Graduate Student AG Seminar

Title: Manin's Conjecture

Abstract: Finding solutions to polynomial equations has seen a long history in mathematics. In algebraic geometry, this translates to  the study of rational points on algebraic varieties. In this talk, I will focus on one aspect of the area: the distribution of rational points on Fano varieties. I will talk about the formulation of Manin's conjecture and a heuristic by Batyrev on the asymptotic formula on the growth of rational points.

Title: Exceptional sets in Manin's Conjecture

Abstract: Let X be a smooth Fano variety, Manin's conjecture predicts a counting formula for the growth of rational points on X. In particular, one has to delete an exceptional set from X to make the formula work. Continuing my last talk, I will discuss the geometric invariants arising in the formula and how to remove such an exceptional set from X based on these invariants.

Due to Distinguished Lecture Series.

Title: Intersection theory and intrinsic normal cone

Due to Number Theory seminar.

Title: Intersection theory and intrinsic normal cone

Title: Connections between ReLU neural network and toric geometry

Abstract: The theory of elliptic pairs provides useful conditions to determine polyhedrality of the pseudo-effective cone, which give rise to interesting arithmetic questions when reducing the variety modulo p. In this talk, we examine one such case, namely the blow-up of 9 points in P^2 lying on the nodal cubic, and study the density of primes p for which the pseudo-effective cone of the reduction of X modulo p is polyhedral.

Paper link: https://arxiv.org/abs/2311.16281

Abstract: The theory of elliptic pairs provides useful conditions to determine polyhedrality of the pseudo-effective cone, which give rise to interesting arithmetic questions when reducing the variety modulo p. In this talk, we examine one such case, namely the blow-up of 9 points in P^2 lying on the nodal cubic, and study the density of primes p for which the pseudo-effective cone of the reduction of X modulo p is polyhedral.

Paper link: https://arxiv.org/abs/2311.16281

Title: Intersection Theory on the Moduli Space of Curves

Title: The art of compactifying group schemes 

Abstract: Given a group scheme G, a compactification of G is the data of a nice embedding of G into a smooth projective scheme. In this talk, we will do an extremely shoddy job of exploring such compactification and what they look like in specific examples.

Title: Intro to Brill-Noether Theory: Curves, Divisors and Maps to Projective Space.

Abstract: Many advancements were made in Brill-Noether Theory in the 1980s; the Brill-Noether theorem and its proof being at the forefront. In this talk, we aim to introduce Brill-Noether theory, establish some fundamentals and state and motivate the Brill-Noether theorem. Time permitting, we will sketch a more recent proof of the theorem that uses degeneration. Be warned, it will surely be a very sketchy proof.  Most of this talk is motivated by Isabel Vogt's lecture series at the SLMath Summer School on Curves.

Title: An Introduction to K-Stability

Abstract: The introduction of K-Stability in algebraic geometry has led to a revolution of sorts. After Yau proved the Calabi conjecture for canonically-polarised and what are now called Calabi-Yau manifolds, the case for Fanos remained unsolved for over 40 years. There was some obstruction for Fano varieties to admit a certain prescribed scalar curvature metric. After decades of work, the obstruction was found to be K-Stability. In this talk, we will recall the various relevant notions of complex geometry, give an overview of the history of its development, and go through some examples, using modern techniques