Talks
Highlighted talks are marked with an asterisk

* Symmetries of differential equations
Tomorrow’s Mathematicians Today, IMA @ Virtual 2024
[slides] [recording] Symmetries often gives us greater insight into differential equations. For example, Noether’s theorem tells us that symmetries of an action give us invariants. However, this idea is applicable to differentiable equations that do not necessarily come from variational principles. Symmetries can also tell us a lot more beyond generating invariants. We would try to gain a more elementary and fundamental understanding of symmetries by studying the symmetries of elementary ODEs and PDEs (e.g. Laplace, Heat, Wave). 
The isoperimetric and Sobolev inequality
Members Talk, The Oxford Invariants @ Oxford 2024
[slides] The isoperimetric inequality on a plane states that among all curves of a given length, it is the circle that encloses the maximum area. While the theorem seems obvious, a satisfactory proof was only first obtained in the 19th century. We would demonstrate that is in fact equivalent to Sobolev inequality, which is often used in functional analysis. 
Symmetries of differential equations
Oxford Junior MathPhys Journal Club @ Oxford 2024
[slides] Symmetries often gives us greater insight into differential equations. For example, Noether’s theorem tells us that symmetries of an action give us invariants. However, this idea is applicable to differentiable equations that do not necessarily come from variational principles. Symmetries can also tell us a lot more beyond generating invariants. We would try to gain a more elementary and fundamental understanding of symmetries by studying the symmetries of elementary ODEs and PDEs (e.g. Laplace, Heat, Wave). 
* Structures closed under intersections or unions
Members Talk, The Oxford Invariants @ Oxford 2023
[slides] [blog post] Mathematical structures that are closed under countable or uncountable intersections give rise to a minimal structure generated by some set. We would go through examples in algebra, topology and measure theory to explore this idea. 
Showing preservation of properties under multiplication via difference of squares
Tomorrow’s Mathematicians Today, IMA @ Virtual 2023
[slides] [recording] [blog post] If you want to show a property is preserved under multiplication, you could first try to show that the property is preserved under linear combinations and squaring, then use difference of squares to establish preservation of that property under multiplication. Using this, we could produce simpler proofs of algebra of limits, product rule, and that products of Riemann integrable function are integrable. 
Showing preservation of properties under multiplication via difference of squares
Second Year Seminar, Balliol College @ Oxford 2022
[slides] [blog post] If you want to show a property is preserved under multiplication, you could first try to show that the property is preserved under linear combinations and squaring, then use difference of squares to establish preservation of that property under multiplication. Using this, we could produce simpler proofs of algebra of limits, product rule, and that products of Riemann integrable function are integrable.