Analysis, Applied Math and Geometry Seminar
Fall 2026
- Location: South Engineering 118
- Time: Tuesdays, 11:00-11:50 am
- Organizer: Maria Alfonseca-Cubero
15 September 2026
Stephanie Juntunen:
Floating bodies in equilibrium with variable densities
Abstract: In his 1822 memoir, Dupin established three fundamental laws for floating bodies. In the first talk, we will give a generalization these laws to the entire R^n with a probability density, and discuss their connection to Ulam’s Problem 19.
In the following talk, we will further explore probability densities with compact support and address a question similar to Ulam’s.
22 September 2026
Stephanie Juntunen:
Floating bodies in equilibrium with variable densities (Part 2)
29 September 2026
Reynold Fregoli:
Rational Approximations and Dynamics in the space of lattices
Abstract: Diophantine approximation asks how well real numbers can be approximated by rational numbers. A classical way to study this question is through continued fractions, but there is another point of view that connects it to dynamics on the space of lattices. In this talk, I will introduce this correspondence in its simplest setting. We will see how rational approximation to a real number can be encoded by short vectors in a lattice, and how the action of the diagonal flow translates Diophantine properties into geometric properties of trajectories.
This talk is intended as a gentle introduction to the subject, in preparation for my second talk next week.
6 October 2026
Reynold Fregoli:
Rational Approximations and Dynamics in the space of lattices (Part 2)
Abstract: In this second talk, I will revisit the correspondence, introduced in my first talk, between Diophantine properties of real numbers and dynamics on the space of lattices. I will explain some of the basic geometry underlying this correspondence, focusing on the role of Minkowski's Convex Body Theorem. I will then give a brief introduction to ergodic theory and show how ergodicity of the geodesic flow implies that the set of badly approximable real numbers has Lebesgue measure zero.
13 October 2026
Faraad Armwood:
Circle Summing Near Transverse Intersections
Abstract: In this talk, we discuss the original circle sum, an additional environment in which the sum can be realized, and some newer environments that may be possible. This work is from conversations with my advisors, Dr. Josef Dorfmeister and Dr. Jeremy Van-Morris.
20 October 2026
Reynold Fregoli:
Rational Approximations and Dynamics in the space of lattices (Part 3)
Abstract: In this talk, I will continue the study of the correspondence between Diophantine approximation and dynamics on the space of lattices, focusing mainly on the latter. I will construct the natural invariant measure on the space of lattices and explain why the geodesic flow is ergodic. As the culmination of this series of talks, I will show how ergodicity, together with the dynamical correspondence developed in the previous talks, implies that the set of badly approximable real numbers has Lebesgue measure zero.
27 October 2026
Azer Akhmedov:
Homogeneous Dynamics and Margulis Program
Abstract: I will build on Reynold Fregoli's recent series of presentations. I'll show that the Oppenheim Conjecture (solved by Gregory Margulis in 1986) and the Littlewood Conjecture (still open, but with substantial progress made by Einsiedler, Katok, and Lindenstrauss in 2006) are instances of a broader program proposed by Margulis.
3 November 2026
Jimmy Thorne
Structure, Ordering, and Embeddings of Groups
Abstract: This talk will be the first of a two-part series on group orderability and groups of diffeomorphisms of the interval ([0,1]).
In the first talk, I will introduce the necessary background on diffeomorphism groups and their connections to group orderability. In the second talk, the following week, I will discuss my current research on determining which torsion-free abelian-by-cyclic groups embed into these diffeomorphism groups, along with some results concerning one-relator groups.