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Triangle areas determined by arrangements of planar linesFeb 08 2019A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in ... More

An analogue of the Szemeredi Regularity Lemma for bounded degree graphsSep 17 2008Apr 18 2009We show that a sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous pieces. The result can be viewed as a "finitarization" of the classical Farrell-Varadarajan Ergodic Decomposition Theorem.

Adaptive Majority Problems for Restricted Query Graphs and for Weighted SetsMar 20 2019Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study ... More

On directed local chromatic number, shift graphs, and Borsuk-like graphsJun 16 2009We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying ... More

Colorful subgraphs in Kneser-like graphsDec 01 2005Combining Ky Fan's theorem with ideas of Greene and Matousek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored ... More

Sofic equivalence relationsJun 19 2009We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of L\"uck, Sauer and Wegner hold for treeable equivalence relations.

The combinatorial costAug 18 2006We study the combinatorial analogues of the classical invariants of measurable equivalence relations. We introduce the notion of cost and $\beta$-invariants (the analogue of the first $L^2$-Betti number introduced by Gaboriau) for sequences of finite ... More

The Cayley isomorphism property for groups of order p^3qDec 28 2013For every prime $p > 3$ and for every prime $q>p^3$ we prove that $\mathbb{Z}_q \times \mathbb{Z}_p^3$ is a DCI-group.

The tail of the crossing probability in near-critical percolation --- an appendix to Ahlberg & Steif [arXiv:1405.7144]Jul 16 2015May 02 2016We answer a question of Ahlberg and Steif (2014) by finding the tail behaviour of the crossing probability in near-critical planar percolation. Interestingly, this superexponentially small behaviour is different from the case of dynamical percolation, ... More

On Induced Subgraphs of Finite Graphs not Containing Large Empty and Complete SubgraphsNov 16 2012In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G does not contain ... More

Characterizing circles by a convex combinatorial propertyNov 28 2016Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that $K_1\subseteq \mathbb R^2$ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\preal\to\preal$. Kira Adaricheva and Madina ... More

On algebras that almost have finite dimensional representationsNov 21 2003We introduce the notion of almost finite dimensionality of algebras and study its connection with the classical finiteness conditions.

Extremal metrics and K-stabilityOct 18 2004Apr 02 2007We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. ... More

Characterizing fully principal congruence representable distributive latticesJun 11 2017Motivated by a recent paper of G. Gr\"atzer, a finite distributive lattice $D$ is said to be fully principal congruence representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible elements of $D$, there ... More

On the set of principal congruences in a distributive congruence lattice of an algebraMay 30 2017Jun 29 2017Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ represents the inclusion $Q\subseteq D$ by principal congruences if the congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal congruences of $A$ corresponds ... More

Shifted Jacobi polynomials and Delannoy numbersSep 30 2009Dec 24 2009We express a weighted generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago, to all ... More

The amenability and non-amenability of skew fieldsNov 21 2003We investigate the amenability of skew filed extensions of the complex numbers. We prove that all skew fields of finite Gelfand-Kirillov transcendence degree are amenable. However there are both amenable and non-amenable skew fields of infinite Gelfand-Kirillov ... More

Almost maximally almost-periodic group topologies determined by T-sequencesMar 31 2005Nov 27 2005A sequence $\{a_n\}$ in a group $G$ is a {\em $T$-sequence} if there is a Hausdorff group topology $\tau$ on $G$ such that $a_n\stackrel\tau\longrightarrow 0$. In this paper, we provide several sufficient conditions for a sequence in an abelian group ... More

Precompact abelian groups and topological annihilatorsFeb 10 2005Jun 05 2005For a compact Hausdorff abelian group K and its subgroup H, one defines the g-closure g(H) of H in K as the subgroup consisting of $\chi \in K$ such that $\chi(a_n)\longrightarrow 0$ in T=R/Z for every sequence {a_n} in $\hat K$ (the Pontryagin dual of ... More

$L^2$-spectral invariants and quasi-crystal graphsJul 07 2006Introducing and studying the pattern frequency algebra, we prove the analogue of L\"uck's approximation theorems on $L^2$-spectral invariants in the case of aperiodic order. These results imply a uniform convergence theorem for the integrated density ... More

Hurwitzian continued fractions containing a repeated constant and an arithmetic progressionNov 12 2012May 25 2013We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and ... More

Remark on the Calabi flow with bounded curvatureSep 12 2012In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric.

The number of permutations with k inversionsFeb 10 2010Let $n\geq 1$, $0\leq t\leq {n \choose 2}$ be arbitrary integers. Define the numbers $I_n(t)$ as the number of permutations of $[n]$ with $t$ inversions. Let $n,d\geq 1$ and $0\leq t\leq (d-1)n$ be arbitrary integers. Define {\em the polynomial coefficients} ... More

QCNN: Quantile Convolutional Neural NetworkAug 21 2019A dilated causal one-dimensional convolutional neural network architecture is proposed for quantile regression. The model can forecast any arbitrary quantile, and it can be trained jointly on multiple similar time series. An application to Value at Risk ... More

Hermitian codes from higher degree placesJun 20 2012Matthews and Michel investigated the minimum distances in certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. ... More

The partial $C^0$-estimate along the continuity methodOct 31 2013Apr 03 2015We prove that the partial $C^0$-estimate holds for metrics along Aubin's continuity method for finding K\"ahler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on ... More

The short toric polynomialAug 26 2010Jun 22 2011We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining ... More

Determination of the upper and lower bound of masslimit of degenerate fermionic dark matter objectsNov 19 2007We give a gravitational upper limit for the mass of static degenerate fermionic dark matter objects. The treatment we use includes fully relativistic equations for describing the static solutions of these objects. We study the influence of the annihilation ... More

A Bollobás-type theorem for affine subspacesDec 03 2015Let $W$ denote the $n$-dimensional affine space over the finite field $\mathbb F_q$. We prove here a Bollob\'as-type upper bound in the case of the set of affine subspaces. We give a construction of a pair of families of affine subspaces, which shows ... More

Spontaneous Symmetry Breaking in SO(3) Gauge Theory to Discrete SubgroupsJan 01 1997A systematical description of possible symmetry breakings in the SO(3) gauge theory and an algorithmical method to construct SU(2) or SO(3) invariant Higgs potentials in an arbitrary irreducible representation is given. We close our paper with the explicit ... More

Alternation acyclic tournamentsApr 24 2017Nov 17 2018We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. ... More

Circles and crossing planar compact convex setsFeb 18 2018Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that whenever $K_1\subseteq \mathbb R^2$ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. Fejes-T\'oth. A theorem of L. Fejes-T\'oth from ... More

Characterizing circles by a convex combinatorial propertyNov 28 2016Jul 23 2017Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that $K_1\subseteq \mathbb R^2$ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\mathbb R^2\to\mathbb R^2$. Kira Adaricheva ... More

An easy way to a theorem of Kira Adaricheva and Madina Bolat on convexity and circlesOct 08 2016May 23 2017Kira Adaricheva and Madina Bolat have recently proved that if $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$ is included in the convex hull of $U_k\cup(\{A_0,A_1, ... More

On colorful edge triples in edge-colored complete graphsDec 17 2018An edge-coloring of the complete graph $K_n$ we call $F$-caring if it leaves no $F$-subgraph of $K_n$ monochromatic and at the same time every subset of $|V(F)|$ vertices contains in it at least one completely multicolored version of $F$. For the first ... More

On blowing up extremal Kähler manifoldsOct 25 2010Feb 02 2011We show that the blowup of an extremal Kahler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kahler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo-Pacard-Singer. We also ... More

A note on finite lattices with many congruencesDec 17 2017By a twenty year old result of Ralph Freese, an $n$-element lattice $L$ has at most $2^{n-1}$ congruences. We prove that if $L$ has less than $2^{n-1}$ congruences, then it has at most $2^{n-2}$ congruences. Also, we describe the $n$-element lattices ... More

A simultaneous representation of a group and a bounded poset with lattice automorphisms and principal congruencesAug 18 2015Aug 21 2015Given a poset $P$ with at least two elements and a group $G$, there exists a selfdual lattice of length 16 such that the collection of its principal congruences is order isomorphic to $P$ while its automorphism group to $G$.

Characterization of matrix types of ultramatricial algebrasJun 15 2004Jul 21 2005A dimension group is a partially ordered countable group such that (1) every finite subset is contained in an ordered subgroup which is a finite direct power of Z and (2) the group has an order unit i.e. a positive element u such that every group element ... More

Spectral sets in $\Z_{p^2qr}$ tileJul 09 2019We prove the every spectral set in $\Z_{p^2qr}$ tiles, where $p$, $q$ and $r$ are primes, which is a special case of Fuglede's conjecture for cyclic groups.

The partial $C^0$-estimate along the continuity methodOct 31 2013Sep 22 2017We prove that the partial $C^0$-estimate holds for metrics along Aubin's continuity method for finding K\"ahler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on ... More

Notes on duality theories of abelian groupsMay 05 2006In this notebook, I present duality theory (or theories) of abelian groups with some categorical and categorical topological flavour. I consider writing this notebook as a longer-term project, and its current content and presentation is "under development." ... More

Meixner polynomials of the second kind and quantum algebras representing su(1,1)Sep 24 2009Nov 11 2009We show how Viennot's combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges on the matrix entries in powers of certain operators in a representation of su(1,1). Our results link these calculations ... More

Weak convergence of finite graphs, integrated density of states and a Cheeger type inequalitySep 03 2005May 23 2006In \cite{Elek} we proved that the limit of a weakly convergent sequence of finite graphs can be viewed as a graphing or a continuous field of infinite graphs. Thus one can associate a type $II_1$-von Neumann algebra to such graph sequences. We show that ... More

Sunflowers and $L$-intersecting familiesJan 19 2016Let $f(k,r,s)$ stand for the least number so that if $\cal F$ is an arbitrary $k$-uniform, $L$-intersecting set system, where $|L|=s$, and $\cal F$ has more than $f(k,r,s)$ elements, then $\cal F$ contains a sunflower with $r$ petals. We give an upper ... More

Eighty-three sublattices and planarityJan 03 2019Jul 02 2019Let $L$ be a finite $n$-element lattice. We prove that if $L$ has at least $83\cdot 2^{n-8}$ sublattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar lattice with exactly $83\cdot 2^{n-8}-1$ sublattices.

Hereditarily non-topologizable groupsMar 21 2006A group G is non-topologizable if the only Hausdorff group topology that G admits is the discrete one. Is there an infinite group G such that H/N is non-topologizable for every subgroup H <= G and every normal subgroup N <| H? We show that a solution ... More

Extremal Kähler metricsMay 19 2014This paper is a survey of some recent progress on the study of Calabi's extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some ... More

Uniqueness of some Calabi-Yau metrics on $\mathbf{C}^n$Jun 26 2019We consider the Calabi-Yau metrics on $\mathbf{C}^n$ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone $\mathbf{C}\times A_1$ at infinity for the $(n-1)$-dimensional Stenzel cone $A_1$. We show that up to scaling and ... More

Fully non-linear elliptic equations on compact Hermitian manifoldsJan 12 2015Apr 23 2015We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Amp\`ere, Hessian ... More

Finite convex geometries of circlesDec 14 2012May 21 2013Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H Edelman in 1980 under the name "anti-exchange closure system". We ... More

On sequentially h-complete groupsOct 22 2002Jan 15 2004A topological group $G$ is {\em sequentially $h$-complete} if all the continuous homomorphic images of $G$ are sequentially complete. In this paper we give necessary and sufficient conditions on a complete group for being compact, using the language of ... More

Alternation acyclic tournamentsApr 24 2017Apr 16 2019We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. ... More

Lifted closure operatorsFeb 19 2005Feb 28 2005In this paper, we study the properties of closure operators obtained as initial lifts along a reflector, and compactness with respect to them in particular. Applications in the areas of topology, topological groups and topological *-algebras are presented. ... More

An explicit Ramsey graphDec 08 2014Jan 05 2015Explicit construction of Ramsey graphs has remained a challenging open problem for a long time. Frankl--Wilson \cite{FW}, Alon \cite{A} and Grolmusz \cite{G2} gave the best explicit constructions of graphs on $m$ vertices with no clique or independent ... More

Difference sets and power residuesJan 19 2018Jan 31 2018Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb Z_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb Z_p})^n$ be an arbitrary subset such that $$ \{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq ... More

Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed AlgorithmsDec 18 2018The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and local-global graph ... More

Ergodic properties of subcritical multitype Galton-Watson processesFeb 22 2014Apr 26 2016We show the existance of the stationary distributions of subcritical multitype Galton-Watson processes without any conditions on the mean matrix of the offspring distributions. Some additional properties of the stationary distribution are also proven. ... More

Eighty-three sublattices and planarityJan 03 2019Let $L$ be a finite $n$-element lattice. We prove that if $L$ has at least $83\cdot 2^{n-8}$ sublattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar lattice with exactly $83\cdot 2^{n-8}-1$ sublattices.

Diagrams and rectangular extensions of planar semimodular latticesDec 15 2014In 2009, G. Gr\"atzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of special diagrams ... More

Comments on "Cavitons and spontaneous hot flow anomalies in a hybrid-Vlasov global magnetospheric simulation" by Blanco-Cano et al. (2018)Jan 22 2019Blanco-Cano et al. (2018) intended to find a type of transient event in the solar wind before the terrestrial bow shock using a special type of simulation. However, the simulation results cannot reproduce the main features of the event. Not only are the ... More

Blowing up extremal Kähler manifolds IIFeb 04 2013Mar 06 2013This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal K\"ahler manifolds. We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds. As an application we prove ... More

One hundred twenty-seven subsemilattices and planarityJun 28 2019Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ sublattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$ sublattices.

Construction of locally plane graphs with many edgesOct 28 2011A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most $k$ in a geometric graph $G$ is self-intersecting we call $G$ $k$-locally plane. The main result of this paper is a construction of $k$-locally ... More

Greatest lower bounds on the Ricci curvature of Fano manifoldsMar 31 2009On a Fano manifold M we study the supremum of the possible t such that there is a K\"ahler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding K\"ahler-Einstein ... More

On Induced Subgraphs of Finite Graphs not Containing Large Empty and Complete SubgraphsNov 16 2012Aug 18 2017In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G does not contain ... More

One hundred twenty-seven subsemilattices and planarityJun 28 2019Jul 01 2019Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ subsemilattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$ subsemilattices. ... More

About sunflowersApr 26 2018May 11 2018Alon, Shpilka and Umans considered the following version of usual sunflower-free subset: a subset $\mbox{$\cal F$}\subseteq \{1,\ldots ,D\}^n$ for $D>2$ is sunflower-free if for every distinct triple $x,y,z\in \mbox{$\cal F$}$ there exists a coordinate ... More

Efron's coins and the Linial arrangementNov 14 2015Jun 14 2016We characterize the tournaments that are dominance graphs of sets of (unfair) coins in which each coin displays its larger side with greater probability. The class of these tournaments coincides with the class of tournaments whose vertices can be numbered ... More

Quasiplanar diagrams and slim semimodular latticesDec 31 2012A (Hasse) diagram of a finite partially ordered set (poset) P will be called quasiplanar if for any two incomparable elements u and v, either v is on the left of all maximal chains containing u, or v is on the right of all these chains. Every planar diagram ... More

A generalization of Croot-Lev-Pach's Lemma and a new upper bound for the size of difference sets in polynomial ringsMar 14 2018Mar 20 2018Croot, Lev and Pach used a new polynomial technique to give a new exponential upper bound for the size of three-term progression-free subsets in the groups $(\mathbb Z _4)^n$. The main tool in proving their striking result is a simple lemma about polynomials, ... More

On universal continuous actions on the Cantor setMar 14 2018Mar 16 2018Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: ... More

A remark on conical Kähler-Einstein metricsNov 12 2012We give some non-existence results for K\"ahler-Einstein metrics with conical singularities along a divisor on Fano manifolds. In particular we show that the maximal possible cone angle is in general smaller than the invariant R(M). We study this discrepancy ... More

Degenerations of $\mathbf{C}^n$ and Calabi-Yau metricsJun 01 2017Apr 02 2019We construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated singularities ... More

Inequalities for two systems of subspaces with prescribed intersectionsOct 10 2016Let $W$ denote a linear space over a fixed field ${\mathbb F}$. We define the notions of weak $ISP$-system and weak $(u,v)$-system $\cal S=\{(U_i,V_i):~ 1\leq i\leq m\}$ of subspaces of $W$. We give upper bounds for the size of weak $ISP$-systems and ... More

Degenerations of $\mathbf{C}^n$ and Calabi-Yau metricsJun 01 2017We construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated singularities ... More

The asymptotic number of planar, slim, semimodular lattice diagramsJun 16 2012A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of these lattices of size ... More

A generalization of the Erdős-Ko-Rado TheoremDec 17 2015Dec 18 2015Our main result is a new upper bound for the size of k-uniform, L-intersecting families of sets, where L contains only positive integers. We characterize extremal families in this setting. Our proof is based on the Ray-Chaudhuri--Wilson Theorem. As an ... More

Finite semilattices with many congruencesJan 04 2018For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq 4$. Furthermore, ... More

A second look at the toric h-polynomial of a cubical complexFeb 18 2010Jul 06 2010We provide an explicit formula for the toric $h$-contribution of each cubical shelling component, and a new combinatorial model to prove Clara Chan's result on the non-negativity of these contributions. Our model allows for a variant of the Gessel-Shapiro ... More

The Kahler-Ricci flow and K-stabilityMar 11 2008Jan 15 2009We consider the K\"ahler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a K\"ahler-Einstein metric. ... More

Lattices embeddable in three-generated latticesDec 12 2015We prove that every finite lattice L can be embedded in a three-generated finite lattice K. We also prove that every algebraic lattice with accessible cardinality is a complete sublattice of an appropriate algebraic lattice K such that K is completely ... More

Geometry of splice-quotient singularitiesDec 23 2008We obtain a new important basic result on splice-quotient singularities in an elegant combinatorial-geometric way: every level of the divisorial filtration of the ring of functions is generated by monomials of the defining coordinate functions. The elegant ... More

Extremal metrics and K-stability (PhD thesis)Oct 31 2006In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to ... More

Free minimal actions of countable groups with invariant probability measuresMay 28 2018Jul 06 2018We prove that for any countable group G there exists a free minimal continuous action of G on the Cantor set admitting an invariant Borel probability measure.

Coordinatization of join-distributive latticesAug 17 2012Oct 12 2012Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since then. Let L be ... More

Linear equations for the number of intervals which are isomorphic with Boolean lattices and the Dehn--Sommerville equationsFeb 10 2010Let $P$ be a finite poset. Let $L:=J(P)$ denote the lattice of order ideals of $P$. Let $b_i(L)$ denote the number of Boolean intervals of $L$ of rank $i$. We construct a simple graph $G(P)$ from our poset $P$. Denote by $f_i(P)$ the number of the cliques ... More

Group-labeled light dual multinets in the projective plane (with Appendix)Sep 30 2017In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the characteristic $\mathbb{K}$, ... More

Calculating Ultra-Strong and Extended Solutions for Nine Men's Morris, Morabaraba, and LaskerJul 31 2014Mar 15 2015The strong solutions of Nine Men's Morris and its variant, Lasker Morris are well-known results (the starting positions are draws). We re-examined both of these games, and calculated extended strong solutions for them. By this we mean the game-theoretic ... More

3-nets realizing a diassociative loop in a projective planeMar 01 2016A \textit{$3$-net} of order $n$ is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size $n$, such that every point incident with two lines from distinct classes is incident with exactly one line ... More

Finding hidden Borel subgroups of the general linear groupMay 23 2011We present a quantum algorithm for solving the hidden subgroup problem in the general linear group over a finite field where the hidden subgroup is promised to be a conjugate of the group of the invertible lower triangular matrices. The complexity of ... More

Corner percolation on $\mathbb{Z}^2$ and the square root of 17Jul 22 2005Sep 24 2008We consider a four-vertex model introduced by B\'{a}lint T\'{o}th: a dependent bond percolation model on $\mathbb{Z}^2$ in which every edge is present with probability 1/2 and each vertex has exactly two incident edges, perpendicular to each other. We ... More

Shattering bounds for tuple systemsDec 03 2015Let $\mbox{ V}(n,d,q)$ stand for the $q$--ary Hamming spheres. Let $\mbox{ C}\subseteq (q)^n$ denote a tuple system such that $\mbox{ C}\subseteq \cup_{i=0}^s \mbox{ V}(n,d_i,q)$, where $d_1<\ldots <d_s$. We give here a general upper bound on the size ... More

Entropy of non-extremal STU black holes: the F-invariant unveiledAug 26 2015Apr 11 2016We find a manifestly U-duality invariant formula for the Bekenstein-Hawking entropy of the most general 4 dimensional, stationary, asymptoticaly flat, non-extremal STU black holes constructed recently by Chow and Comp\`ere. The expression is entirely ... More

An easy way to a theorem of Kira Adaricheva and Madina Bolat on convexity and circlesOct 08 2016Kira Adaricheva and Madina Bolat have recently proved that if $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$ is included in the convex hull of $U_k\cup(\{A_0,A_1, ... More

Optimal test-configurations for toric varietiesSep 17 2007On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an ... More

The Calabi functional on a ruled surfaceMar 19 2007We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow ... More

Lattices with many congruences are planarJul 22 2018Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly $2^{n-5}$ congruences. ... More

Representing some families of monotone maps by principal lattice congruencesMar 30 2014Sep 05 2014For a lattice L with 0 and 1, let Princ L denote the ordered set of principal congruences of L. For {0,1}-sublattices A subseteq B of L, congruence generation defines a natural map from Princ A to Princ B. In this way, we obtain a small category of bounded ... More

The surface of a lattice polytopeFeb 09 2010Feb 26 2010My main results are simple formulas for the surface area of d-dimensional lattice polytopes using Ehrhart theory.