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Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted $\mathcal{A}$ and $\mathcal{B}$) that are constructed from a nondegenerate quasihomogeneous polynomial $W$ and a related group of symmetries…

In a recent paper Flohr, Grabow and Koehn conjectured that the characters of the logarithmic conformal field theory c_{k,1}, of central charge c=1-6(k-1)^2/k, admit fermionic representations labelled by the Lie algebra D_k. In this note we…

Let $I\subset S$ be a graded ideal of a standard graded polynomial ring $S$ with coefficients in a field $K$, and let $\text{v}(I)$ be the $\text{v}$-number of $I$. In previous work, we showed that for any graded ideal $I\subset S$…

Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper we perform a thorough study of general (nonlinear) canonical transformations…

The Hamiltonian reduction $\mathcal N///T$ of the nilpotent cone in $\mathfrak{sl}_n$ by the torus of diagonal matrices is a Nakajima quiver variety which admits a symplectic resolution $\widetilde{\mathcal N///T}$, and the corresponding…

Bass and Pardoux (1987) deduce from the Krein-Rutman theorem a reverse ergodic theorem for a sub-probability transition function, which turns out to be a key tool in proving uniqueness of reflecting Brownian Motion in cones in Kwon and…

We present a new approach to construct $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideas originally stemming from PBW-filtrations.…

Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let f be a regular self-map on X defined over K, let V be a subvariety of X defined over K, and let x be a…

To a pair $(A,s)$ consisting of a smooth, cyclic $A_\infty$-algebra $A$ and a splitting $s$ of the Hodge filtration on its Hochschild homology Costello (2005) associates an invariant which conjecturally generalizes the total descendant…

We consider the Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in $\C^k$ with normal crossings. We construct two potential functions (of first and second kind) of variables…

In this paper we determine, for all $r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting $r$-tuples of nilpotent elements of $\mathfrak{gl}_n$. Our main result is that in characteristic $\neq…

We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a…

It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley--Reisner ring (i.e. a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely…

Rota's basis conjecture (RBC) states that given a collection B of n bases in a matroid M of rank n, one can always find n disjoint rainbow bases with respect to B. We show that if M is a matroid having n + k elements, then one can construct…

Let q_1, ..., q_n be some variables and set K:=Z[q_1, ..., q_n]/(q_1q_2...q_n). We show that there exists a K-bilinear product \star on H^*(F_n;Z)\otimes K which is uniquely determined by some quantum cohomology like properties (most…

We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and…

In this paper we perform in a manifestly $SO(n-1,1)$ [or, alternatively $SO(n)$] covariant fashion, the canonical analysis of general relativity in $n$ dimensions written as a constrained $BF$ theory. Since the Lagrangian action of the…

We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and…

In this paper we prove that every Khovanov homology associated to a Frobenius algebra of rank $2$ can be modified in such a way as to produce a TQFT on oriented links, that is a monoidal functor from the category of cobordisms of oriented…

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This…