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A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the $\mathfrak{p}$-local…

For a tuple of $k+1$ convex polytopes $(A, B,\ldots, B)$ we solve the so-called effective membership problem, i.e. for a tuple $f=(f_1,\ldots, f_k)$ of polynomials satisfying some certain properties of generality and having Newton polytope…

Let $G$ be a finite group and let $\pi$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $\pi$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $\pi$-Hall subgroup if, and only if, $G$ has…

The detailed derivation of a theory for transport in quasi-two-dimensional metals, with small-angle elastic scattering and angle-independent inelastic scattering is presented. The transport equation is solved for a model Fermi surface…

Let $\mathbb C$ be the set of complex numbers, and let $\mathcal P$ be a collection of complex polynomial maps in several variables. Assuming at least one $P\in\mathcal P$ depends on at least two variables, we classify all possibilities for…

Illumination complexes are examples of 'flat polyhedral complexes' which arise if several copies of a convex polyhedron (convex body) Q are glued together along some of their common faces (closed convex subsets of their boundaries). A…

We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $\Delta$ denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its…

A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds.…

Using a well-ordering on the reals, one can prove there exists a partition of the three-dimensional Euclidean space into unit circles (PUC). We show that the converse does not hold: there exist models of $\mathsf{ZF}$ without a…

We define the representation ring of a saturated fusion system $\mathcal F$ as the Grothendieck ring of the semiring of $\mathcal F$-stable representations, and study the dimension functions of $\mathcal F$-stable representations using the…

Let $R\subset F$ be an extension of real closed fields and ${\mathcal S}(M,R)$ the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^n$. We prove that every $R$-homomorphism $\varphi:{\mathcal S}(M,R)\to F$ is…

We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras.The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained via…

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance…

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb…

To a finite group G one can associate a tower of wreath products S_n[G]. It is well known that the graded direct sum of the Grothendieck groups of the categories of finite dimensional complex representations of these groups can be given the…

We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of…

Let T be the unit circle in the complex plane C. This paper proves the existence of analytic structure in a compact subset K of T X C^n, where K has so-called "lineally convex" or "hypoconvex" fibers over T. It also addresses a related…

Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$. Let $V$ be an irreducible ${\mathbb F}H$-module and $U$ a quotient of the induced ${\mathbb F}G$-module $V\kern-3pt\uparrow$. We describe the structure of $U$, which is…

We study large classes of real-valued analytic functions that naturally emerge in the understanding of Dulac's problem, which addresses the finiteness of limit cycles in planar differential equations. Building on a Maximum Modulus-type…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and…