Adjoining roots in homotopy theory

具有实系数的有界重数非结式系统空间的同伦稳定性

Homotopy stability for spaces of triples of real polynomials without common roots

本文研究具有实系数的有界重数非结式系统空间的同伦稳定性，解决了(m,n)=(3,1)的边界情况，证明其同伦稳定性并确定稳定范围，提及另一边界情况(m,n)=(1,3)待后续研究。

Towers of MU-algebras and the generalized Hopkins-Miller theorem

Our results are of three types. First we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions.A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of MU, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable K(n)-localizations o…

Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes

For a log scheme locally of finite type over $\mathbb{C}$, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over $\mathbb{C}$, another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over $\mathbb{C}$, these three notions agree. In particular, we construct a comparison …

The homotopy type of spaces of real resultants with bounded multiplicity

For positive integers $d,m,n\geq 1$ with $(m,n)\not= (1,1)$ and $\Bbb K=\Bbb R$ or $\Bbb C$, let $Q^{d,m}_{n}(\Bbb K)$ denote the space of $m$-tuples $(f_1(z),\cdots ,f_m(z))\in \Bbb K [z]^m$ of $\Bbb K$-coefficients monic polynomials of the same degree $d$ such that polynomials $\{f_k(z)\}_{k=1}^m$ have no common {\it real} root of multiplicity $\geq n$ (but may have complex common root of any multiplicity). %% These spaces can be regarded as one of generalizations of the spaces defined and studied by Arnold and Vassiliev, and they may be also considered as the real analogues of the spaces st…

The Bredon-Landweber region in $C_2$-equivariant stable homotopy groups

We use the $C_2$-equivariant Adams spectral sequence to compute part of the $C_2$-equivariant stable homotopy groups $π^{C_2}_{n,n}$. This allows us to recover results of Bredon and Landweber on the image of the geometric fixed-points map from the equivariant homotopy group $π^{C_2}_{n,n}$ to the classical $π_0$. We also recover results of Mahowald and Ravenel on the Mahowald root invariants of the elements $2^k$.

来自Weyl多胞体的环面叠空间的同伦类型

Homotopy Types Of Toric Orbifolds From Weyl Polytopes

本文研究了约化晶体学根系统关联的Weyl多胞体与环面簇，证明了底拓扑空间$X_P/W_K$和$X_{P/W_K}$在根格或权格实张成中同伦等价。