# Graduate Student Seminar

The math grad seminar serves as a space for graduate students to gather, give talks about math they’re doing and thinking about, and discuss ideas with one another. It’s meant to offer an opportunity for students to communicate their mathematical ideas and foster collaboration within the department.

The talks will highlight some mathematical research: either your own original work or an introduction to an area of research you are interested in (or a combination of the two!). For MSc students, we compiled a list of suggested topics for possible talks below. If you’re interested in one of the topics and need some guidance about resources or preparing the talk, please contact one of the organizers. You’re more than welcome to select a topic that is not on the list, please just reach out and let us know. All talks will be 50 minutes in length with 10 minutes at the end for questions and discussion. We encourage everyone to attend and participate, so please consider giving a talk this term. During the seminar, we will serve pizza and refreshments. Afterwards, we invite you to join us for an informal social at the Grad Club. We hope to see you all there!

# Practical information

**Time:**Fridays at 3:30 PM**Location**: MC 107**Organizers**: Alan Flatres, Luuk Verhoeven, Sinan Nurlu, and Mohabat Tarkeshian

# Schedule

**September 16: Oussama Hamza**

__Title__: * Filtrations, arithmetic, and explicit examples in an isotypical context*

__Abstract__: Pro-p groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences (a_n) and (c_n), closely related to a special filtration of a finitely generated pro-p group G, called the Zassenhaus filtration. These sequences give the cardinality of G, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer n such that a_n=0 (or c_n has a polynomial growth) if and only if G is a Lie group over p-adic fields. In 2016, Minac, Rogelstad and Tan inferred an explicit relation between a_n and c_n. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an isotypical context: when the automorphism group of G admits a subgroup of order a prime q dividing p-1. In this talk, we will mostly review some results of Golod, Shafarevic, Koch, Lazard, Minac, Tan and Hamza. We also give several explicit examples in an arithmetical context.

**September 23: Jarl Taxerås Flaten**

Title: *The moduli space of multiplications on a space*

Abstract: Since the mid-50s, various topologists have been interested in counting homotopy classes of multiplications (i.e. H-space structures) on certain spaces. For example, there's a unique multiplication on the circle (complex multiplication), and James showed that there are 12 multiplications on the 3-sphere and 120 on the 7-sphere. No other spheres admit a multiplication, barring the 0-sphere.

We present a formula for the moduli space of multiplications on a pointed object of an ∞-topos. By specializing to the ∞-topos of spaces and counting the path components of these moduli spaces, we recover the numbers just mentioned. These results have been shown in Homotopy Type Theory, which I will give a brief introduction to, and have been formalized in the Coq proof assistant, which I will demonstrate with some live-coding.

**September 30:** Tenoch Morales

__Title__: *Using Fitness Landscapes to understand the shifts in mutation biases *

__Abstract__: Mutations are the engine that drives evolution and adaptation forward in that it generates the variation on which natural selection acts. Although mutations are considered to occur randomly in the genome, we see that in many organisms some types of mutations occur more often than expected under uniformity; these deviations are called mutation biases.

Even though there is no clear description of the biological mechanisms governing the formation of mutation biases, theoretical and experimental work has shown that a shift in mutation biases during the evolutionary process could grant an adaptive advantage to an organism by increasing the sampling of previously poorly explored types of mutations.

In this talk, we will explore the most popular Fitness Landscape models, which map the genotypic space of an organism to its adaptive fitness. With these models, we can simulate the evolutionary process of a population as a walk through the genotypic space towards genotypes with higher fitness, which will help us understand the adaptive effect of shifts in mutation biases at different points on the evolutionary path.

**October 7:** TBA

**October 14: Alejandro Santacruz Hidalgo**

**October 21: Nathan Pagliaroli**

**October 28:** TBD

**November 4:** Reading week (no seminar)

**November 11:** TBD

**November 18:** TBD

**November 25:** TBD

# Suggested topics for MSc students

- Expander graphs and their applications
- Cayley graphs
- Graph invariants (e.g., Tutte polynomial, chromatic polynomial)
- Jacobian groups of graphs and the graph-theoretic Riemann-Roch theorem
- Caratheodory’s theorem in convex geometry
- The Birkhoff-von Neumann theorem in convex geometry
- Permutohedra and generalized permutohedra
- Galois groups of quartic polynomials
- Matroids and examples
- Lattice-based cryptography
- Game theory
- Dynamical systems
- Probability
- Models
- Math-bio (e.g., epidemiology, population dynamics...)
- Machine learning
- Neuroscience models
- Numerical simulations
- Free probability (central limit theorem)
- Weyl integration formula (in random matrix theory)
- Finite spectral triples (classification of)
- Fuzzy spectral triples (classification of)
- Fuzzy sphere
- Gelfand-Naimark theorem
- Spontaneous symmetry breaking (Higgs mechanism)
- Stone-Čech compactification theorem
- Urysohn’s lemma
- Classification of connected Riemann surfaces
- Frobenius theorem (differential topology)
- Flow box theorem
- Holomorphic differential equations and existence of their solutions
- Tychonoff's theorem