Seminar
A geometric approach to understanding ranks of fundamental groups
Box and Nabla Products, Part II
An introduction to the `Box Product Problem', its connection with
nabla products, and an overview of recent results with Hector Barriga
Acosta (UNAM Morelia).
Empirical Families of Number Fields
Abstract:
Number fields with a given Galois group often come in polynomial families. I compare the two main theoretic approaches to such families with empirical studies, suggesting opportunities overlooked due to exclusive focus on best-possible outcomes or simplest normal forms.
Box and Nabla Products, Part I
An introduction to the `Box Product Problem', its connection with
nabla products, and an overview of recent results with Hector Barriga
Acosta (UNAM Morelia).
“An introduction to fixed point theory in Banach spaces. 5.”
“Generalizing the Logistic Map. 2.”
Serre weight conjectures for unitary groups
Abstract:
Modular Hashing and the Collatz conjecture
Choose a positive integer. If it is odd, multiply it by three and add one. If it is even, divide it by two. The Collatz conjecture is a decades old conjecture that states that for all positive integers, if you repeat this process, you will eventually reach one. In computer science, a hash function is a function that is used to map data values into organized ``buckets'', so that the information in the buckets can be easily retrieved later. These two concepts are seemingly unrelated.
Is this Chaos?
The logistic map is a classic recursive sequence defined by $x_{n+1}=rx_{n}(1-x_{n})$ and $r$ is a parameter with $0\leq r \leq 4$. It turns out, this innocent sequence is quite strange when $r$ gets closer to 4. For large enough $r$, chaos can even occur. We will investigate the fixed points of this map as well as some of its periodic orbits, and later we will look into some of the numerics with this sequence.