2024-2025
2025 Hilary Term
03/03/2025 (Week 7) - Fellow's Talk: Decidability of Problems for Finitely Represented Groups
Speaker: Prof. Nikolay Nikolov
Given a finite presentation of a group G it is known that most group theoretic statements about G are in general undecidable. For example there is no algorithm that can tell (in finite time) if G is the trivial group or not. However if we assume that G is an abelian group all such problems are decidable by (usually very fast) algortithms. In this talk I will explain some of the main ideas for these results and discuss the case when G is virtually abelian, (i.e. G contains an abelian subgroup of finite index). Most of the talk should be accessible to first year students.
24/02/2025 (Week 6) - A whistle-stop tour of Probabilistic Combinatorics
Speaker: Agniv Sarkar
We will try to hit the basics of probabilistic combinatorics with a few examples of proofs. I shall also express my opinions about combinatorics at the beginning as well.
17/02/2025 (Week 5) - Five mini talks
Do you desire to give a talk in PedroSoc, but you can't put together a full 1h30 talk? Do you want to test your skills, giving a shorter talk? PedroSoc has a new project: week 5 talks! We will try to get 5 mini-talks (between 15 and 30 minutes) in one meeting! Hoping to see many of your submissions! Any ideas (not necessarily maths) invited!
| Speaker | Content |
|---|---|
| Maeve Dever | Squaring the Circle A tour on how Greeks thought about the problem of constructing a square that has the same area of a circle by compasses and straightedge. Spoiler alert: this is impossible. |
| Alex Dong | Maths of how to solve the Rubix Cube We study the symmetry of Rubix Cube using group theory and counting. |
| Yuhan Wang | Group Theory and Quantum Mechanics We are going through how Group Repredentstion Theory is able to simplify calculations within Quantum Mechanics and allow us to deduce some profound results. This would be through analysing the Greater Orthogonality Theorem and analysing the implications of those results. |
| Dominic Seymour | Lubrication theory and Viscous Fingers A talk to showcase how mathematics could be used to study the shape of liquid when injected into a thin layer. |
| Yifu Zhang | What day is it today (or another day)? A light talk to explain the mathemagic behind finding which date is today. |
03/02/2025 (Week 4) - Riesz-Markov Representation Theorem
Speaker: Shing Fung Chan
We shall prove the Riesz-Markov Representation Theorem, which states that any functional of the space of continuous, compactly supported function are Lebesgue Integrals. This shall motivate another way of defining the Lebesgue Integral.
03/02/2025 (Week 3) - Lebesgue Measures and Integrals
Speaker: Shing Fung Chan
We will go through the standard way to construct the Lebesgue measure and Lebesgue integral on the real line, and why are these constructions are the better ones.
27/01/2025 (Week 2) - Algebraic Topology
Speaker: Morgan Healey
In this talk we will be motivating a special class of topological groups. Our canonical example will be the p-adic integers, but we will see many more and understand how they connect to the theory of finite groups. We will also see the importance of topology in number theory and algebraic geometry. Finally we will touch on cohomology and the usefulness of category theory in maths.
20/01/2025 (Week 1) - Fermat Last Theorem for Polynomials
Speaker: Pedro Lack
We shall start the year by proving the Fermat's last theorem (for polynomials)! It is very exciting, still should be quite easy to follow (no prerequisites)!
2024 Michaelmas Term
25/11/2024 (Week 8) - 8 ⟂ ∞: How to become JCR President: what deep data analysis on democracies can say about College politics?"
Speaker: Ardeel Hussain
Given that we are not a maths society, but rather a Pedro society, the Pedro was happy to announce the project 8 ⟂ ∞, in which PedroSoc always reserves week 8 to topics perpendicular to the infinite (that is earthly and material topics). In order to inaugurate this project, I am happy to announce that Ardeel Hussain will be our guest speaker and he will present on the topic: "How to become JCR president: what deep data analysis on democracies can say about College politics". As usual, we are meeting on Monday at 7 in 90 High Street! I am waiting for all of you!
25/11/2024 (Week 7) - Hasse Theorem II
Speaker: Pedro Lack
We shall go further and prove if a quadratic Diophantine equation has a solution in the ring of polynomials. This relies on the notion of Hilbert Symbol and results of quadratic reciprocity.
18/11/2024 (Week 6) - Hasse Theorem I: what does it feel like to invent mathematics?
Speaker: Pedro Lack
We know how check if a linear Diophantine equation for integers has solution. Can we do more? We will start by creating the p-adic numbers, before proving a version of the Hasse Theorem that determines if linear Diophantine equation has a solution in the ring of polynomials.
11/11/2024 (Week 5) - Permutation and Primes
Speaker: Yifu Zhang
We shall explore a surprising connection between permutation and primes. A highlight is the shockingly similar Central Limit Theorems of primes (Erdos-Kac) and permutations (permutations).
04/11/2024 (Week 4) - Law of Iterated Logarithm Part II
Speaker: Samuel Chun Hei Lam
With the tools built from the previous talk, we shall now complete the proof of the classical law of iterated logarithms. We shall see how the Borel-Cantelli lemma and the Chernoff bound give half of the proof. Then we will do a little bit more heavy-lifting to obtain the second half of the proof (which may also lead to the Central Limit Theorem for free).
03/06/2024 (Week 3) - Law of Iterated Logarithm I: demytifying almost sure convergence
Speaker: Samuel Chun Hei Lam
In the sequel of two talks we shall understand and proof the classical result of law of iterated logarithm in probability theory. For this talk we shall understand what it meant for an event to happen almost surely, and develop the tool of Chernoff bounds (a stronger notion of convergence in probability) to establish almost sure convergence of a properly scaled standard random walk.
21/10/2024 (Week 2) - Sieve Theory
Speaker: Gareth
We will look at how we could carefully design the sieves of natural numbers (the famous one being the Sieve of Eratosthenes) to give an upper bound of the number of prime numbers that satisfies certain properties. You will be surprised that a little trick would be the foundations of reasonably strong results to a handful of famous problems in analytic number theory.
14/10/2024 (Week 1) - Bertrand's Postulate
Speaker: Pedro Lack
We shall go through the Ramanujan's and Erdő's proof of the Bertrand's Postulate, which states that for any integers m, there exists prime number such that m/2 < p < m-1. This is a good starting point if you want to have a feeling on what analytic number theory is about.