Related papers: Strict Bott-Samelson Resolutions of Schubert Varie…

We generalize Standard Monomial Theory (SMT) to intersections of Schubert varieties and opposite Schubert varieties; such varieties are called Richardson varieties. The aim of this article is to get closer to a geometric interpretation of…

Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a "Giambelli formula" expressing…

As local and nonlocal reductions of a discrete second-order Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr\"odinger type equations are considered. Through the bilinearization reduction method, we construct double…

In this thesis, we investigate several aspects of models with a multiplicity of Higgs bosons. We start by focusing on models with two Higgs doublets. We claim that one of the most used models in the literature -- the so-called real…

We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than…

Heterotic orbifolds provide promising constructions of MSSM-like models in string theory. We investigate the connection of such orbifold models with smooth Calabi-Yau compactifications by examining resolutions of the T^6/Z6-II orbifold…

Motivated by a recent random pipe dream model, we study a family of probability distributions on \(S_n\) arising from Bott--Samelson varieties over finite fields. More precisely, for a word \(R\), we consider the Bott--Samelson map…

We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…

Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of $u_t = u_{xx} + e^{u^m}$ with the…

Sharpness-aware minimization (SAM) improves generalization of various deep learning tasks. Motivated by popular architectures such as LoRA, we explore the implicit regularization of SAM for scale-invariant problems involving two groups of…

Two discretizations of the vector nonlinear Schrodinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other…

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…

For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T^1_X:=ext^1(Omega_X,O_X). A variety is semi-smooth if its singularities are…

We review the recent progress on the long-time behavior for a general class of focusing $L^2$-critical nonlinear Schr\"odinger equations (NLS) with lower order perturbations. Two canonical models are the stochastic NLS driven by linear…

Results on stability of tautological sheaves on Hilbert schemes of points are extended to higher dimensions and transferred to abelian surfaces and to the restriction of tautological sheaves to generalised Kummer varieties. This provides a…

In this paper, given a compact Kcsc orbifolds of any dimension and with nontrivial holomorphic vector fields, we find sufficient conditions on the position of singular points in order to admit a Kcsc desingularization, generalizing the…

We construct an MSSM with three generations from the heterotic string compactified on a smooth 6D internal manifold using Abelian gauge fluxes only. The compactification space is obtained as a resolution of the T6/Z2xZ2xZ2-free orbifold.…

We consider the self-dual Chern-Simons-Schr\"odinger equation (CSS) under equivariant symmetry, which is a $L^{2}$-critical equation. It is known that (CSS) admits solitons and finite-time blow-up solutions. In this paper, we show soliton…

We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the…

In this paper we study the behavior of a class of mild solutions of the homogeneous and isotropic bosonic Boltzmann-Nordheim equation near the blow-up. We obtain some estimates on the blow-up rate of the solutions and prove that, as long as…