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We study the back stable Schubert calculus of the infinite flag variety. Our main results are:
– a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;
– a novel definition of double and triple Stanley symmetric functions;
– a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;
– the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;
– the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;
– equivariant Pieri rules for the homology of the infinite Grassmannian;
– homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.
We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.
The purpose of this note is to correct a mistake in the article “A curve selection lemma in spaces of arcs and the image of the Nash map” Compositio Math. 142 (2006), 119–130. It is due to an overlooked hypothesis in the definition of generically stable subset of the space of arcs X∞ of a variety X defined over a perfect field k.
Building upon ideas of the second and third authors, we prove that at least
$2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$
values of the Riemann zeta function at odd integers between 3 and
$s$
are irrational, where
$\unicode[STIX]{x1D700}$
is any positive real number and
$s$
is large enough in terms of
$\unicode[STIX]{x1D700}$
. This lower bound is asymptotically larger than any power of
$\log s$
; it improves on the bound
$(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$
that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.
We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$-theory. The Milnor $K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.
Let
$G$
be a connected split reductive group over a finite field
$\mathbb{F}_{q}$
and
$X$
a smooth projective geometrically connected curve over
$\mathbb{F}_{q}$
. The
$\ell$
-adic cohomology of stacks of
$G$
-shtukas is a generalization of the space of automorphic forms with compact support over the function field of
$X$
. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.
We revisit the paper [Automorphy lifting for residually reducible$l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$,
where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.
The commutative differential graded algebra $A_{\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal {I}}(X)$ of $A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal {I}$ to model $E_{\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\mathcal {I}$-dgas. We define a functor $A^{\mathcal {I}}$ from simplicial sets to commutative $\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty }$ dga of cochains. The functor $A^{\mathcal {I}}$ shares many properties of $A_{\mathrm {PL}}$, and can be viewed as a generalization of $A_{\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.
We study a class of two-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.