We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

Abelian Surfaces over totally real fields are Potentially Modular

A Deletion-Contraction Relation for the Chromatic Symmetric Function

This paper contributes to a programme initiated by the first author: `How much information about a graph is revealed in its Potts partition function?'. We show that the W-polynomial distinguishes non-isomorphic weighted trees of a good family. The framework developed to do so also allows us to show that the W-polynomial distinguishes non-isomorphic caterpillars. This establishes Stanley's isomorphism conjecture for caterpillars, an extensively studied problem.

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Isomorphism of Weighted Trees and Stanley's Isomorphism Conjecture for Caterpillars

Flux compactification of IIB string theory associates special points in Calabi-Yau moduli space to choices of (pairs of) integral three-form fluxes. In this paper, we propose that supersymmetric flux vacua are modular. That is, to a supersymmetric flux vacuum arising in a variety defined over $\mathbb{Q}$, we associate a two-dimensional Galois representation that we conjecture to be modular. We provide numerical evidence for our conjecture by examining flux vacua arising on the octic hypersurface in $\mathbb{P}^{4}(1,1,2,2,2)$.

Supersymmetric Flux Compactifications and Calabi-Yau Modularity

We investigate how the motive of hyper-Kahler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular modul...

https://www.worldscientific.com/doi/pdf/10.1142/S0219199720500340

On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties

It is a long-standing conjecture of Stanley that the chromatic symmetric function distinguishes unrooted trees. Previously, the best computational result came from Russell, who proved this for all trees with at most $25$ vertices. In this paper, we present a novel probabilistic algorithm which may be used to check the conjecture more efficiently. Applying it, we verify the conjecture for all trees with up to $28$ vertices.

On an Algorithm for Comparing the Chromatic Symmetric Functions of Trees

1.1 Definitions Let X be a smooth projective variety of dimension d and let H∗(−) be a Weil cohomology theory. In particular, we have Poincaré duality, a Künneth formula, a cycle map, and the Hard Lefschetz theorem. Definition 1.1. A correspondence u ∈ H∗(X × Y ) is a correspondence when viewed as a map u : H∗(X) → H∗(Y ) under the isomorphisms given by the Künneth formula and Poincaré duality: H∗(X × Y ) ∼= H∗(X)⊗H∗(Y ) ∼= Hom(H∗(X),K)⊗H∗(Y ) ∼= Hom(H∗(X), H∗(Y )). We use A∗(X) to denote the Q-subspace spanned by the cycle class map from the Chow ring toH∗(X). Correspondences contained in A∗(X × Y ) are called algebraic.

The Standard Conjectures

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The distinguishing number of a graph was introduced by Albertson and Collins as a measure of the amount of symmetry contained in the graph. Tymoczko extended this definition to faithful group actions on sets; taking the set to be the vertex set of a graph and the group to be the automorphism group of the graph allows one to recover the previous definition. Since then, several authors have studied properties of the distinguishing number as well as extensions of the notion. In this paper, we first answer a few open questions regarding the distinguishing number. Next we turn to generalizations regarding the labeling of Cartesian powers of a set and the different subgroups that can be obtained through labelings. We then introduce a new partially ordered set on partitions that follows naturally from extending the theory of distinguishing numbers to that of distinguishing partitions. Then we investigate the groups obtainable from partitioning Cartesian powers of a set in more detail and show how the original notion of the distinguishing number of a graph can be recovered in this way. Next, we introduce a polynomial and a symmetric function generalization of the distinguishing number. Finally, we present a large number of open questions and problems for further research.

Distinguishing Numbers and Generalizations

The cyclic sieving phenomenon is a well-studied occurrence in combinatorics appearing when a cyclic group acts on a finite set. In this paper, we demonstrate a natural extension of this theory to finite abelian groups. We also present a similar result for dihedral groups and suggest approaches for natural generalizations to nonabelian groups.

Caleb Ji

Papers

On an Algorithm for Comparing the Chromatic Symmetric Functions of Trees

The Standard Conjectures

On an Algorithm for Comparing the Chromatic Symmetric Functions of Trees

Distinguishing Numbers and Generalizations

The sieving phenomenon for finite groups