Pure Math Honours presentations

Abstract:

Honours presentations, in order, of Joshua Graham, Edmond Gao, Kevin Tran, and Madhav Padmakumar.

Each talk is 20 minutes long followed by 5 minutes of questions and by 5 minutes break.

We will start at 2:05.

Speaker: Joshua Graham

Title: Higher Algebraic K-theory of Rings

Abstract: The focus of this talk will be looking at how lower K-groups of rings are defined by algebraic means and how we extend the definition to higher K-groups in a functorial manner. I will also briefly talk briefly about K-theory of the integers and of finite fields.

Speaker: Edmond Gao

Title: Futoshiki - A Latin Square Puzzle Variant

Abstract: The act of filling in the empty cells of a partial Latin square to form a Latin square has become a popular logic puzzle over the last few decades. In that time, many variants with additional conditions have emerged, the most popular being the Sudoku puzzle. Both Latin squares and Sudoku have been studied extensively by combinatorialists, however other variants of Latin square puzzles remain mathematically untouched. One such variant is the Futoshiki puzzle, which incorporates inequalities into Latin squares.

It is surprising that a puzzle that is so mathematical in nature is barely present in mathematical literature. I aim to change this, exploring Futoshiki puzzles from a combinatorial perspective. I define and introduce features and characteristics of the variant, and extend results about Latin squares to this special case.

Speaker: Kevin Tran

Title: Combinatorial and arithmetic properties of Hilbert cubes

Abstract: In the area of additive combinatorics and number theory, Hilbert cubes are sets defined to have a strong additive structure. Following the standard paradigm of additive combinatorics, our goal is to show the absence of any multiplicative structure. Results in this form include those about the intersection of Hilbert cubes in a ring with multiplicative subgroups of this ring, or those about powers in and prime divisors of elements of Hilbert cubes of integers. We also show that the results in "Hilbert cubes meet arithmetic sets" by Hegyvári and Pach (2020), which aim to show that Hilbert cubes in finite fields are "uncorrelated" with reciprocals of sumsets, are essentially void, as they are no stronger than a trivial bound. Instead, we suggest a different approach which allows us to get nontrivial versions of these results.

Speaker: Madhav Padmakumar

Title: Measuring and Computing Optimal Generators for Homology Groups

Abstract: The field of topological data analysis typically concerns itself with imposing a topological space on a dataset, then finding topological invariants. This talk will define the problem of "Optimal" generators for those topological features, as well as how to compute them. I will outline the Smith Normal Form reduction algorithm for computing homology groups and the persistent homology algorithm for extracting topological features from datasets. I will then explain the algorithm for computing optimal generators, with examples and potential applications.