Quite an old joke I heard from one TV serial long time ago (I forgot what’s the title though) :

Q : What is a dessert that half of its height is 1.57
A : A piece of pi(e).

lol… quite funny isn’t.
Well, we all know, perhaps since we’re at elementary school, that Pi or π is […]

E8 is first of all the largest exceptional root system, which is a set of vectors in an 8-dimensional real vector space satisfying certain properties. Root systems were classified by Wilhelm Killing in the 1890s. He found 4 infinite classes of Lie algebras, labelled An, Bn, Cn, and Dn, where n=1,2,3…. He […]

A flat complete causal Lorentzian manifold is called {\it strictly causal} if
the past and the future of each its point are closed near this point. We
consider strictly causal manifolds with unipotent holonomy groups and assign to
a manifold of this type four nonnegative integers (a signature) and a parabola
in the cone of positive definite matrices. Two manifolds are equivalent if and
only if their signatures coincides and the corresponding parabolas are equal
(up to a suitable automorphism of the cone and an affine change of variable).
Also, we give necessary and sufficient conditions, which distinguish parabolas
of this type among all parabolas in the cone.

We give a short proof of the (known) result that there are no Kaehler
structures on exotic tori. This yields a negative solution to a problem posed
by Benson and Gordon. W discuss the symplectic version of the problem and
analyze results which yield an evidence for the conjecture that there are no
symplectic structures on exotic tori.

We show that certain submanifolds of generalized complex manifolds (“weak
branes”) admit a natural quotient which inherits a generalized complex
structure. This is analog to quotienting coisotropic submanifolds of symplectic
manifolds. In particular Gualtieri’s generalized complex submanifolds
(“branes”) quotient to space-filling branes. Along the way we perform
reductions by foliations (i.e. no group action is involved) for exact Courant
algebroids - interpreting the reduced \v{S}evera class - and for Dirac
structures.

To every convex $d$-polytope with the dual graph $G$ a matrix is associated.
The matrix is shown to be a discrete Schr\”odinger operator on $G$ with the
second least eigenvalue of multiplicity $d$. This implies that the Colin de
Verdi\`ere parameter of $G$ is greater or equal $d$. The construction
generalizes the one given by Lov\’asz in the case $d = 3$.

In this paper we consider successive iterations of the first-order
differential operations in space ${\bf R}^3.$

We show that classical Wilczynski–Se-ashi invariants of linear systems of
ordinary differential equations are generalized in a natural way to contact
invariants of non-linear ODEs. We explore geometric structures associated with
equations that have vanishing generalized Wilczynski invariants and establish
relationship of such equations with deformation theory of rational curves on
complex algebraic surfaces.

In this paper we present a recurrent relation for counting meaningful
compositions of the higher-order differential operations on the space $R^{n}$
(n=3,4,…) and extract the non-trivial compositions of order higher than two.

A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost
nup}, if $(D, C) > 0$ for every very general curve $C$ on $M$. An algebraic
variety $X$ is of {\it almost general type}, if there exists a projective
variety $M$ with only terminal singularities such that the canonical divisor
$K_M$ is almost nup and that $M$ is birationally equivalent to $X$. We prove
that, a complex algebraic variety is of almost general type if and only if it
is neither uniruled nor covered by any family of varieties being birationally
equivalent to minimal varieties with numerically trivial canonical divisors,
under the minimal model conjecture. Furthermore we prove that, for a projective
variety $X$ with only terminal singularities, $X$ is of almost general type if
and only if the canonical divisor $K_X$ is almost nup, under the minimal model
conjecture.

We give an introduction to Tropical Geometry and prove some results in
Tropical Intersection Theory. The first part of this paper is an introduction
to tropical geometry aimed at researchers in Algebraic Geometry from the point
of view of degenerations of varieties using projective not-necessarily-normal
toric varieties. The second part is a foundational account of tropical
intersection theory with proofs of some new theorems relating it to classical
intersection theory.

Revised version includes corrections and new proofs relating on lattice
indices and the associated cycle.

Under some positivity assumptions, extension properties of rationally
connected fibrations from a submanifold to its ambient variety are studied.
Given a family of rational curves on a complex projective manifold X inducing a
covering family on a submanifold Y with ample normal bundle in X, the main
results relate, under suitable conditions, the associated rational connected
fiber structures on X and on Y. Applications of these results include an
extension theorem for Mori contractions of fiber type and a classification
theorem in the case Y has a structure of projective bundle or quadric
fibration.