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Real Zeros of SONC PolynomialsSep 15 2019We provide a complete and explicit characterization of the real zeros of sums of nonnegative circuit (SONC) polynomials, a recent certificate for nonnegative polynomials independent of sums of squares. As a consequence, we derive an exact determination ... More

Groups, graphs, and hypergraphs: average sizes of kernels of generic matrices with support constraintsAug 26 2019We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity ... More

Rational Motions with Generic Trajectories of Low DegreeJul 26 2019The trajectories of a rational motion given by a polynomial of degree n in the dual quaternion model of rigid body displacements are generically of degree 2n. In this article we study those exceptional motions whose trajectory degree is lower. An algebraic ... More

A motivic global Torelli theorem for isogenous K3 surfacesJul 25 2019We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective ... More

Trifocal Relative Pose from Lines at Points and its Efficient SolutionMar 23 2019Mar 28 2019We present a new minimal problem for relative pose estimation mixing point features with lines incident at points observed in three views and its efficient homotopy continuation solver. We demonstrate the generality of the approach by analyzing and solving ... More

Trifocal Relative Pose from Lines at Points and its Efficient SolutionMar 23 2019Apr 16 2019We present a new minimal problem for relative pose estimation mixing point features with lines incident at points observed in three views and its efficient homotopy continuation solver. We demonstrate the generality of the approach by analyzing and solving ... More

Einstein Metrics, Harmonic Forms, and Conformally Kaehler GeometryMar 03 2019Mar 24 2019The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are ... More

Einstein Metrics, Harmonic Forms, and Conformally Kaehler GeometryMar 03 2019The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are ... More

Asymptotics for the expected number of nodal components for random lemniscatesFeb 22 2019We determine the true asymptotic behaviour for the expected number of connected components for a model of random lemniscates proposed recently by Lerario and Lundberg. These are defined as the subsets of the Riemann sphere, where the absolute value of ... More

Characterization of polynomials whose large powers have fully positive coefficientsFeb 09 2019We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial ... More

Ramification divisors of general projectionsJan 06 2019We study the ramification divisors of projections of a smooth projective variety onto a linear subspace of the same dimension. We prove that the ramification divisors vary in a maximal dimensional family for a large class of varieties. Going further, ... More

The smooth torus orbit closures in the GrassmanniansDec 30 2018It is known that for the natural algebraic torus actions on the Grassmannians, the closures of torus orbits are toric varieties, and that these toric varieties are smooth if and only if the corresponding matroid polytopes are simple. We prove that simple ... More

On singularity properties of convolutions of algebraic morphisms - the general case (with an appendix joint with Gady Kozma)Nov 24 2018Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $G$ be a algebraic $K$-group. Given two algebraic morphisms $\varphi:X\rightarrow G$ and $\psi:Y\rightarrow G$, we define their convolution $\varphi*\psi:X\times Y\to ... More

The CR geometry of weighted extremal Kahler and Sasaki metricsOct 24 2018Jan 04 2019We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics ... More

Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen typeOct 17 2018We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with ... More

A New Sparse SOS Decomposition Algorithm Based on Term SparsitySep 28 2018Jan 22 2019A new sparse SOS decomposition algorithm is proposed based on a new sparsity pattern, called cross sparsity patterns. The new sparsity pattern focuses on the sparsity of terms and thus is different from the well-known correlative sparsity pattern which ... More

A Physical Perspective on Control Points and Polar Forms: Bézier Curves, Angular Momentum and Harmonic OscillatorsSep 19 2018Bernstein polynomials and B\'ezier curves play an important role in computer-aided geometric design and numerical analysis, and their study relates to mathematical fields such as abstract algebra, algebraic geometry and probability theory. We describe ... More

Definable retractions and a non-Archimedean Tietze--Urysohn theorem over Henselian valued fieldsAug 29 2018Mar 30 2019We prove the existence of definable retractions onto arbitrary closed subsets of $K^{n}$ definable over Henselian valued fields $K$. Hence directly follows non-Archimedian analogues of the Tietze--Urysohn and Dugundji theorems on extending continuous ... More

Definable retractions and a non-Archimedean Tietze--Urysohn theorem over Henselian valued fieldsAug 29 2018Feb 03 2019We prove the existence of definable retractions onto arbitrary closed subsets of $K^{n}$ definable over Henselian valued fields $K$. Hence directly follows non-Archimedian analogues of the Tietze--Urysohn and Dugundji theorems on extending continuous ... More

Generic torus orbit closures in Schubert varietiesJul 09 2018Jul 30 2018The closure of a generic torus orbit in the flag variety $G/B$ of type $A_{n-1}$ is known to be a permutohedral variety and well studied. In this paper we introduce the notion of a generic torus orbit in the Schubert variety $X_w$ $(w\in \mathfrak{S}_n)$ ... More

Realisation of groups as automorphism groups in categoriesJul 02 2018Oct 15 2018It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely ... More

Non-divisible cycles on products of very general Abelian varietiesJun 24 2018In this paper, we give a recipe for producing infinitely many non-divisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of "field of definition" for cycles in the Chow group ... More

The lemniscate tree of a random polynomialJun 01 2018To each generic complex polynomial $p(z)$ there is associated a labeled binary tree (here referred to as a "lemniscate tree") that encodes the topological type of the graph of $|p(z)|$. The branching structure of the lemniscate tree is determined by the ... More

Automorphism groups of maps, hypermaps and dessinsMay 24 2018A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism ... More

On the Gauss algebra of toric algebrasApr 21 2018Nov 08 2018Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\GG(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We describe the ... More

Abelian networks IV. Dynamics of nonhalting networksApr 10 2018An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. This paper is a continuation of the abelian networks series of Bond and Levine (2016), for which we ... More

On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one halfMar 30 2018Apr 04 2018It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we ... More

A group law on the projective plane with applications in Public Key CryptographyFeb 01 2018Jun 10 2019We present a new group law defined on a subset of the projective plane $\mathbb{F}P^2$ over an arbitrary field $\mathbb{F}$, which lends itself to applications in Public Key Cryptography, in particular to a Diffie-Hellman-like key agreement protocol. ... More

A group law on the projective plane with applications in Public Key CryptographyFeb 01 2018Mar 15 2019We present a new group law defined on a subset of the projective plane $\mathbb{F}P^2$ over an arbitrary field $\mathbb{F}$, which lends itself to applications in Public Key Cryptography, in particular to a Diffie-Hellman-like key agreement protocol. ... More

Global Identifiability of Differential ModelsJan 24 2018Apr 03 2018Many real-world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if it can be uniquely recovered from input and output data. The ... More

A closedness theorem over Henselian valued fields with analytic structureDec 21 2017Jan 07 2018The main purpose of the paper is to establish a closedness theorem over Henselian valued fields $K$ of equicharacteristic zero (not necessarily algebraically closed) with separated analytic structure. It says that every projection with a projective fiber ... More

Distinguished cycles on varieties with motive of abelian type and the Section PropertySep 17 2017Jul 25 2019A remarkable result of Peter O'Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville's splitting principle, ... More

Resolution of singularities and geometric proofs of the Lojasiewicz inequalitiesAug 31 2017Jul 01 2019The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, ... More

Resolution of singularities and geometric proofs of the Lojasiewicz inequalitiesAug 31 2017Dec 20 2018The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, ... More

Resolution of singularities and geometric proofs of the Lojasiewicz inequalitiesAug 31 2017May 02 2019The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, ... More

On line arrangements over fields with $1-ad$ structureAug 20 2017Jan 20 2018In this article we explore local and global gonality principle present in the line arrangements of the plane. We prove two main results, Theorems $[1.1,1.2]$. Firstly we associate invariants such as permutation cycles and local cycles at infinity with ... More

Levi-Kahler reduction of CR structures, products of spheres, and toric geometryAug 17 2017We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler ... More

Graphons arising from graphs definable over finite fieldsJul 19 2017We prove a version of Tao's algebraic regularity lemma for asymptotic classes in the context of graphons. We apply it to study expander difference polynomials over fields with powers of Frobenius.

A closedness theorem and applications in geometry of rational points over Henselian valued fieldsJun 04 2017Aug 30 2018We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$ of equicharacteristic zero. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center ... More

Maximum Number of Common Zeros of Homogeneous Polynomials over Finite FieldsMay 29 2017About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$ elements can ... More

Global smoothing of a subanalytic setMay 17 2017May 29 2018We give rather simple answers to two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set, and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibres by global blowings-up of ... More

Wolf Barth (1942--2016)Apr 24 2017In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his ... More

Freeness of multi-reflection arrangements via primitive vector fieldsMar 27 2017Apr 17 2019In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated ... More

Freeness of multi-reflection arrangements via primitive vector fieldsMar 27 2017Mar 26 2018In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated ... More

On manifolds defined by 4-colourings of simple 3-polytopesMar 20 2017Let $\mathcal{P}$ be the class of combinatorial 3-dimensional simple polytopes $P$, different from a tetrahedron, without 3- and 4-belts of facets. By the results of Pogorelov and Andreev, a polytope $P$ admits a realisation in Lobachevsky space $\mathbb{L}^3$ ... More

Modeling rooted in-trees by finite p-groupsJan 27 2017The aim of this chapter is to provide an adequate graph theoretic framework for the description of periodic bifurcations which have recently been discovered in descendant trees of finite p-groups. The graph theoretic concepts of rooted in-trees with weighted ... More

Tropical plactic algebra, the cloaktic monoid, and semigroup representationsJan 18 2017A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid $\mathcal{P}_n$. This algebra manifests a natural framework for accommodating ... More

On the absolute irreducibility of hyperplane sections of generalized Fermat varieties in $\Bbb{P}^3$ and the conjecture on exceptional APN functions: the Kasami-Welch degree caseDec 18 2016Let $f$ be a function on a finite field $F$. The decomposition of the generalized Fermat variety $X$ defined by the multivariate polynomial of degree $n$, $\phi(x,y,z)=f(x)+f(y)+f(z)$ in $\Bbb{P}^3(\overline{\mathbb{F}}_2)$, plays a crucial role in the ... More

A specialization property of indexDec 02 2016In this article we study certain specialization properties of the index of varieties defined by Koll\'{a}r.

On the binomial edge ideals of proper interval graphsNov 30 2016We prove several cases of the Betti number conjecture for the binomial edge ideal $J_G$ of a proper interval graph $G$ (also known as closed graph). Namely, we show that this conjecture is true for the linear strand of $J_G$, and true in general for any ... More

Kodaira fibrations and beyond: methods for moduli theoryNov 21 2016Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic ... More

Parameter spaces for algebraic equivalenceOct 20 2016A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles ... More

Parameter spaces for algebraic equivalenceOct 20 2016Jul 11 2017A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles ... More

Identities involving (doubly) symmetric polynomials and integrals over GrassmanniansJul 17 2016Sep 11 2018We obtain identities involving symmetric and doubly symmetric polynomials. These identities provide a way of handling expressions appearing in the Atiyah-Bott-Berline-Vergne formula for Grassmannians. As corollaries, we obtain formulas for integrals over ... More

Identities involving (doubly) symmetric polynomials and integrals over GrassmanniansJul 17 2016Sep 09 2016We obtain identities for symmetric and doubly symmetric polynomials. These identities provide a way of handling expressions appearing in the Atiyah-Bott-Berline-Vergne formula for Grassmannians. As a corollary, we obtain formulas for integrals over Grassmannians ... More

Explicit projective embeddings of standard opens of the Hilbert scheme of pointsMay 24 2016We describe explicitly how certain standard opens of the Hilbert scheme of points are embedded into Grassmannians. The standard opens of the Hilbert scheme that we consider are given as the intersection of a corresponding basic open affine of the Grassmannian ... More

Solving generic nonarchimedean semidefinite programs using stochastic game algorithmsMar 22 2016Jan 06 2018A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean. We provide a solution for a class of semidefinite feasibility problems given ... More

Remarks on Tsfasman-Boguslavsky Conjecture and Higher Weights of Projective Reed-Muller CodesMar 20 2016Apr 18 2016Tsfasman-Boguslavsky Conjecture predicts the maximum number of zeros that a system of linearly independent homogeneous polynomials of the same positive degree with coefficients in a finite field can have in the corresponding projective space. We give ... More

Linear equations on real algebraic surfacesFeb 05 2016Apr 26 2016We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher dimensions.

Progress Towards the Conjecture on APN Functions and Absolutely Irreducible PolynomialsJan 30 2016Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes, because of their good resistance to differential cryptanalysis. An APN function $f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$ is called exceptional ... More

On the geometry of random lemniscatesJan 11 2016We investigate the geometry of a random rational lemniscate $\Gamma$, the level set $\{|r(z)|=1\}$ on the Riemann sphere of the modulus of a random rational function $r$. We assign a probability distribution to the space of rational functions $r=p/q$ ... More

Decentralized Coherent Quantum Control Design for Translation Invariant Linear Quantum Stochastic Networks with Direct CouplingSep 07 2015This paper is concerned with coherent quantum control design for translation invariant networks of identical quantum stochastic systems subjected to external quantum noise. The network is modelled as an open quantum harmonic oscillator and is governed ... More

Artin ApproximationJun 15 2015May 14 2018In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of P{\l}oski, he conjectured that this remains true ... More

On a conjecture of Tsfasman and an inequality of Serre for the number of points on hypersurfaces over finite fieldsMar 10 2015Sep 28 2015We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points on hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of this inequality ... More

Complete intersection vanishing ideals on sets of clutter type over finite fieldsMar 04 2015In this paper we give a classification of complete intersection vanishing ideals on parameterized sets of clutter type over finite fields.

On the irreducible components of globally defined semianalytic setsMar 03 2015Nov 22 2015In this work we present the concept of amenable $C$-semianalytic subset of a real analytic manifold $M$ and study the main properties of this type of sets. Amenable $C$-semianalytic sets can be understood as globally defined semianalytic sets with a neat ... More

On globally defined semianalytic setsMar 03 2015Nov 22 2015In this work we present the concept of $C$-semianalytic subset of a real analytic manifold and more generally of a real analytic space. $C$-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic ... More

The motive of the Hilbert cubeMar 03 2015Oct 05 2016The Hilbert scheme $X^{[3]}$ of length-$3$ subschemes of a smooth projective variety $X$ is known to be smooth and projective. We investigate whether the property of having a multiplicative Chow-Kuenneth decomposition is stable under taking the Hilbert ... More

Tropical curves in sandpile modelsFeb 22 2015Nov 20 2017A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to each of its neighbors. In the study of pattern formation ... More

Vanishing ideals over finite fieldsFeb 19 2015Jun 19 2016Let $\mathbb{F}_q$ be a finite field, let $\mathbb{X}$ be a subset of a projective space ${\mathbb P}^{s-1}$, over the field $\mathbb{F}_q$, parameterized by rational functions, and let $I(\mathbb{X})$ be the vanishing ideal of $\mathbb{X}$. The main ... More

Toric origami structures on quasitoric manifoldsSep 24 2014Dec 08 2014We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete ... More

Lebesgue measure theory and integration theory on non-archimedean real closed fields with archimedean value groupSep 08 2014Feb 02 2016Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results from the classical ... More

Lebesgue measure theory and integration theory on non-archimedean real closed fields with archimedean value groupSep 08 2014Aug 08 2017Given a non-archimedean real closed field with archimedean value group which contains the reals, we establish for the category of semialgebraic sets and functions a full Lebesgue measure and integration theory such that the main results from the classical ... More

Splines, lattice points, and arithmetic matroidsAug 18 2014Sep 15 2014Let $X$ be a $(d\times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \{w \ge 0 : X w = u \}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^d$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. ... More

On the range of the Douglas-Rachford operatorMay 15 2014Jul 31 2014The problem of finding a minimizer of the sum of two convex functions - or, more generally, that of finding a zero of the sum of two maximally monotone operators - is of central importance in variational analysis. Perhaps the most popular method of solving ... More

Algebraic versus homological equivalence for singular varietiesApr 29 2014Let $ Y \subseteq \Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and assume that ... More

Some remarks on biequidimensionality of topological spaces and Noetherian schemesMar 23 2014There are many examples of the fact that dimension and codimension behave somewhat counterintuitively. In EGA it is stated that a topological space is equidimensional, equicodimensional and catenary if and only if every maximal chain of irreducible closed ... More

An extension theorem for Kähler currentsNov 18 2013Jul 02 2014We prove an extension theorem for Kahler currents with analytic singularities in a Kahler class on a complex submanifold of a compact Kahler manifold.

Geometric presentations of Lie groups and their Dehn functionsOct 20 2013We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of these groups. Our work, which also addresses algebraic groups ... More

Geometric presentations of Lie groups and their Dehn functionsOct 20 2013Nov 26 2016We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic ... More

A Thom-Porteous formula for connective K-theory using algebraic cobordismOct 03 2013We prove a formula for the push-forward class of Bott-Samelson resolutions in the algebraic cobordism ring of the flag bundle. We specialise our formula to connective K-theory providing a geometric interpretation to the double beta-polynomials of Fomin ... More

Enumerating Regular Objects associated with Suzuki GroupsSep 20 2013We use the M\"obius function of the simple Suzuki group Sz(q) to enumerate regular objects such as maps, hypermaps, dessins d'enfants and surface coverings with automorphism groups isomorphic to Sz(q).

On quadrilateral orbits in complex algebraic planar billiardsSep 07 2013Jan 27 2014The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex algebraic version of Ivrii's conjecture ... More

Characteristic classes of mixed Hodge modules and applicationsJul 19 2013Sep 20 2013These notes are an extended version of the authors' lectures at the 2013 CMI Workshop "Mixed Hodge Modules and Their Applications". We give an overview, with an emphasis on applications, of recent developments on the interaction between characteristic ... More

Characteristic classes of affine varieties and Plucker formulas for affine morphismsMay 14 2013Jan 24 2016An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is spherical), then many ... More

Kähler currents and null lociApr 18 2013Feb 20 2015We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. ... More

On Hamiltonian flows whose orbits are straight linesApr 11 2013We consider real analytic Hamiltonians whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q$. By a theorem of Moser, every polynomial Hamiltonian of degree 3 reduces to such a $q$-independent ... More

From pentacyclic coordinates to chain geometries, and backApr 04 2013Starting with the classical circle geometry of Sophus Lie, we give a survey about some of the developments in the area of chain geometries during the last three decades.

A characterization of varieties whose universal cover is a bounded symmetric domain without ball factorsFeb 10 2013Mar 25 2014We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism \s of the tensor product T\otimes T' of the tangent bundle T with the cotangent ... More

Special prime Fano fourfolds of degree 10 and index 2Feb 06 2013Feb 25 2014Mukai proved that most prime Fano fourfolds of degree 10 and index 2 are contained in a Grassmannian G(2,5). They are all unirational and some are rational, as remarked by Roth in 1949. We show that their middle cohomology is of K3 type and that their ... More

Vanishing ideals over complete multipartite graphsFeb 04 2013Sep 28 2013We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{\alpha_1,...,\alpha_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for this lattice ... More

Intermediate extension of Chow motives of Abelian typeNov 22 2012Sep 23 2016In this article, we give an unconditional construction of a motivic analogue of the intermediate extension in the context of Chow motives of Abelian type. Our main application concerns intermediate extensions of Chow motives associated to Kuga families ... More

Depths and Cohen-Macaulay Properties of Path IdealsNov 20 2012Given a tree T on n vertices, there is an associated ideal I of a polynomial ring in n variables over a field, generated by all paths of a fixed length of T. We show that such an ideal always satisfies the Konig property and classify all trees for which ... More

Algebraic Geometry of Matrix Product StatesOct 10 2012Sep 10 2014We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state ... More

A Nullstellensatz for Łojasiewicz idealsOct 09 2012For an ideal of smooth functions that is either {\L}ojasiewicz or weakly {\L}ojasiewicz, we give a complete characterization of the ideal of functions vanishing on its variety in terms of the global {\L}ojasiewicz radical and Whitney closure. We also ... More

The index of an algebraic varietySep 13 2012Let K be the field of fractions of a Henselian discrete valuation ring O_K. Let X_K/K be a smooth proper geometrically connected scheme admitting a regular model X/O_K. We show that the index \delta(X_K/K) of X_K/K can be explicitly computed using data ... More

Voronoi tilings hidden in crystals - The case of maximal abelian coveringsApr 30 2012Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application ... More

Automorphism groups of Grassmann codesApr 25 2012May 20 2013We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of ... More

Algebraic cycles and fibrationsMar 12 2012Sep 24 2013Let $f : X -> B$ be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of $X$ in terms of the Chow groups of $B$ and of the fibers of $f$. One of the applications concerns quadric ... More

Mixed DiscriminantsDec 05 2011The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the ... More