Related papers: DR/DZ equivalence conjecture and tautological rela…

Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, is algebraic. We show that this holds for all degree $k$ away from the…

We show that various loci of stable curves of sufficiently large genus admitting degree $d$ covers of positive genus curves define non-tautological algebraic cycles on $\overline{\mathcal{M}}_{g,N}$, assuming the non-vanishing of the $d$-th…

We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of $g$ copies of a product of spheres $S^d \times S^d$ and an odd $d$, we find numerous…

Reflexive dg categories were introduced by Kuznetsov and Shinder to abstract the duality between bounded and perfect derived categories. In particular this duality relates their Hochschild cohomologies, autoequivalence groups, and…

Poisson-Lie (PL) T-duality has received much attention over the last five years in connection with integrable string worldsheet theories. At the level of the worldsheet, the algebraic structure underpinning these connections is made…

A $D_{\infty}$-topological Markov chain can be represented by a pair of zero-one square matrices, which is called a flip pair. We introduce the concepts of $D_{\infty}$-strong shift equivalence and $D_{\infty}$-shift equivalence, which are…

We study 2+1D toroidal compactifications of M-theory with twists in the U-duality group. These compactifications realize many symmetric-manifolds from the classification of 2+1D extended supergravity moduli-spaces. We then focus on the…

We consider unitary Shimura varieties at places where the totally real field ramifies over $\mbQ$. Our first result constructs comparison isomorphisms between absolute and relative local models in this context, which relies on a…

We extend the definition of relative Gromov--Witten invariants with negative contact orders to all genera. Then we show that relative Gromov--Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are…

We prove a relative Lefschetz-Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$-category of cohomological correspondences. We show that local…

We show that the $p$-adic KZ connection associated with the family of curves $y^q=(t-z_1)\dots (t-z_{qg+1})$ has an invariant subbundle of rank $g$, while the corresponding complex KZ connection has no nontrivial proper subbundles due to…

We study the rational homology of the Deligne--Mumford compactification $\overline{\mathcal M}_{g,n}$ of the moduli space of stable curves via a family of Morse functions, namely the $\text{sys}_T$ functions. Exploiting the geometric and…

We introduce proof terms for string rewrite systems and, using these, show that various notions of equivalence on reductions known from the literature can be viewed as different perspectives on the notion of causal equivalence. In…

Dualities of M-theory are used to determine the exact dependence on the coupling constant of the D^6R^4 interaction of the IIA and IIB superstring effective action. Upon lifting to eleven dimensions this determines the coefficient of the…

We employ the $1/2$-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Faber's formula for proportionalities of kappa-classes on $\mathcal{M}_g$, $g\geq 2$. We…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

We show that a pair of almost commuting self-adjoint, symmetric matrices is close to a pair of commuting self-adjoint, symmetric matrices (in a uniform way). Moreover we prove that the same holds with self-dual in place of symmetric. The…

We present a conjecture for the Chow ring of the universal moduli stack of bundles over hyperelliptic curves and prove it for rank and genus two. Consequently, we obtain explicit generators and relations to conclude that the Chow ring is…

Using twisted nearby cycles, we define a new notion of slopes for complex holonomic D-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly…

We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer about equality of 2-rank…