Unlikely Intersections Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 20, 2022
Tangli Ge :
2 p.m. in 636 SEO
Abstract
Let A be an abelian variety over a number field. The Mordell—Lang conjecture (resp. the Bogomolov conjecture) states that a general subvariety of A contains few rational points (resp. few points of small Néron—Tate height). They were proved in the late 20th century. Later, Poonen and Zhang independently showed that the equidistribution theorem can be used to merge two results into a stronger one, which is called the Mordell—Lang plus Bogomolov. In this talk, I will explain how to get a uniform version of this combined theorem. This talk is partially based on the joint work with Gao and Kühne.
Oct. 18, 2022
Sebastian Eterovic :
2 p.m. in 636 SEO
Abstract
In this talk I will present a strong counterpart to the Zilber-Pink conjecture. Zilber-Pink predicts that if X is a proper subvariety of a special kind of variety S and X is not contained in a proper special subvariety of S, then the union of the unlikely intersections of X with the proper special subvarieties of S is not Zariski dense in X. We will see that the likely intersections of X (which are defined in a slightly more delicate way than one might naively expect) are Euclidean dense in X. This is joint work with Tom Scanlon.
Nov. 8, 2022
Gabriel Dill :
2 p.m. in Zoom
Abstract
I will present joint work with Fabrizio Barroero (Roma Tre), in which we develop the axiomatic framework of distinguished and very distinguished categories for studying unlikely intersections (over a fixed algebraically closed ground field of characteristic 0). In distinguished categories, one can define special and weakly special subvarieties and formulate a Zilber-Pink statement. (Semi-)Abelian varieties and algebraic tori as well as pure and mixed Shimura varieties all form distinguished categories and the aforementioned notions specialize to the usual ones in these cases. We prove that the Zilber-Pink statement for a very distinguished category implies the same statement over any algebraically closed field extension of the ground field. This yields new instances of the Zilber-Pink conjecture in semiabelian varieties as well as in (mixed) Shimura varieties. Time permitting, I will finish by mentioning work in progress on generalized Hecke orbits in distinguished categories.
Jan. 24, 2023
Patrick Ingram :
2 p.m. in 636 SEO
Patrick Ingram :
2 p.m. in 636 SEO
Abstract
A result of Malmquist establishes the rationality of solutions to non-Riccati differential equations of the form f’=R(z,f), where R is a rational function, while an analogous result of Yanagihara places restrictions on solutions to the difference equation f(z+1)=R(z,f). In this talk we will look at the problem of bounding the number of solutions to such equations, and if time allows we will also mention an improvement to Nevanlinna’s Second Main Theorem for solutions to difference equations.
Feb. 16, 2023
Jamie Juul :
2 p.m. in 636 SEO
Abstract
We study the Galois groups of the extensions $K((f'\circ f^n)^{-1}(0))/K$ where $K$ is a number field for polynomials $f(x)\in K[x]$. We use the results to study the proportion of primes for which $f$ has a $p$-adic attracting periodic point for a "typical" $f$ and apply the statement to the split case of the Dynamical Mordell-Lang Conjecture.
April 18, 2023
Vahagn Aslanyan :
2 p.m. in 636 SEO
Abstract
There are three important conjectures at the boundary between model theory, arithmetic geometry, and complex geometry -- Schanuel, Zilber-Pink, and Existential Closedness. I will discuss these conjectures for the modular j-function and mostly focus on novel versions incorporating the derivatives of j. I will then explain the relationship between them including a new link which is visible only when we consider the derivatives. At the end I will present the functional/differential variants of these conjectures which are all theorems.
Sept. 12, 2023
Aaron Levin :
2 p.m. in 636 SEO
Abstract
The classical Weil height machine associates heights to divisors on a projective variety. I will give a brief introduction to how this machinery extends to objects (closed subschemes) in higher codimension, due to Silverman, and present various ways to interpret the heights. We will then discuss several recent results in Diophantine approximation in which these ideas play a prominent and central role.
Sept. 26, 2023
Ananth Shankar :
2 p.m. in 636 SEO
Abstract
I will talk about the folklore question of whether there exist abelian varieties (of dimension $\geq4$) not isogenous to Jacobians, in the context of unlikely intersections. I will also touch upon how this question relates to the André--Oort conjecture (now, a theorem). If time permits, I will describe why we expect unlikely intersections to occur over $\overline{\mathbb{F}_p}$, even though we don't expect unlikely intersections in characteristic zero.
Nov. 1, 2023
Max Weinreich :
3 p.m. in 1227 SEO
Abstract
Billiards is a dynamical system that models the behavior of a point particle bouncing around some region. If the region is a plane region bounded by an algebraic curve, then we can use tools from algebraic geometry to study the billiards map in that region. In this talk, we explain how to view billiards as a complex algebraic correspondence, and we investigate its dynamical degree, a difficult-to-compute invariant that measures the asymptotic growth rate of the degrees of the iterates. We show that the dynamical degree of billiards in a general plane curve of degree $d$ is approximately $2d^2$.
Nov. 14, 2023
Leo Jimenez :
4 p.m. in 636 SEO
Abstract
An ordinary algebraic differential equation is said to be internal to the constants if its general solution is obtained as a rational function of finitely many of its solutions and finitely many constant terms. Any such equation has an algebraic group acting as its Galois group. In this talk, I will use decomposition theorems for algebraic groups to show that some internal equations (do not) split into a product of internal equations. The methods are model-theoretic and could be applied to other contexts. This is a joint work in progress with Christine Eagles.
Nov. 26, 2024
Jenn Park :
2 p.m. in 636 SEO
Abstract
This work is joint with Francesca Balestrieri, Kevin Destagnol, Julian Lyczak, and Nick Rome. Manin-Peyre conjecture has been the subject of intense research in the past few decades, and it has been studied from various perspectives including algebra, arithmetic, analytic, and logic. We provide a general framework for the Manin-Peyre conjecture for for the symmetric square of any del Pezzo surface X, and prove the conjecture for the infinite family of nonsplit quadric surfaces. Previously, there were only two examples in the literature: P^2 and P^1 x P^1. In order to achieve the predicted asymptotic, we show that a type II thin set of a new flavour must be removed. A key tool we develop and that can be applied to further examples is a result for summing multiplicative functions and Euler products over quadratic extensions. To establish our counting result for the specific family of quadric surfaces, we improve existing lattice point counting results in the literature and make crucial use of a novel form of lattice point counting.
Dec. 3, 2024
James Freitag :
2 p.m. in 636 SEO
Abstract
We will give a survey of nonstandard methods for problems in commutative algebra. This talk is planned to be fairly accessible, but we will assume the audience understands what an ultrafilter is.
April 15, 2025
Gyujin Oh :
2 p.m. in 636 SEO
Abstract
For a smooth projective curve over a finite field, we construct the p-adic analytic moduli stack of isocrystals and study its geometry. This is a crystalline analogue of the moduli of integrable connections; in particular, it admits a Frobenius pullback endomorphism, even though the moduli space is a characteristic 0 object. We will illustrate, with the concrete example of rank 1 moduli, how its interesting geometry can be used to count (the p-adic analogues of) rank 1 local systems over a curve. Joint work with Koji Shimizu.