Related papers: Rational motivic path spaces and Kim's relative un…

Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a separated scheme $X$ smooth, geometrically connected, and of finite type over $k$ and a smooth proper morphism $f:Y\rightarrow X$. The main result of this paper…

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. $\mathrm{Hom}(\pi,G)_0$ denotes the null-component of the space of homomorphisms from $\pi$ to $G$, and $\mathrm{map}_*(B\pi,BG)_0$ denotes the null-component of…

We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was…

The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and…

For a reductive group $G$, we prove that complex irreducible rigid $G$-local systems with quasi-unipotent monodromies and finite order abelianization on a smooth curve are motivic, generalizing a theorem of Katz for $GL_n$. We do so by…

In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of R\'emond's generalisation of…

For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the…

We develop a motivic framework for Feynman integrals of one-loop graphs in momentum space. Its advantage compared to the already existing framework in Feynman representation is that it naturally includes graphs with cuts. To each such…

Simple boundary expressions for the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The method is by virtual localization on the moduli space of maps to the projective…

A key triviality result for support extension maps for motivic $\mathbb{A}^1$-homotopies of cellular motivic spaces $S$ over a DVR spectrum $B$ is proven. Combining with earlier known results on Gersten complex and the K-theory motivic…

In this essay we study various notions of projective space (and other schemes) over $\mathbb{F}_{1^\ell}$, with $\mathbb{F}_1$ denoting the field with one element. Our leading motivation is the "Hiden Points Principle," which shows a huge…

Let X be a smooth projective curve of genus >1 over a field K which is finitely generated over the rationals. The section conjecture in Grothendieck's anabelian geometry says that the sections of the canonical projection from the arithmetic…

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that…

In the large rank limit, for any nonexceptional affine algebra, the graded branching multiplicities known as one-dimensional sums, are conjectured to have a simple relationship with those of type A, which are known as generalized Kostka…

Given a finite group $\Gamma$, we prove results on the distribution of the prime-to-$q|\Gamma|$ part of fundamental groups of $\Gamma$-covers of the projective line $\mathbb P^1_{\mathbb F_q}$ over a finite field $\mathbb F_q$ as…

Let k be a field, let G be a finite group and let T be a split k-torus on which G acts multiplicatively, and for every m greater than 1 denote by T[m] the m-torsion subgroup of T. Under a suitable assumption on m, we show that the motivic…

The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and…

We revisit the low-energy effective $U(1)$ action of topologically twisted $\mathcal N=2$ SYM theory with gauge group of rank one on a generic oriented smooth 4-manifold $X$ with nontrivial fundamental group. After including a specific new…

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the…

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational…