Wilson病(WD)是一种伴发原发性铜代谢障碍为特征的常染色体隐性遗传性疾病，病种稀少，对其早期发现和确定合适的治疗策略尤为重要，WD的发病机制与ATP7B基因突变及铜代谢障碍密切相关，临床表现具有差异性，适时选择肝移植对延长患者的生存时间和提高生存质量具有重要意义。

Wilson病与肝移植

Having determined E*(E) for E = MU, MSp, K and KO we turn now to a determination of E*(E)and E*(E) for E = H(ℤ2). Although historically H*(H(ℤ2); ℤ2) and H*(H(ℤ2); ℤ2) were known as much as fifteen years before KO*(KO), for example, we shall see that the calculation of these Hopf algebras is at least as difficult as the calculation we just did for KO. In theory we could proceed as follows: for each n we have a fibration H(ℤ2,n-1) → PH(ℤ2, n) → H(ℤ2, n), and the total space PH(ℤ2,n)is contractible (the path space PY is contractible for any Y). Thus in the Serre spectral sequence for this fibration E∞pq= 0 if (p,q) ≠ (0,0). With a little luck this should enable us to determine H*(H (ℤ2, n); ℤ2) from a knowledge of H*(H(ℤ2,n - 1); ℤ2). The process could begin at n=1 since H*(H(ℤ2,1); ℤ2) ≅ H*(RP∞; ℤ), which is a ℤ2-vector space with basis {xi:i⩾1}, xi∈ Hi(RP∞; ℤ2). Then we would have Hq(H(ℤ2); ℤ2)≅ dirlimn Hq+n(H(ℤ2,n); ℤ2). This is precisely how Cartan did determine this algebra (see [28]) using some heavy guns from homological algebra. We shall take a different approach, however; we shall construct some specific cohomology operations—the Steenrod squares Sqi—and show that they generate the algebra A* = H*(H(ℤ2); ℤ2). It will then not be too difficult to determine the dual A* = H*(H(ℤ2); ℤ2). We shall also indicate the results for H(ℤp),p an odd prime.

The Steenrod Algebra and its Dual

We survey computations of stable motivic homotopy groups over various fields. The main tools are the motivic Adams spectral sequence, the motivic Adams-Novikov spectral sequence, and the effective slice spectral sequence. We state some projects for future study.

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Motivic stable homotopy groups

We define the Chow $t$-structure on the $\infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, c\heartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum is algebraic. 
Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To all base fields other than just $C$; To the entire $\infty$-category of motivic spectra $SH(k)$, rather than a subcategory containing only certain cellular objects. 
We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field $k$ using the Postnikov tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.

The Chow $t$-structure on the $\infty$-category of motivic spectra

The motivic Mahowald invariant was introduced in \cite{Qui19a} and \cite{Qui19b} to study periodicity in the $\mathbb{C}$- and $\mathbb{R}$-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field $F$ of characteristic not two and use it to study periodicity in the $F$-motivic stable stems. In particular, we construct lifts of some of Adams' classical $v_1$-periodic families \cite{Ada66} and identify them as the motivic Mahowald invariants of powers of $2+\rho \eta$.

Motivic Mahowald invariants over general base fields

We prove strong convergence results for the motivic Adams spectral sequence of the sphere spectrum over fields with finite virtual cohomological dimension at the prime 2, and over arbitrary fields at odd primes. We show that the motivic Adams spectral sequence is not strongly convergent over number fields. As applications we give bounds on the exponents of the $(\ell,\eta)$-completed motivic stable stems, and calculate the zeroth $(\ell,\eta)$-completed motivic stable stems.

Strong convergence in the motivic Adams spectral sequence

We give recursive formulas for the generating elements in the Milnor basis of the mod 2 motivic Steenrod algebra.

Recursive formulas for the motivic Milnor basis

We compute the motivic homotopy groups of algebraic cobordism over number fields, the motivic homotopy groups of 2-complete algebraic cobordism over the real numbers and rings of $2$-integers and the motivic homotopy groups of mod 2 motivic Morava $K$-theory over fields with low virtual cohomological dimension. As an application we relate the order of the algebraic cobordism groups of rings of 2-integers to special values of Dedekind $\zeta$-functions of totally real abelian number fields.

Jonas Irgens Kylling

Papers

Strong convergence in the motivic Adams spectral sequence

Recursive formulas for the motivic Milnor basis

Recursive formulas for the motivic Milnor basis

Algebraic cobordism of number fields