The Étale locus in complete local rings

Let R R be a complete local ring and let Q Q be a prime ideal of R R . It is determined precisely which conditions on R R are equivalent to the existence of a complete unramified regular local ring A A and an element g ∈ A − Q g\in A-Q such that R R is a finite A A -module and A g ⟶ R g A_g\longrightarrow R_g is étale . A number of other properties of the possible embeddings A ⟶ R A\longrightarrow R are developed in the process including the determination of precisely which fields can be coefficient fields in the Cohen-Gabber Theorem.

Magic squares of squares over a finite field

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

Anti-symplectic involutions on rational symplectic 4-manifolds

This is an expanded version of the talk given by the first author at the conference “Topology, Geometry, and Dynamics: Rokhlin – 100”. The purpose of this talk was to explain our current results on the classification of rational symplectic 4-manifolds equipped with an anti-symplectic involution. A detailed exposition will appear elsewhere.

𝐵-rigidity of the property to be an almost Pogorelov polytope

Toric topology assigns to each n n -dimensional combinatorial simple convex polytope P P with m m facets an ( m + n ) (m+n) -dimensional moment-angle manifold Z P \mathcal {Z}_P with an action of the compact torus T m T^m such that Z P / T m \mathcal {Z}_P/T^m is a convex polytope of combinatorial type P P . We study the notion of B B -rigidity. A property of a polytope P P is called B B -rigid if each simple n n -polytope Q Q such that the graded rings H ∗ ( Z P , Z ) H^*(\mathcal {Z}_P,\mathbb Z) and H ∗ ( Z Q , Z ) H^*(\mathcal {Z}_Q,\mathbb Z) are isomorphic also has this property. We study families of 3 3 -dimensional polytopes defined by their cyclic k k -edge-connectivity. These families include flag polytopes and Pogorelov polytopes, that is, polytopes realizable as bounded right-angled polytopes in Lobachevsky space L 3 \mathbb L^3 . Pogorelov polytopes include fullerenes—simple polytopes with only pentagonal and hexagonal faces. It is known that the properties to be flag and to be Pogorelov are B B -rigid. We focus on almost Pogorelov polytopes, which are strongly cyclically 4 4 -edge-connected polytopes. They correspond to right-angled polytopes of finite volume in L 3 \mathbb L^3 . There is a subfamily of ideal almost Pogorelov polytopes corresponding to ideal right-angled polytopes. We prove that the properties to be an almost Pogorelov polytope and to be an ideal almost Pogorelov polytope are B B -rigid. As a corollary, we obtain that the 3 3 -dimensional associahedron A s 3 As^3 and permutohedron P e 3 Pe^3 are B B -rigid. We generalize methods known for Pogorelov polytopes. We obtain results on B B -rigidity of subsets in H ∗ ( Z P , Z ) H^*(\mathcal {Z}_P,\mathbb Z) and prove an analog of the so-called separable circuit condition (SCC).

Notes on endomorphisms, local cohomology and completion

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

Poincaré polynomials of generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety G / B G/B of type A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

Dolbeault cohomology of complex manifolds with torus action

We describe the basic Dolbeault cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a DGA model for the ordinary Dolbeault cohomology algebra. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kähler (equivalently, polytopal) case using a foliated analogue of toric blow-up.

Ascent properties for test modules

We investigate modules for which vanishing of Tor-modules implies finiteness of homological dimensions (e.g., projective dimension and G-dimension). In particular, we answer a question of O. Celikbas and Sather-Wagstaff about ascent properties of such modules over residually algebraic flat local ring homomorphisms. To accomplish this, we consider ascent and descent properties over local ring homomorphisms of finite flat dimension, and for flat extensions of finite dimensional differential graded algebras.

A user’s guide to basic knot and link theory

We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible to a non-specialist or a student. The simplest invariants naturally appear in an attempt to unknot a knot or unlink a link. Then we present certain ‘skein’ recursive relations for the simplest invariants, which allow us to introduce stronger invariants. We state the Vassiliev–Kontsevich theorem in a way convenient for calculating the invariants themselves, not only the dimension of the space of the invariants. No prerequisites are required; we give rigorous definitions of the main notions in a way not obstructing intuitive understanding.

A survey on extensions of Riemannian manifolds and Bartnik mass estimates

Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds ( Σ ≅ S 2 , g ) (\Sigma \cong \mathbb {S}^2,g) , with g g satisfying λ 1 ≔ λ 1 ( − Δ g + K ( g ) ) > 0 \lambda _1 ≔\lambda _1(-\Delta _g + K(g))>0 , where λ 1 \lambda _1 is the first eigenvalue of the operator − Δ g + K ( g ) -\Delta _g+K(g) and K ( g ) K(g) is the Gaussian curvature of g g , with control on the ADM mass of the extension. Remarkably, this procedure allowed them to compute the Bartnik mass in this so-called minimal case; the Bartnik mass is a notion of quasi-local mass in General Relativity which is very challenging to compute. In this survey, we describe the Mantoulidis–Schoen construction, its impact and influence in subsequent research related to Bartnik mass estimates when the minimality assumption is dropped, and its adaptation to other settings of interest in General Relativity.