We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.

Using motivic integration theory and the notion of riso-triviality, we introduce two new objects in the framework of definable nonarchimedean geometry: a convenient partial preorder $\preccurlyeq$ on the set of constructible motivic functions, and an invariant $V_0$, nonarchimedean substitute for the number of connected components. We then give several applications based on $\preccurlyeq$ and $V_0$: we obtain the existence of nonarchimedean substitutes of real measure geometric invariants $V_i$, called the Vitushkin variations, and we establish the nonarchimedean counterpart of a real inequality involving $\preccurlyeq$, the metric entropy and our invariants $V_i$. We also prove the nonarchimedean Cauchy–Crofton formula for definable sets of dimension $d$, relating $V_0$ (and $V_d$) and the motivic measure in dimension $d$.

In their 1988 paper ‘Gluing of perverse sheaves and discrete series representations’, D. Kazhdan and G. Laumon constructed an abelian category $\mathcal{A}$ associated to a reductive group G over a finite field with the aim of using it to construct discrete series representations of the finite Chevalley group $G(\mathbb{F}_q)$. The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproved in 2001 by R. Bezrukavnikov and A. Polishchuk, who found a counterexample in the case $G = SL_3$. Polishchuk then made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension, and suggested that this conjecture could lead toward a proof that Kazhdan and Laumon’s construction is well defined. He proved this conjecture in Types $A_1, A_2, A_3$, and $B_2$. In the present paper, we prove Polishchuk’s conjecture for all types, and prove that Kazhdan and Laumon’s construction is indeed well defined, giving a new geometric construction of discrete series representations of $G(\mathbb{F}_q)$.

Restriction is a natural quasi-order on d-way tensors. We establish a remarkable aspect of this quasi-order in the case of tensors over a fixed finite field; namely, that it is a well-quasi-order: it admits no infinite antichains and no infinite strictly decreasing sequences. This result, reminiscent of the graph minor theorem, has important consequences for an arbitrary restriction-closed tensor property X. For instance, X admits a characterisation by finitely many forbidden restrictions and can be tested by looking at subtensors of a fixed size. Our proof involves an induction over polynomial generic representations, establishes a generalisation of the tensor restriction theorem to other such representations (e.g., homogeneous polynomials of a fixed degree), and also describes the coarse structure of any restriction-closed property.

The Newell–Littlewood (NL) numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood–Richardson (LR) coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: a description by Extended Horn inequalities (as conjectured in part II of this series), where, using a result of King’s, this description is controlled by the saturated LR-cone and thereby recursive, just like the Horn inequalities; a minimal list of defining linear inequalities; an eigenvalue interpretation; and a factorization of Newell–Littlewood numbers, on the boundary.

We develop a local-to-global formalism for constructing Calabi–Yau structures for global sections of constructible sheaves or cosheaves of differential graded categories. The required data (a morphism between the sheafified Hochschild homology with the topological dualizing sheaf, satisfying a nondegeneracy condition) specializes to the classical notion of orientation when applied to the category of local systems on a manifold. We apply this construction to the cosheaves on arboreal skeleta arising in the microlocal approach to the A-model.

We study the Macdonald intersection polynomials $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}[X;q,t]$, which are indexed by $k$-tuples of partitions $\mu ^{(1)},\dots ,\mu ^{(k)}$. These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia–Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection polynomials. Additionally, we unveil a remarkable connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the character $\nabla e_{k-1}$ of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia, Haiman, and Tesler. Furthermore, we establish a connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the shuffle formula $D_{k-1}[X;q,t]$, utilizing novel combinatorial tools such as the column exchange rule and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.