**5-min lightning talks I (3 - 3:30 PM):**

*Emily Casey**Measuring Flatness*

Abstract: In calculus, we say that the graph of a function is nice if at each point the function has a derivative. When is the boundary of a domain, which is not necessarily the graph of a function, nice? We discuss two geometric functions that help us determine when the boundary of a domain will have a tangent line.

*Suh Young Choi**A Brief Life of Hypatia*

Abstract: Hypatia is best known for being the first female mathematician from the ancient Mediterranean for whom we have a surviving record. She was one of the foremost mathematicians, philosophers, and educators of her time, working and living in Alexandria at a time when the scholarly reputation of the city was beginning to fade. Her murder in 415 sent shockwaves through the intellectual and political communities in Alexandria because of the prominent position she held as a leader of both scholarship and education. This talk gives an overview of her life and circumstances, her contributions to mathematics, and her legacy.

*Be'eri Greenfeld**Noetherian Rings, Lie Algebras and Noncommutative Geometry: The Sierra-Walton Theorem*

Abstract: We give an overview of the Sierra-Walton theorem on the non-Noetherianity of universal enveloping algebras of many infinite-dimensional Lie algebras. This breakthrough made a significant progress toward a wide open problem in noncommutative algebra, and was achieved by a junction of ring theory, (noncommutative) algebraic geometry and infinite-dimensional Lie theory.

*Kaitlynn Lilly**A Numerical Riemann-Hilbert Approach to the Computation of Transform Pairs*

Abstract: This research presents a unified methodology integrating spectral theory, Riemann-Hilbert problems, and inverse scattering theory to efficiently derive, and numerically implement, transform pairs associated to time-evolution variable-coefficient partial differential equations (PDEs). More specifically, the approach combines analytical formulae with iterative ODE and Riemann-Hilbert methods to efficiently evaluate the forward and inverse transforms, giving a hybrid analytical-numerical method for such PDEs. The method is demonstrated on transforms arising in the solution of the time-dependent Schrӧdinger and Dirac equations, producing an accurate and stable time evolution method that does not require time stepping.

**Plenary talk followed by advice for early career mathematicians (3:30 - 4:30 PM):**

*Bianca Viray**The Interplay Between Geometry and Arithmetic*

Abstract: In this talk, I will introduce some of the overarching questions in the study of rational points, and how they motivate my current research. The talk will be accessible to a general audience.

**5-min lightning talks II (4:30 - 5 PM):**

*Julie Curtis**The Integer Decomposition Property in Smooth Lattice Polytopes*

Abstract: Polytopes are higher dimensional versions of polygons that live in Euclidean space. We can combine two polytopes via their Minkowski sum and examine the resulting polytope to determine if has what's called the Integer Decomposition Property, IDP. In this talk, we define what it means for a polytope to have the IDP and under what conditions a polytope will be IDP.

*Sara Ford**Locally Gentle Algebras*

Abstract: A brief introduction to the notions of path algebras with relations, locally gentle algebras, and the kinds of questions that we want to answer about them.

*Grace O'Brien**Graph Theory*

Abstract: This talk will be a speed-run of the topics I've been learning about this quarter including some basic definitions in spectral graph theory and a graph property called conformal rigidity.

*Xiaowen Zhu**Mathematical Aspects of Topological Insulators and Moire Materials*

Abstract: Topological insulators (TIs) are 2D materials that act as insulators within their bulk while demonstrating robust states along their edges. One anticipated key properties of TIs is the robustness of this characteristics in relation to changes in the shape of their edges. This talk will explore the influence of edge shape on the properties of TIs. Specifically, we will offer a general, intuitive condition for this property to hold, along with a counterexample demonstrating its absence. Additionally, we will present a quantitative version of this property with a potential physical interpretation. This talk is based on a joint work with Alexis Drouot.

**Free food (After 5 PM)**