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In this work we analyze the main properties of the Zariski and maximal spectra of the ring ({{mathcal {S}}}^r(M)) of differentiable semialgebraic functions of class ({{mathcal {C}}}^r) on a semialgebraic set (Msubset {{mathbb {R}}}^m). Denote ({{mathcal {S}}}^0(M)) the ring of semialgebraic functions on M that admit a continuous extension to an open semialgebraic neighborhood of M in ({text {Cl}}(M)). This ring is the real closure of ({{mathcal {S}}}^r(M)). If M is locally compact, the ring ({{mathcal {S}}}^r(M)) enjoys a Łojasiewicz’s Nullstellensatz, which becomes a crucial tool. Despite ({{mathcal {S}}}^r(M)) is not real closed for (rge 1), the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring ({{mathcal {S}}}^0(M)). In addition, the quotients of ({{mathcal {S}}}^r(M)) by its prime ideals have real closed fields of fractions, so the ring ({{mathcal {S}}}^r(M)) is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of ({{mathcal {S}}}^r(M)) and ({{mathcal {S}}}^0(M)) guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring ({{mathcal {S}}}^r(M)) is a Gelfand ring and its Krull dimension is equal to (dim (M)). We also show similar properties for the ring ({{mathcal {S}}}^{r*}(M)) of differentiable bounded semialgebraic functions. In addition, we confront the ring ({mathcal S}^{infty }(M)) of differentiable semialgebraic functions of class ({{mathcal {C}}}^{infty }) with the ring ({{mathcal {N}}}(M)) of Nash functions on M.

Let X be a generic quasi-symmetric representation of a connected reductive group G. The GIT quotient stack (mathfrak {X}=[X^text {ss}(ell )/G]) with respect to a generic (ell ) is a (stacky) crepant resolution of the affine quotient X/G, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of X/G. Halpern-Leistner and Sam showed that the derived category ({{textrm{D}}^{textrm{b}}}({text {coh}}mathfrak {X})) is equivalent to certain subcategories of ({{textrm{D}}^{textrm{b}}}({text {coh}}[X/G])), which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When G is a torus, it turns out that the exchanges are nothing but iterated Iyama–Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama–Wemyss mutations induce a group action of the fundamental group (pi _1(mathbb {P}^1,backslash {0,1,infty })) on the derived category of a Calabi-Yau complete intersection in a weighted projective space.

The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin’s conjecture in its original form would turn out to be incorrect.

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Kollár (Publ. Res. Inst. Math. Sci. 40(3):689–708, 2004).

We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to (U_q(mathfrak {s}mathfrak {l}_2)) at roots of unity and we show that many of the classical results on representations of quivers can be generalised to this setting. Namely, we prove a generalised Gabriel’s theorem for Coxeter quivers that encompasses all Coxeter–Dynkin diagrams—including the non-crystallographic types H and I. Moreover, a similar relation between reflection functors and Coxeter theory is used to show that the indecomposable representations correspond bijectively to the (extended) positive roots of Coxeter root systems over fusion rings.

Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order.

In this note we establish the existence of a new type of rigidity of symplectic embeddings coming from obligatory intersections with symplectic planes. In particular, we prove that if a Euclidean ball is symplectically embedded in the Euclidean unit ball, then it must intersect a sufficiently fine grid of two-codimensional pairwise disjoint symplectic planes. Inspired by analogous terminology for Lagrangian submanifolds, we refer to these obstructions as symplectic barriers.

Motivated by constructions appearing in mirror symmetry, we study special representatives of complexified Kähler classes, which extend the notions of constant scalar curvature and extremal representatives for usual Kähler classes. In particular, we provide a moment map interpretation, discuss a possible correspondence with compactified Landau–Ginzburg models, and prove existence results for such special complexified Kähler forms and their large volume limits in certain toric cases.

We study the category of (textbf{GL})-equivariant modules over the infinite exterior algebra in positive characteristic. Our main structural result is a shift theorem à la Nagpal. Using this, we obtain a Church–Ellenberg type bound for the Castelnuovo–Mumford regularity. We also prove finiteness results for local cohomology.

The author has recently introduced the class of ( CNED ) sets in Euclidean space, generalizing the classical notion of ( NED ) sets, and shown that they are quasiconformally removable. A set E is ( CNED ) if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting E at countably many points. We prove that several classes of sets that were known to be removable are also ( CNED ), including sets of (sigma )-finite Hausdorff ((n-1))-measure and boundaries of domains with n-integrable quasihyperbolic distance. Thus, this work puts in common framework many known results on the problem of quasiconformal removability and suggests that the ( CNED ) condition should also be necessary for removability. We give a new necessary and sufficient criterion for closed sets to be (C)NED. Applying this criterion, we show that countable unions of closed (C)NED sets are (C)NED. Therefore we enlarge significantly the known classes of quasiconformally removable sets.