Related papers: An Approach with Toric Varieties for Singular Lear…

We develop a mathematical framework to address a broad class of metric and preference learning problems within a Hilbert space. We obtain a novel representer theorem for the simultaneous task of metric and preference learning. Our key…

In this paper, we study the strong convergence of an algorithm to solve the variational inequality problem which extends(Thong et al, Numerical Algorithms. 78, 1045-1060 (2018)). We have reduced and refined some of their algorithm's…

This is a first graduate course in algebraic geometry. It aims to give the student a lift up into the subject at the research level, with lots of interesting topics taken from the classification of surfaces, and a human-oriented discussion…

We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…

The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…

This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of…

We consider the motivic Hall algebra of coherent sheaves over an irreducible reduced projective curve of arithmetic genus $1$. We introduce the composition subalgebra in the singular curve case, and show that it is isomorphic to the…

We develop algorithms to compute two versions of the motivic Hilbert zeta function for curve singularities: the classical version, applicable to singularities with a monomial valuation semigroup or to singular curves defined by…

Historically, probability theory has been studied for a long time, and Kolmogorov, Levy Ito Kiyoshi, and others have mathematically developed modern probability in conjunction with measurement theory. On the other hand, commutative algebra…

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of…

A theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started since a…

Any ruled surface in Euclidean 3-space is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra of type (0,3,1)). Combining this classical framework and Singularity Theory, we characterize…

We make progress in understanding the geometry associated to the Generalized Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories (SCFTs), by exploiting the connection between Hanany-Witten transitions and the…

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this paper we explore this correspondence to classify smooth lattice polytopes…

The integral variation map and algebraic monodromy of isolated plane curve singularities are important homological invariants of the singularity which are still far from being completely understood. This work provides effective ways of…

Mathematical equivalence between statistical mechanics and machine learning theory has been known since the 20th century, and research based on this equivalence has provided novel methodologies in both theoretical physics and statistical…

The paper is based on a talk. Complete exposition is given in "Equivariant Hirzebruch class for singular varieties". Starting from the classical theory we describe Hirzebruch class and the related Todd genus of a complex singular algebraic…

As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in…

We show how well known tools of algebraic geometry for the study of finite sets can be fruitfully applied to the study of Waring decompositions of symmetric tensors (forms). We mainly focus on the uniqueness of a given decomposition (the…

We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in…