Related papers: Resolution of Singularities -- Seattle Lecture

This paper demonstrates that singularities form in the classical $(5+1)$-dimensional, co-rotational Skyrme model. It was recently proven by Chen, Sch\"orkhuber, and the author that the strong field limit of the $(5+1)$-dimensional,…

This is a Thesis submitted for the degree of Doctor Philosophiae at S.I.S.S.A./I.S.A.S.

We show here that molecular resolution is inherently hybrid in terms of relative separation: If molecules are close to each other, they must be characterized by a fine-grained (geometrically detailed) model, yet if molecules are far from…

We give a simple proof of the splitting lemma in singularity theory, also known as generalized Morse lemma, for formal power series over arbitrary fields. Our proof for the uniqueness of the residual part in any characteristic is new and…

This is the author's 2004 Master's thesis at Iowa State University, done under the supervision of Roger D. Maddux. It provides a background in relation algebras. Three results from the literature are demonstrated in full: (i.) RRA is a…

This work resolves the open problem of strong singularity ($\alpha(z)> 1$) in nonlocal Kirchhoff-type equations with variable exponents through five original theorems that collectively establish a comprehensive theory. Beginning with…

This is the text of a series of three lectures given at the CMA of the Australian National University on the recent solution of the square root problem for divergence form elliptic operators, a long-standing conjecture posed by Kato in the…

In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-\Delta)^\sigma u = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $\sigma \in (0, 1)$…

This chapter provides a hands-on tutorial on the important technique known as self-reducibility. Through a series of "Challenge Problems" that are theorems that the reader will---after being given definitions and tools---try to prove, the…

We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…

In this paper we provide a formalism, Sudoku logic, in which a solution is logically deducible if for every cell of the grid we can provably exclude all but a single option. We prove that the deductive system of Sudoku logic is sound and…

We consider manifolds with conic singularites that are isometric to $\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the…

This paper is a follow-up to our joint paper with I. Agol, P. Storm and K. Whyte "Finiteness of arithmetic hyperbolic reflection groups". The main purpose is to investigate the effective side of the method developed there and its possible…

We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved…

We prove the existence and uniqueness of a strong solution for an SDE on a semi-axis with singularities at the point 0. The result obtained yields, for example, the strong uniqueness of non-negative solutions to SDEs governing Bessel…

In this paper, we prove several generalizations and applications of a fixed point theorem. This theorem is used to prove the existence and uniqueness of solutions of the linear sparse matrix problem considered.

In this paper, we attempt to resolve the singularities of the zero variety of a $C^{\infty}$ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero…

These notes are an extended version of a series of lectures given at the CIME Summer School in Cetraro in June 2022. The goal is to explain questions about optimal functional inequalities on the example of the sharp Sobolev inequality and…

Let $n\geq 3$, $0\le m<\frac{n-2}{n}$, $\rho_1>0$, $\beta>\beta_0^{(m)}=\frac{m\rho_1}{n-2-nm}$, $\alpha_m=\frac{2\beta+\rho_1}{1-m}$ and $\alpha=2\beta+\rho_1$. For any $\lambda>0$, we prove the uniqueness of radially symmetric solution…

These are Notes prepared for nine lectures given at the Mathematical Sciences Research Institute, MSRI, Berkeley during the period January--March 1995. It is a pleasant duty to record here my gratitude to MSRI, and its staff, for making…