Conjectural Description Research Articles

Chaotic Dynamics of Spatially Homogeneous Spacetimes

In the 1970’s, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright–Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright–Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows.

Regular Functions on the 𝐾-nilpotent cone

Let G G be a complex reductive algebraic group with Lie algebra g \mathfrak {g} and let G R G_{\mathbb {R}} be a real form of G G with maximal compact subgroup K R K_{\mathbb {R}} . Associated to G R G_{\mathbb {R}} is a K × C × K \times \mathbb {C}^{\times } -invariant subvariety N θ \mathcal {N}_{\theta } of the (usual) nilpotent cone N ⊂ g ∗ \mathcal {N} \subset \mathfrak {g}^* . In this article, we will derive a formula for the ring of regular functions C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] as a representation of K × C × K \times \mathbb {C}^{\times } . Some motivation comes from Hodge theory. In [Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences, Int. Press, Somerville, MA, 2011, pp. 397–420], Schmid and Vilonen use ideas from Saito’s theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If G R G_{\mathbb {R}} is split, and X X is the spherical principal series representation of infinitesimal character 0 0 , then conjecturally g r ( X ) ≃ C [ N θ ] gr(X) \simeq \mathbb {C}[\mathcal {N}_{\theta }] as representations of K × C × K \times \mathbb {C}^{\times } . So a formula for C [ N θ ] \mathbb {C}[\mathcal {N}_{\theta }] is an essential ingredient for computing Hodge filtrations.

A Serre weight conjecture for geometric Hilbert modular forms in characteristic $p$

Let p be a prime and F a totally real field in which p is unramified. We consider mod p Hilbert modular forms for F , defined as sections of automorphic line bundles on Hilbert modular varieties of level prime to p in characteristic p . For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without parity hypotheses), we associate a two-dimensional representation of the absolute Galois group of F , and we give a conjectural description of the set of weights of all eigenforms from which it arises. This conjecture can be viewed as a “geometric” variant of the “algebraic” Serre weight conjecture of Buzzard–Diamond–Jarvis, in the spirit of Edixhoven's variant of Serre's original conjecture in the case F = \mathbb{Q} . We develop techniques for studying the set of weights giving rise to a fixed Galois representation, and prove results in support of the conjecture, including cases of partial weight 1.

Orthogonal root numbers of tempered parameters

We show that an orthogonal root number of a tempered L-parameter varphi decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a central involution of Langlands’s central character for varphi . The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil–Deligne representations arising in a conjectural description of the Plancherel measure.

On a class of generalized Fermat equations of signature (2,2n,3)

We consider the Diophantine equation 7x2+y2n=4z3. We determine all solutions to this equation for n=2,3,4 and 5. We formulate a Kraus type criterion for showing that the Diophantine equation 7x2+y2p=4z3 has no non-trivial proper integer solutions for specific primes p>7. We computationally verify the criterion for all primes 7<p<109, p≠13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7x2+y2p=4z3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x2+7y2n=4z3, determining all families of solutions for n=2 and 3, as well as giving a (mostly) conjectural description of the solutions for n=4 and primes n≥5.

Gromov\u2013Witten theory of K3 surfaces and a Kaneko\u2013Zagier equation for Jacobi forms

We prove the existence of quasi-Jacobi form solutions for an analogue of the Kaneko–Zagier differential equation for Jacobi forms. The transformation properties of the solutions under the Jacobi group are derived. A special feature of the solutions is the polynomial dependence of the index parameter. The results yield an explicit conjectural description for all double ramification cycle integrals in the Gromov–Witten theory of K3 surfaces.

Stable Pair Invariants of Local Calabi–Yau 4-folds

Abstract In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ using Gromov–Witten theory. Recently, Cao–Maulik–Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao–Maulik–Toda conjectures for low-degree curve classes and find connections to Carlsson–Okounkov numbers. Some of our verifications involve genus zero Gopakumar–Vafa type invariants recently determined in the context of the log-local principle by Bousseau–Brini–van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao–Maulik–Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$.

Depth-graded motivic multiple zeta values

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.

Koszul algebras and quadratic duals in Galois cohomology

We investigate the Galois cohomology of finitely generated maximal pro-p quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with the p-Zassenhaus filtration on the group. This paper also contains a survey of Koszul property in Galois cohomology and its relation with absolute Galois groups.

Special components of Noether–Lefschetz loci

We take a sum $$C_1+rC_2, r\in {\mathbb {Q}}$$ of a line $$C_1$$ and a complete intersection curve $$C_2$$ of type (3, 3) inside a smooth surface of degree 8 and with $$C_1\cap C_2=\emptyset $$ . We gather evidences to the fact that for all except a finite number of $$r$$ , the Noether–Lefschetz loci attached to the cohomology classes of $$C_1+rC_2$$ are distinct 31 codimensional subvarieties intersecting each other in a 32 codimensional subvariety of the ambient space. The maximum codimension for components of the Noether–Lefschetz locus in this case is 35, and hence, we provide a conjectural description of a counterexample to a conjecture of J. Harris. The methods used in this paper also produce in a rigorous way an infinite number of general components passing through the point representing the Fermat surface of degree $$\le 9$$ , and many non-reduced components for such degrees.

Moment Maps and Galois Orbits in Quantum Information Theory

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb{C}^d$. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm{U}(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

GEOMETRY OF KOTTWITZ–VIEHMANN VARIETIES

AbstractWe study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

ON THE HUMPHREYS CONJECTURE ON SUPPORT VARIETIES OF TILTING MODULES

Let G be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number. The of a G-module M is a certain closed subvariety of the nilpotent cone of G, defined in terms of cohomology for the first Frobenius kernel of G. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for G=SLn in earlier work of the second author. In this paper, we show that for any G, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when p is sufficiently large. We also prove variants of these statements involving relative support varieties.

On exceptional sets in Manin’s conjecture

In this survey paper, we study Manin’s conjecture from a geometric perspective. The focus of the paper is the recent conjectural description of the exceptional set in Manin’s conjecture due to Lehmann–Sengupta–Tanimoto. After giving an extensive background, we give a precise description of this set and compute it in many examples.

The Langlands parameter of a simple supercuspidal representation: symplectic groups

Let $$\pi $$ be a simple supercuspidal representation of the symplectic group $${\mathrm {Sp}}_{2l}(F)$$, over a p-adic field F. In this work, we explicitly compute the Rankin–Selberg $$\gamma $$-factor of rank-1 twists of $$\pi $$. We then completely determine the Langlands parameter of $$\pi $$, if $$p \ne 2$$. In the case that $$F = \mathbb {Q}_2$$, we give a conjectural description of the functorial lift of $$\pi $$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.

THE HALL ALGEBRAS OF SURFACES I

We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with $m$ marked intervals and we give a conjectural description of the Hall algebras of all surfaces with enough marked intervals. Then we use a functoriality result to show that a graded version of the HOMFLY-PT skein relation holds among certain arcs in the Hall algebras of general surfaces.

Syzygies of Determinantal Thickenings and Representations of the General Linear Lie Superalgebra

We let S denote the ring of polynomial functions on the space of m x n matrices, and consider the action of the group GL = GL_m x GL_n via row and column operations on the matrix entries. For a GL-invariant ideal I in S we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I is the ideal generated by the GL-orbit of a highest weight vector, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution.

DAHA and plane curve singularities

We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities. This generalizes the Cherednik-Danilenko conjecture on the Betti numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of superpolynomials of torus knots and that by Gorsky-Mazin for their constant term. The paper mainly focuses on non-torus algebraic knots. A connection with the conjecture due to Oblomkov-Rasmussen-Shende is possible, but our approach is different. A motivic version of our conjecture is related to p-adic orbital A-type integrals for anisotropic centralizers.

Quadruply-graded colored homology of knots

We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of arbitrary knots. We propose an explicit conjectural description for the rectangular colored homology of torus knots, and identify the new gradings in this context. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLYPT homology as the Hochschild homology of the category of branes in a Landau–Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work.

Weil-Étale Cohomology and Zeta-Values of Proper Regular Arithmetic Schemes

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme \mathcal{X} at any integer n in terms of Weil-étale cohomology complexes. This extends work of S. Lichtenbaum [Compos. Math. 141, No. 3, 689–702 (2005; Zbl 1073.14024)] and T. Geisser [Math. Ann. 330, No. 4, 665–692 (2004; Zbl 1069.14021)] for \mathcal{X} of characteristic p , of S. Lichtenbaum [Ann. Math. (2) 170, No. 2, 657–683 (2009; Zbl 1278.14029)] for \mathcal{X}=\mathrm{Spec}(\mathcal{O}_F) and n=0 where F is a number field, and of the second author for arbitrary \mathcal{X} and n=0 [B. Morin, Duke Math. J. 163, No. 7, 1263–1336 (2014; Zbl 06303878)]. We show that our conjecture is compatible with the Tamagawa number conjecture of S. Bloch and K. Kato [Prog. Math. 86, 333–400 (1990; Zbl 0768.14001)], and J.-M. Fontaine and B. Perrin-Riou [Proc. Symp. Pure Math. 55, 599–706 (1994; Zbl 0821.14013)] if \mathcal{X} is smooth over \mathrm{Spec}(\mathcal{O}_F) , and hence that it holds in cases where the Tamagawa number conjecture is known.