Authors:Sergio Conti; Adriana Garroni; Stefan Müller Pages: 3 - 17 Abstract: We consider target-space homogenization for energies defined on partitions parametrized by a discrete lattice \(\mathcal {B}\subset \mathbb {R}^N\) . For a small \(\sigma >0\) , the variable is a piecewise constant function taking values in \(\sigma \mathcal {B}\) , and the energy depends on the jumps and their orientation. In the limit as \(\sigma \rightarrow 0\) we obtain a homogenized functional defined on functions of bounded variation. This result is relevant in the study of dislocation structures in plastically deformed crystals. We review recent literature on the topic and propose our limiting effective energy as a continuum model for strain-gradient plasticity. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0083-z Issue No:Vol. 10, No. 1 (2017)

Authors:Giuseppe Molteni Pages: 19 - 28 Abstract: Let \(\psi _{\mathbb {{{K}}}}\) be the Chebyshev function of a number field \(\mathbb {{{K}}}\) , and let \(\psi ^{(1)}_\mathbb {{{K}}}(x):=\int _{0}^{x}\psi _\mathbb {{{K}}}(t)\,\mathrm {d}t\) and \(\psi ^{(2)}_\mathbb {{{K}}}(x):=2\int _{0}^{x}\psi ^{(1)}_\mathbb {{{K}}}(t)\,\mathrm {d}t\) . Assuming the truth of Riemann’s hypothesis for Dedekind’s zeta function of \(\mathbb {{{K}}}\) it is possible to prove explicit bounds for \( \psi _\mathbb {{{K}}}(x) - x \) , \( \psi ^{(1)}_\mathbb {{{K}}}(x) - \tfrac{x^2}{2} \) and \( \psi ^{(2)}_\mathbb {{{K}}}(x) - \tfrac{x^3}{3} \) . These results can be used to prove the existence of many ideals with a small norms and to produce an extremely efficient algorithm for the computation of the residue of Dedekind’s zeta function. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0084-y Issue No:Vol. 10, No. 1 (2017)

Authors:Alberto Perelli Pages: 29 - 53 Abstract: This is an expanded version of the author’s lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of Hamburger, Hecke and Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0085-x Issue No:Vol. 10, No. 1 (2017)

Authors:A. Figalli Pages: 55 - 74 Abstract: In this note we discuss a new recent approach, based on transportation techniques, to obtain universality results in random matrix theory. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0087-8 Issue No:Vol. 10, No. 1 (2017)

Authors:Andrea Malchiodi Pages: 75 - 97 Abstract: We consider a class of Liouville equations motivated by uniformization problems in a singular setting, as well as from models in Mathematical Physics. We will study existence from a variational point of view, using suitable improvements of the Moser-Trudinger inequality. These improvements allow to study the concentration properties of conformal volume, and to reduce the problem to studying explicit finite-dimensional objects. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0092-y Issue No:Vol. 10, No. 1 (2017)

Authors:Giovanna Carnovale; Iulian I. Simion Pages: 99 - 112 Abstract: A conjecture of De Concini Kac and Procesi provides a bound on the minimal possible dimension of an irreducible module for quantized enveloping algebras at an odd root of unity. We pose the problem of the existence of modules whose dimension equals this bound. We show that this question cannot have a positive answer in full generality and discuss variants of this question. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0094-9 Issue No:Vol. 10, No. 1 (2017)

Authors:R. E. Caflisch; F. Gargano; M. Sammartino; V. Sciacca Pages: 113 - 141 Abstract: We consider the Euler- \(\alpha \) regularization of the Birkhoff–Rott equation and compare its solutions with the dynamics of the non regularized vortex-sheet. For a flow induced by an infinite array of planar vortex-sheets we analyze the complex singularities of the solutions.Through the singularity tracking method we show that the regularized solution has several complex singularities that approach the real axis. We relate their presence to the formation of two high-curvature points in the vortex sheet during the roll-up phenomenon. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0097-6 Issue No:Vol. 10, No. 1 (2017)

Authors:Marzia Bisi Pages: 143 - 158 Abstract: We review the main results on some basic kinetic models for wealth distribution in a simple market economy, with interaction rules involving random variables to take into account effects due to market risks. Then, we investigate in more detail long time behavior of a model which includes the taxation phenomenon and the redistribution of collected wealth according to proper criterions. Finally, we propose a new class of kinetic equations in which agent’s trading propensity varies according to the personal amount of wealth. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0099-4 Issue No:Vol. 10, No. 1 (2017)

Authors:Vladimir Gol’dshtein; Valerii Pchelintsev; Alexander Ukhlov Abstract: In this paper we obtain lower estimates for the first non-trivial eigenvalue of the p-Laplace Neumann operator in bounded simply connected planar domains \(\varOmega \subset {\mathbb {R}}^2\) . This study is based on a quasiconformal version of the universal two-weight Poincaré–Sobolev inequalities obtained in our previous papers for conformal weights and its non weighted version for so-called K-quasiconformal \(\alpha \) -regular domains. The main technical tool is the geometric theory of composition operators in relation with the Brennan’s conjecture for (quasi)conformal mappings. PubDate: 2017-05-17 DOI: 10.1007/s40574-017-0127-z

Authors:Luca Motto Ros Abstract: We survey some old and new results concerning the classification of complete metric spaces up to isometry, a theme initiated by Gromov, Vershik and others. All theorems concerning separable spaces appeared in various papers in the last twenty years: here we tried to present them in a unitary and organic way, sometimes with new and/or simplified proofs. The results concerning non-separable spaces (and, to some extent, the setup and techniques used to handle them) are instead new, and suggest new lines of investigation in this area of research. PubDate: 2017-04-27 DOI: 10.1007/s40574-017-0125-1

Authors:Margherita Lelli-Chiesa Abstract: We survey some results concerning Severi varieties and variation in moduli of curves lying on K3 surfaces or on abelian surfaces. A number of open problems is listed and some work in progress is mentioned. PubDate: 2017-04-22 DOI: 10.1007/s40574-017-0126-0

Authors:Andrea Vaccaro; Matteo Viale Abstract: It is common knowledge in the set theory community that there exists a duality relating the commutative \(C^*\) -algebras with the family of \(\mathsf {B}\) -names for complex numbers in a boolean valued model for set theory \(V^{\mathsf {B}}\) . Several aspects of this correlation have been considered in works of the late 1970s and early 1980s, for example by Takeuti (Two Applications of Logic to Mathematics. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, Kanô Memorial Lectures, vol 3. Publications of the Mathematical Society of Japan, No. 13, 1978) and Fourman et al. (eds.) (Applications of sheaves. In: Lecture Notes in Mathematics, vol 753. Springer, Berlin, 1979), and by Jech (Trans Am Math Soc 289(1):133–162, 1985). Generalizing Jech’s results, we extend this duality so as to be able to describe the family of boolean names for elements of any given Polish space Y (such as the complex numbers) in a boolean valued model for set theory \(V^\mathsf {B}\) as a space \(C^+(X,Y)\) consisting of functions f whose domain X is the Stone space of \(\mathsf {B}\) , and whose range is contained in Y modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of \(C^+(X,Y)\) . PubDate: 2017-04-12 DOI: 10.1007/s40574-017-0124-2

Authors:Benjamin F. Dribus; J. W. Hoffman; Sen Yang Abstract: We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch–Gersten–Quillen-type resolutions of algebraic K-theory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their “tangent sequences,” expressed in terms of absolute Kähler differentials. More generally, we place Green and Griffiths’ concrete geometric approach to the infinitesimal theory of Chow groups in a natural and formally rigorous structural context, expressed in terms of nonconnective K-theory, negative cyclic homology, and the relative algebraic Chern character. PubDate: 2017-04-07 DOI: 10.1007/s40574-017-0123-3

Authors:Stefano Filipazzi Abstract: We show that there exist smooth surfaces violating Generic Vanishing in any characteristic \(p \ge 3\) . As a corollary, we recover a result of Hacon and Kovács, producing counterexamples to Generic Vanishing in dimension 3 and higher. PubDate: 2017-04-05 DOI: 10.1007/s40574-017-0120-6

Authors:Luca Migliorini Abstract: We give an introduction to non-abelian Hodge theory for curves with the aim of stating the \(P = W\) conjecture both in its original cohomological version and in the more recent geometric one, and proposing a strategy to relate the two conjectures. PubDate: 2017-03-30 DOI: 10.1007/s40574-017-0122-4

Authors:Giovanni Cimatti Abstract: The elliptic boundary value problem governing the steady electrical heating of a conductor of heat and electricity, the so-called thermistor problem, $$\begin{aligned}&{\nabla }\cdot ({\sigma }(u){\nabla }\phi )=0\ {\quad \hbox {in}\ {\Omega }}\quad \phi =\phi _b\ {\quad \hbox {on}\ {\Gamma }}\\&{\nabla }\cdot ({\kappa }(u){\nabla }u)=-{\sigma }(u) {\nabla }\phi ^2\ {\quad \hbox {in}\ {\Omega }}\quad u=0\ {\quad \hbox {on}\ {\Gamma }}, \end{aligned}$$ where \({\sigma }(u)\) is the temperature dependent electric conductivity and \({\kappa }(u)\) the thermal conductivity, admits a reinterpretation in the framework of general relativity if we choose \({\sigma }(u)=e^u\) , \({\kappa }(u)=1\) and, in addition, \({\Omega }\) is a domain of \({\mathbf{R}^3}\) axially symmetric whereas the function \(\phi _b\) , in a cylindrical coordinate system \({\rho },z,{\varphi }\) , is independent of \({\varphi }\) . The same analytical methods relevant in the thermistor problem can be used in this new context. PubDate: 2017-03-28 DOI: 10.1007/s40574-017-0121-5

Authors:Julie Decaup; Adrien Dubouloz Abstract: We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular automorphisms outside Q, there are exactly two such lines: the restriction of a smooth conic osculating Q at a rational point and the restriction of the tangent line to Q at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed embeddings of the affine line in the complement of Q. We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the automorphism group of the complement of Q over a field of arbitrary characteristic. PubDate: 2017-03-18 DOI: 10.1007/s40574-017-0119-z

Authors:Francesco Bastianelli Abstract: This survey retraces the author’s talk at the Workshop Birational geometry of surfaces, Rome, January 11–15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces. PubDate: 2017-01-27 DOI: 10.1007/s40574-017-0116-2

Authors:Mauro Maccioni Abstract: I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than \(2c+1\) , if d is odd, and t is greater or equal than \(\max (3,2c+1)\) , if d is even, where c is the number of ovals in the zero locus of f. About binary forms, I prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms. PubDate: 2017-01-09 DOI: 10.1007/s40574-016-0112-y