Related papers: The Nash problem from a geometric and topological …
The Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was…
The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real…
We show that the second Jacobian ideal of a hypersurface can be decomposed such that a power of the Jacobian ideal becomes a factor. As an application of the decomposition, we present an elementary proof establishing that the second Nash…
We generalize the notion of Monk's schema in such a way to integrate finite dimensions. This allows us to lift a plathora of deep results proved for finite dimensions to the infinite dimensional case, like the solution to problem 2.12 in…
We overview the main ideas and techniques of the functional-analytical approach to some extremal problems of convex geometry that stem from the Queen Dido problem.
We show that if the Atiyah Jones conjecture holds for a surface $X,$ then it also holds for the blow-up of $X$ at a point. Since the conjecture is known to hold for ${\mathbb P}^2$ and for ruled surfaces, it follows that the conjecture is…
Recently, I. Kossovskiy and R. Shafikov have settled the so-called Dimension Conjecture, which characterizes spherical hypersurfaces in ${\mathbb C}^2$ via the dimension of the algebra of infinitesimal automorphisms. In this note, we…
For generalized Nash equilibrium problems (GNEP) with shared constraints we focus on the notion of normalized Nash equilibrium in the nonconvex setting. The property of nondegeneracy for normalized Nash equilibria is introduced.…
In this paper, we prove a conjecture of Schnell in the surface case.
The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between…
In this paper, we provide some remarks on the scalar curvature rigidity theorem of Brendle and Marques in \cite{BrendleMarques}. The main result is that Brendle and Marques' theorem holds on a geodesic ball larger than that specified in…
We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…
We prove De Giorgi-Nash-Moser Theory using a geometric approach.
Perception research provides strong evidence in favor of part based representation of shapes in human visual system. Despite considerable differences among different theories in terms of how part boundaries are found, there is substantial…
We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify…
This paper introduces a notion of generalised geometric logic. Connections of generalised geometric logic with L-topological system and L-topological space are established.
The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of…
The more recent paper "Generic strange duality for K3 surfaces" by the authors contains stronger results.
We prove the Strengthened Hanna Neumann Conjecture. We give a more direct cohomological interpretation of the conjecture in terms of "typical" covering maps, and use graph Galois theory to "symmetrize" the conjecture. The conjecture is then…
We revisit the notion of equations describing pseudospherical surfaces, starting from the works by Sasaki, whose roots were influenced by the AKNS system, the works by Chern and Tenenblat, until current research topics in the field relating…