We compute the group $\mathrm{LM}_{2,2}^4$ of link homotopy classes of link maps of two $2$-spheres into $4$-space. It turns out to be free abelian, generated by geometric constructions applied to the Fenn--Rolfsen link map and detected by two self-intersection invariants introduced by Kirk in this setting. As a corollary, we show that any link map with one topologically embedded component is link homotopic to the unlink.

Our proof introduces a new basic link homotopy, which we call a Whitney homotopy, that shrinks an embedded Whitney sphere constructed from four copies of a Whitney disk. Freedman's disk embedding theorem is applied to get the necessary embedded Whitney disks, after constructing sufficiently many accessory spheres as algebraic duals for immersed Whitney disks. To construct these accessory spheres and immersed Whitney disks we use the algebra of metabolic forms over the group ring $\mathbb{Z}[\mathbb{Z}]$, and we introduce a number of new $4$-dimensional constructions, including maneuvers involving the boundary arcs of Whitney disks.

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation$$x_1^2+x_2^2+\cdots+x_n^2=ax_1x_2\cdots x_n+k.$$When $n\ge 4$, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent $\beta$ that is not in general an integer when $n\ge 4$. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space.

Let ($\sigma_q,\ldots,\sigma_d)$ be a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as $n\to \infty$, these permutations viewed as operators on the $n-1$ dimensional vector space $\{(x_1,\ldots , x_n) \in \mathbb{C}^n, \sum x_1 = 0\}$, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refined trace method.

As a byproduct, we show that the non-trivial eigenvalues of random $n$-lifts of a fixed based graphs approximately achieve the Alon-Boppana bound with high probability in the large $n$ limit. This result generalizes Friedman's Theorem stating that with high probability, the Schreier graph generated by a finite number of independent random permutations is close to Ramanujan.

Finally, we extend our results to tensor products of random permutation matrices. This extension is especially relevant in the context of quantum expanders.

To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision in algebraic $K$-theory. The construction of this new ring spectrum is categorical and hence allows us to determine the failure of excision for any localizing invariant in place of $K$-theory.

As immediate consequences we obtain an improved version of Suslin's excision result in $K$-theory, generalizations of results of Geisser and Hesselholt on torsion in (bi)relative $K$-groups, and a generalized version of pro-excision for $K$-theory. Furthermore, we show that any truncating invariant satisfies excision, nilinvariance, and cdh-descent. Examples of truncating invariants include the fibre of the cyclotomic trace, the fibre of the rational Goodwillie--Jones Chern character, periodic cyclic homology in characteristic zero, and homotopy $K$-theory.

Various of the results we obtain have been known previously, though most of them in weaker forms and with less direct proofs.

We prove that the map on knot Floer homology induced by a ribbon concordance is injective. As a consequence, we prove that the Seifert genus is monotonic under ribbon concordance. Generalizing theorems of Gabai and Scharlemann, we also prove that the Seifert genus is super-additive under band connected sums of arbitrarily many knots. Our results give evidence for a conjecture of Gordon that ribbon concordance is a partial order on the set of knots.

In this paper, we show that every $(2^{n-1}+1)$-vertex induced subgraph of the n-dimensional cube graph has maximum degree at least $\sqrt{n}$. This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.