Conjecture In Theory Research Articles

Towards a classification of connected components of the strata of $k$-differentials

A k -differential on a Riemann surface is a section of the k -th power of the canonical bundle. Loci of k -differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of k -differentials. The classification of connected components of the strata of k -differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich-Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of k -differentials for general k . As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of k -differentials by generalizing the hyperelliptic structure and spin parity for higher k . We also describe an approach to determine explicitly parities of k -differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale k -differentials introduced by Bainbridge-Chen-Gendron-Grushevsky-Möller for k = 1 and extended by Costantini-Möller-Zachhuber for all k .

Quasi-isolated blocks and the Alperin–McKay conjecture

AbstractThe Alperin–McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin–McKay conjecture holds if the so-called inductive Alperin–McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper, we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this, we obtain a proof of the Alperin–McKay conjecture for$2$-blocks of finite groups with abelian defect.

Vaught's conjecture for theories admitting finite monomorphic decompositions

An infinite linear order with finitely many unary relations (colors), $\langle X,{ \lt }, U_0,\dots ,U_{n-1} \rangle $, is a <em>good colored linear order</em> iff the largest convex partition of the set $X$ refining the partition generated by the sets $U

The statistical mechanics of near-BPS black holes

Due to the failure of thermodynamics for low temperature near-extremal black holes, it has long been conjectured that a ‘thermodynamic mass gap’ exists between an extremal black hole and the lightest near-extremal state. For non-supersymmetric near-extremal black holes in Einstein gravity with an AdS 2 throat, no such gap was found. Rather, at that energy scale, the spectrum exhibits a continuum of states, up to non-perturbative corrections. In this paper, we compute the partition function of near-BPS black holes in supergravity where the emergent, broken, symmetry is PSU(1, 1|2). To reliably compute this partition function, we show that the gravitational path integral can be reduced to that of a supersymmetric extension of the Schwarzian theory, which we define and exactly quantize. In contrast to the non-supersymmetric case, we find that black holes in supergravity have a mass gap and a large extremal black hole degeneracy consistent with the Bekenstein–Hawking area. Our results verify a plethora of string theory conjectures, concerning the scale of the mass gap and the counting of extremal micro-states.

Milnor-Witt homotopy sheaves and Morel generalized transfers

We explore a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf and prove that the conjecture is true with rational coefficients. Moreover, we study the relations between (contracted) homotopy sheaves, sheaves with Morel generalized transfers and Milnor-Witt homotopy sheaves, and prove an equivalence of categories. As applications, we describe the essential image of the canonical functor that forgets Milnor-Witt transfers and use these results to discuss the conservativity conjecture in motivic homotopy theory due to Bachmann and Yakerson.

Jordan decomposition for the Alperin–McKay conjecture

Späth showed that the Alperin–McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin–McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnafé–Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.

Tackling the SDC in AdS with CFTs

We study the Swampland Distance Conjecture for supersymmetric theories with AdS5 backgrounds and fixed radius through their mathcal{N} = 2 SCFT holographic duals. By the Maldacena-Zhiboedov theorem, around a large class of infinite-distance points there must exist a tower of exponentially massless higher-spin fields in the bulk, for which we find bounds on the decay rate in terms of the conformal data. We discuss the origin of this tower in the gravity side for type IIB compactification on S5 and its orbifolds, and comment about more general cases.

An infinite family of counterexamples to a conjecture on positivity

Recently, G. Mason has produced a counterexample of order 128 to a conjecture in conformal field theory and tensor category theory in [2]. Here we easily produce an infinite family of counterexamples, the smallest of which has order 72.

Some remarks on Kida’s formula when $$\mu \ne 0$$

The Kida's formula in classical Iwasawa theory relates the Iwasawa $\lambda$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $\lambda$-invariants of Selmer groups under an appropriate $\mu=0$ assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when $\mu\neq 0$. The conjectural component comes from the so-called $\mathfrak{M}_H(G)$-conjecture in noncommutative Iwasawa theory.

Spanning subspace configurations

A spanning configuration in the complex vector space $${{\mathbb {C}}}^k$$ is a sequence $$(W_1, \dots , W_r)$$ of linear subspaces of $${{\mathbb {C}}}^k$$ such that $$W_1 + \cdots + W_r = {{\mathbb {C}}}^k$$ . We present the integral cohomology of the moduli space of spanning configurations in $${{\mathbb {C}}}^k$$ corresponding to a given sequence of subspace dimensions. This simultaneously generalizes the classical presentation of the cohomology of partial flag varieties and the more recent presentation of a variety of spanning line configurations defined by the author and Pawlowski. This latter variety of spanning line configurations plays the role of the flag variety for the Haglund–Remmel–Wilson Delta Conjecture of symmetric function theory.

Solution to the index conjecture in zero-sum theory

The index conjecture in zero-sum theory states that if gcd⁡(n,6)=1 then every minimal zero-sum sequence of length 4 modulo n has index 1. In this paper, we prove a version of this conjecture for all n>N for some absolute constant N.

On the Multiplicity One Conjecture in min-max theory

We prove that in a closed manifold of dimension between $3$ and $7$ with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.

Duality and supersymmetry constraints on the weak gravity conjecture

Positivity bounds coming from consistency of UV scattering amplitudes are not always sufficient to prove the weak gravity conjecture for theories beyond Einstein-Maxwell. Additional ingredients about the UV may be necessary to exclude those regions of parameter space which are naïvely in conflict with the predictions of the weak gravity conjecture. In this paper we explore the consequences of imposing additional symmetries inherited from the UV theory on higher-derivative operators for Einstein-Maxwell-dilaton-axion theory. Using black hole thermodynamics, for a preserved SL(2, ℝ) symmetry we find that the weak gravity conjecture then does follow from positivity bounds. For a preserved O(d, d; ℝ) symmetry we find a simple condition on the two Wilson coefficients which ensures the positivity of corrections to the charge-to-mass ratio and that follows from the null energy condition alone. We find that imposing supersymmetry on top of either of these symmetries gives corrections which vanish identically, as expected for BPS states.

Quantizing derived mapping stacks

In this paper we discuss several topological and geometric invariants obtained by quantizing [Formula: see text]-models. More precisely, we do not quantize the entire mapping stack of fields, but rather only the substack of low energy fields. The theory restricted to this substack can be presented Lie theoretically and the problem is reduced to perturbative gauge theory. Throughout, we make extensive use of derived symplectic geometry and the BV formalism of Costello and Gwilliam. Finally, we frame the AJ conjecture in knot theory as a question of quantizing character stacks.

Prime and coprime values of polynomials

The Schinzel Hypothesis is a celebrated conjecture in number theory linking polynomial values and prime numbers. In the same vein we investigate the common divisors of values P_1(n),\ldots, P_s(n) of several polynomials. We deduce this coprime version of the Schinzel Hypothesis: under some natural assumption, coprime polynomials assume coprime values at infinitely many integers. Consequences include a version "modulo an integer" of the original Schinzel Hypothesis, with the Goldbach conjecture, again modulo an integer, as a special case.

A brief history of Tarskian algebraic logic with new perspectives and innovations

We take a magical tour in algebraic logic, which is the natural interface between universal algebra and mathematical logic, starting from classical results on neat embeddings due to Andreḱa, Henkin, Nemeti, Monk and Tarski, all the way to recent novel results in algebraic logic using so-called rainbow constructions. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. Other topics dealt with include the interaction of algebraic and modal logic, the so-called (central still active) finitizability problem, Godels’s incompleteness Theorem in guarded fragments, counting the number of subvarieties of $$\textsf {RCA}_{\omega }$$ which is reminiscent of Shelah’s classification theory and the interaction of algebraic logic and descriptive set theory as means to approach Vaught’s conjecture in model theory. The paper has a survey character but it contains new results and new approaches to old ones (such as the interaction of algebraic logic and descriptive set theory).

$$\mathrm P=\mathrm W$$ Phenomena

In this paper, we describe recent work towards the mirror $$\mathrm P=\mathrm W$$ conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi–Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [1] and de Cataldo and Migliorini [2]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror $$\mathrm P=\mathrm W$$ conjecture provides a possible bridge between the mirror $$\mathrm P=\mathrm W$$ conjecture and the well-known $$\mathrm P=\mathrm W$$ conjecture in non-Abelian Hodge theory.

A model for finding transition-minors

The well known cycle double cover conjecture in graph theory is strongly related to the compatible circuit decomposition problem. A recent result by Fleischner et al. (2018) gives a sufficient condition for the existence of a compatible circuit decomposition in a transitioned 2-connected Eulerian graph, which is based on an extension of the definition of K5-minors to transitioned graphs. Graphs satisfying this condition are called SUD-K5-minor-free graphs. In this work we formulate a generalization of this property by replacing the K5 by a 4-regular transitioned graph H, which is part of the input. Furthermore, we consider the decision problem of checking for two given graphs if the extended property holds. We prove that this problem is NP-complete and fixed parameter tractable with the size of H as parameter. We then formulate an equivalent problem, present a mathematical model for it, and prove its correctness. This mathematical model is then translated into a mixed integer linear program (MIP) for solving it in practice. Computational results show that the MIP formulation can be solved for small instances in reasonable time. In our computations we found snarks with perfect matchings whose contraction leads to SUD-K5-minor-free graphs that contain K5-minors. Furthermore, we verified that there exists a perfect pseudo-matching whose contraction leads to a SUD-K5-minor-free graph for all snarks with up to 22 vertices.

Packing Nearly Optimal Ramsey R(3,t) Graphs

In 1995 Kim famously proved the Ramsey bound R(3,t) \ge c t^2/\log t by constructing an n-vertex graph that is triangle-free and has independence number at most C \sqrt{n \log n}. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph K_n into a packing of such nearly optimal Ramsey R(3,t) graphs. More precisely, for any \epsilon>0 we find an edge-disjoint collection (G_i)_i of n-vertex graphs G_i \subseteq K_n such that (a) each G_i is triangle-free and has independence number at most C_\epsilon \sqrt{n \log n}, and (b) the union of all the G_i contains at least (1-\epsilon)\binom{n}{2} edges. Our algorithmic proof proceeds by sequentially choosing the graphs G_i via a semi-random (i.e., Rodl nibble type) variation of the triangle-free process. As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo (concerning a Ramsey-type parameter introduced by Burr, Erdos, Lovasz in 1976). Namely, denoting by s_r(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H=K_3 and establish that s_r(K_3) = \Theta(r^2 \log r).

Income Diversification and Performance: Should banks trade

The global banking sector continues to grapple with increasing nonperforming loans, unprecedented growth in financial innovations and competition which have distorted interest income stream. In response, banks are now searching for innovative income generating avenues to as a survival strategy. The portfolio theory conjectures that income diversification reduces income volatility and improves profitability. However extant literature shows mixed results. It’s from this background the study sought to investigate the effect of income diversification on bank performance. Using 310 annual observations drawn from 31 Kenyan commercial banks for the period 2008–2017, the study found that income diversification had a positive and significant effect on bank performance. Therefore, commercial banks are advised to consider diversifying in non-lending activities to better their financial performance. In view of this, the study has implications for bank regulators, scholars and practitioners.