Student Profile: Kshitija Vaidya

Kshitija Vaidya

Monash University

Biography

Kshitija has just finished her third year in a Bachelor of Science and Bachelor of Global Studies double degree at Monash University. She has completed a minor in Physics and is working towards her majors in Pure Mathematics and International Relations. Kshitija’s name was included on the Faculty of Science Dean’s Honours List for 2016 and she was also awarded a Chris Ash Prize in 2017 for Linear Algebra. This year, her passion for problem-solving led her to participate in the Simon Marais Mathematics Competition.

Throughout her undergraduate degree Kshitija has developed an interest in various areas of mathematics including group theory, ring theory and functional analysis. For the remainder of her degree she hopes to further pursue these areas and explore others. An aspiring researcher in Pure Mathematics, Kshitija intends to undertake an Honours year in 2020.
She is excited by the prospect of a research career because it involves uncovering beautiful mathematical truths.

Cocyclic Hadamard Matrices

A Hadamard matrix of order n is an n by n arrangement with entries of either plus or minus one, such that its columns satisfy a combinatorial property called orthogonality. They arose in the mid-19th century through mathematician’s interest in examining maximaldeterminant square matrices. Nowadays Hadamard matrices are one of the most prominent objects in combinatorics and design theory, admitting applications in cryptography and information security. Their mathematical driving force is the famous Hadamard conjecture, which asserts that for every positive integer n there is a Hadamard matrix of order 4n. A well-known algebraic approach for constructing Hadamard matrices is the recent theory of cocyclic Hadamard matrices. This theory establishes a central link between cohomology of finite groups and combinatorics. Research on cocyclic Hadamard matrices focuses on finding families of groups for whose second cohomology group can be used to construct Hadamard matrices. In this project we will investigate this construction for certain finite abelian groups and dihedral groups. In particular, we will compute explicitly two-cocycle representatives for the corresponding cohomology groups, giving rise to potentially new cocyclic Hadmard matrices.

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