We focus on finding the upper bound for some set of points in maximal coverage anchored packing. 4. Diocesan Boys’ School Structure of Critical Groups of Circulant Graphs The critical group of a graph is defined as the torsion subgroup of the cokernel of the Laplacian matrix of the graph. In this paper, we investigated the critical groups of two classes of unitary circulant graphs, which are Cayley graphs on the group of integers modulo n, with connecting set being the set of units modulo n. The explicit group structure of such graphs when n is product of two distinct primes and when n is a prime power, are computed using Ramanujan Sums. Furthermore, we investigated the critical groups of circulant graphs with fixed connecting sets, and expressed one of the components of the group as the greatest common divisor of real and imaginary parts of Chebyshev Polynomials. 5. G.T. (Ellen Yeung) College Quantum Computing: Adapting Shor’s Algorithm to the Problem of CLT Group Size Finding Shor’s algorithm, first proposed by Peter Shor in 1994, is a wellknown quantum algorithm for the discrete logarithm problem and for factoring large integers. Via running on a quantum computer, it is able to have a much better asymptotic runtime than state-of-the-art algorithms on classical computers. This report builds upon the Shor’s algorithm to construct a quantum algorithm for finding the order of CLT groups (groups satisfying the converse of Lagrange’s theorem) with significant advantage over classical algorithms.