Golod-Shafarevich type theorems and potential algebras

Potential algebras feature in the minimal model program and noncommutative
resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Groebner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential.
We consider two-generated potential algebras, and prove that they can not have dimension smaller than 8, using Groebner bases arguments, and arguing in terms of associated truncated algebra. We derive from the improved version of the Golod-Shafarevich theorem,that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree n>=3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class P_n of potential algebras with homogeneous potential of degree n+1>= 4, the minimal Hilbert series is H_n=1/1-2t+2t^n-t^{n+1}, so they are all infinite dimensional. Moreover, growth could be polynomial (but at least quadratic) for the potential of degree 4, and is always exponential for potential of degree starting from 5.
For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invaria

Divisor Braids

We study a novel type of braid groups on a closed orientable surface Σ. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on Σ; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles over Σ. The elements of these braid groups, which we call divisor braids, have coloured strands that are allowed to intersect according to rules specified by a graph Γ. In situations where there is more than one strand of each colour, we show that the corresponding braid group admits a metabelian presentation as a central extension of the free Abelian group H_1(Σ;Z)^r, where r is the number of colours, and describe its Abelian commutator. This computation relies crucially on producing a link invariant (of closed divisor braids) in the three-manifold S^1×Σ for each graph Γ. We also describe the von Neumann algebras associated to these groups in terms of rings that are familiar from noncommutative geometry.

One question from the Polishchuk and Positselski book on Quadratic algebras

In the book 'Quadratic algebras' due to Polishchuk and Positselski algebras with small number of generators (n=2,3) is considered. For some number of relations
r possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the
case of three generators n=3 and three relations r=6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebra A with dim A_1=dim A_2=3 we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not.

On the proof of the homology conjecture for monomial non-unital algebras

We consider the bar complex of a monomial non-unital associative algebra A=k / (w_1,...,w_t). For any fixed monomial w=x_1..x_n in A one can define certain subcomplex of the Bar complex of A. It was conjectured in [3] that homology of this complex is at most one. We prove here this conjecture, and describe the place where this nontrivial homology appears in terms of length of the Dyck path associated to a given word in A.

Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy

Conservative second-order gravitational self-force on circular orbits and the effective one-body formalism

Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?

We examine two claims from the paper "Formality Conjecture" (Ascona 1996): specifically, that 1) a certain tetrahedral graph flow preserves the class of (real-analytic) Poisson structures, and that 2) another tetrahedral graph flow vanishes at every such Poisson structure.
By using twelve Poisson structures with high-order polynomial coefficients
as explicit counterexamples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at the Poisson bi-vectors.
The counterexamples at hand themselves suggest a correction to the formula for the "exotic" flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance (1:6) for which the flow does preserve the space of Poisson bi-vectors.

BADLY APPROXIMABLE VECTORS AND FRACTALS DEFINED BY CONFORMAL DYNAMICAL SYSTEMS

We prove that if J is the limit set of an irreducible conformal iterated function system (with
either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff
dimension in J. The same is true if J is the radial Julia set of an irreducible meromorphic function (either
rational or transcendental). The method of proof is to find subsets of J that support absolutely friendly
and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of
Broderick, Kleinbock, Reich, Weiss, and the second-named author (’12) by showing that every hyperplane
diffuse set supports an absolutely decaying measure.

Real Analyticity for random dynamics of transcendental functions

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh \cite{Rug08} leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron-Frobenius operator are assumed to converge.
We also provide a Bowen's formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function.
Our main application states real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.

SU(N) transitions in M-theory on Calabi-Yau fourfolds and background fluxes

We study M-theory on a Calabi-Yau fourfold with a smooth surface S of AN−1 singularities. The resulting three-dimensional theory has a =2 SU(N) gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional =1 SU(N) gauge theory on the surface S. A variant of the Vafa-Witten equations governs the moduli space of the gauge theory, which, for a trivial SU(N) principal bundle over S, admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi-Yau fourfolds, respectively. We find that the deformed Calabi-Yau fourfold associated to the Higgs branch requires for consistency a non-trivial four-form background flux in M-theory. The flat directions of the flux-induced superpotential are in agreement with the gauge theory prediction for the moduli space of the Higgs branch. We illustrate our findings with explicit examples that realize the Coulomb and Higgs phase transition in Calabi-Yau fourfolds embedded in weighted projective spaces. We generalize and enlarge this class of examples to Calabi-Yau fourfolds embedded in toric varieties with an AN−1 singularity in codimension two.

New gravitational self-force analytical results for eccentric orbits around a Schwarzschild black hole

High post-Newtonian order gravitational self-force analytical results for eccentric orbits around a Kerr black hole

Intermittency in the Hodgkin-Huxley model

We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals.

The Langlands-Shahidi method over function fields: the Ramanujan Conjecture and the Riemann Hypothesis for the unitary groups

On étudie la méthode de Langlands-Shahidi sur les corps de fonctions de caractéristique p. On prouve la fonctorialité de Langlands globale et locale des groupes unitaires vers les groupes linéaires pour les représentations génériques. Supposant connue la conjecture de Shahidi pour les L-paquets modérés, on donne une extension de la définition des fonctions L et des facteurs ε. Enfin, utilisant le travail de L. Lafforgue, on établit la conjecture de Ramanujan et on prouve que les fonctions L automorphes de Langlands-Shahidi satisfont l'hypothèse de Riemann.

Class number problems and Lang conjectures

Given a square-free integer d we introduce an affine hypersurface whose integer points are in one-to-one correspondence with ideal classes of the
quadratic number field Q(\sqrt{d}). Using this we relate class number problems of Gauss to Lang conjectures.

On Emergent Geometry from Entanglement Entropy in Matrix Theory

Using Matrix theory, we compute the entanglement entropy between a supergravity probe and modes on a spherical membrane. We demonstrate that a membrane stretched between the probe and the sphere entangles these modes and leads to an expression for the entanglement entropy that encodes information about local gravitational geometry seen by the probe. We propose in particular that this entanglement entropy measures the rate of convergence of geodesics at the location of the probe.

Perturbative quantum field theory meets number theory

On the conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity