The modular curve X1(N) parametrizes elliptic curves (whose points form an abelian group under a geometric group law) with a marked point of order N. A point on X1(N) is isolated if, roughly speaking, it is not a member of an infinite family of points parametrized by a geometric object. Elliptic curves without CM naturally give isolated points on X1(N), but the non-CM points remain mysterious. We develop an algorithm to test whether an elliptic curve E/Q without complex multiplication gives rise to an isolated point of any degree on any modular curve of the form X1(N). Running this algorithm on all elliptic curves presently in the L-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for our conjecture that only finitely many rational j-invariants give rise to an isolated point on X1(N) for any N; our conjectured list is explicit. This is joint work with Bourdon, Hashimoto, Keller, Klagsburn, Najman, and Shukla.

Gu, Mihalcea, Sharpe, Xu, Zhang, and Zhou proposed a conjectural presentation of the equivariant quantum K-ring of partial flag varieties, inspired by insights from quantum field theory in physics. In this talk, we will explain a proof of these conjectures based on the Toda presentation obtained by Maeno, Naito, and Sagaki for the complete flag variety Fl(n), as well as Kato’s algebra homomorphism from the quantum K-ring of Fl(n) to a partial flag variety.

The equivariant quantum K-theory ring of a flag variety is a Frobenius algebra equipped with a perfect pairing called the quantum K-metric. It is known that in the classical K-theory ring for a given flag variety the ideal sheaf basis is dual to the Schubert basis with regard to the sheaf Euler characteristic. We define a quantization of the ideal sheaf basis for the equivariant quantum K-theory of cominuscule flag varieties. These quantized ideal sheaves are then dual to the Schubert basis with regard to the quantum K-metric. We provide explicit type-uniform combinatorial formulae for the quantized ideal sheaves in terms of the Schubert basis for any cominuscule flag variety.

Research in math education has produced models for how children construct number. They construct an additive world of numbers through iterations of units of 1; and they construct a multiplicative world of numbers through transformations of units, From this childish perspective, the prime number theorem can be understood as a mapping between worlds: the natural logarithm maps a multiplicatively nested sequence of numbers (up to n) to an additively nested sequence of numbers (n). In this talk, I will share possible approaches to proving the prime number theorem by leveraging this understanding of the theorem. I’m hoping members of the algebra group can provide further insight for completing a truly simple proof of the theorem.

Given a nodal genus one curve and a proper subcurve of genus one, we will construct a contraction that collapses the subcurve to a genus one singularity. We will do this by first introducing residues for curves over local artinian rings and "twists" of these residues by tropical data from the curve, then we will use these residues to explicitly construct the contraction.

Affine geometry plays a crucial role in understanding transformations that preserve geometric structures. A key result is the fundamental theorem of affine geometry, which states that a bijective map between affine spaces that preserves collinearity (i.e., maps lines onto lines) must be a semi-affine map. An interesting application of the fundamental theorem of affine geometry to algebraic coding theory is in characterizing the permutation automorphisms of Reed-Muller codes. In their seminal paper in 1993, Berger and Charpin determined the permutation automorphism groups of Reed-Muller codes, showing that it is the affine general linear group. In this talk, we introduce affine spaces, outline a proof of the fundamental theorem of affine geometry, and explore its applications to coding theory, particularly in the context of Reed-Muller codes and their generalizations. We will also state a related open problem. If there is time and interest, we may outline some recent progress on this open problem. The latter is an ongoing joint work with Eduardo Camps Moreno, Sudhir R. Ghorpade and Hiram H. López.

Let X --> Y be a morphism of varieties and k a field. Consider the set X(k) of k-points, or solutions to the equations defining X with coordinates in k. If X = V(x^2 + y^2 - 1) for example, then the set of Q-points consists of primitive Pythagorean triples and the geometry of X shows there are infinitely many solutions. According to Hindry-Silverman, "Geometry determines Arithmetic." We want to determine the image of the map on k-points X(k) --> Y(k). Suppose X --> Y arises as the fiber over 0 of a family of varieties X' --> Y'. The geometry of the family X' --> Y' can also constrain the type of k-points that land in the image of X(k) --> Y(k), as Campana observed. Abramovich generalized this notion in a beautiful, incomplete manuscript called "birational geometry for number theorists" and gave us permission to realize this vision using log geometry. Log structures on X, Y remember enough about the families X', Y' to constrain the rational points. Joint work with Sara Mehidi, Marta Pieropan, Thibault Poiret.

Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a maps critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit are called post-critically finite (PCF), and these are of great interest in arithmetic dynamics. They are viewed as a dynamical analogue of abelian varieties with complex multiplication and often display interesting dynamical behavior. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction, and these maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials. Using this family, we can begin to explore properties of polynomial maps with two critical points. In this talk we will discuss how this family connects to the more general setting of bicritical polynomials, how we can use it to understand the $p$-adic Mandelbrot set, and how we can use it to determine reduction properties of PCF maps. In particular, we can use this family to provide necessary conditions for the existence of a PCF polynomial of degree d to have persistent bad reduction at a prime $p$. This demonstrates that considering maps with only one additional critical point may be enough to provide complete answers to questions in arithmetic dynamics.

Skein relations were historically first used to construct "quantum" invariants of knots in S^3, and this construction was generalized by Turaev to produce quantizations of certain character varieties of topological surfaces. In this talk we discuss Turaev's construction and some relationships to the elliptic Hall algebra of Burban-Schiffmann, and to the Hall algebra of the Fukaya category of the surface. This will be a survey-style talk, and in particular we will not assume prior knowledge of any of the objects described above.