The monotone wrapped Fukaya category and the open-closed string map

We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points and provide explicit expressions that allow one to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using $A$-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.

In this paper we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac34$ there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than $c\rk K_0(Y).$ To obtain varieties without exceptional collections of line bundles, it suffices to put $c=1.$ On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least $\frac34 \rk K_0(Y).$ The constant $\frac34$ is thus maximal with this property.

It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409.8611 that for punctured surfaces $X$ with stops $\Lambda$ at their boundary, the partially wrapped Fukaya category $\mathcal{W}(X,\Lambda)$ provides a categorical (smooth and proper) resolution of the compact Fukaya category $\mathcal{F}(X)$. Inspired by this analogy, we establish an equivalence between the derived category of modules over the Auslander orders over certain nodal stacky curves and partially wrapped Fukaya categories associated to punctured surfaces of arbitrary genus equipped with stops at their boundary. As an application, we deduce equivalences between derived categories of coherent sheaves (resp. perfect complexes) on such nodal stacky curves and the wrapped (resp. compact) Fukaya categories of punctured surfaces of arbitrary genus.

In this paper we are concerned with Banach A∞-module M over admissible Banach A∞-algebra A. We give some properties of admissible modules and algebras. We study the cohomology of the complex C∞(A,M). We show that the vanishing of cohomology of this complex in certain dimensions implies to the existence of the A∞-module structure.

We study the Picard rank of smooth toric Fano varieties constrained to possess families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen–Fu–Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.

We construct a pair of adjoint functors between the categories of crossed modules of groups and associative algebras and establish an equivalence of categories between module structures over a crossed module of groups and its respective crossed module of associative algebras.

On metric properties of maps between Hamming spaces and related graph homomorphisms

The Loewy and socle structure of the projective indecomposable modules for the group algebras of the Mathieu group M 12 and its automorphism group M 12.2 over an algebraically closed field of characteristic 3 are determined without using computers.

Let M x (resp. Vx) denote the irreducible module (resp. the Weyl module) with highest weight Z of a connected reductive algebraic group G over k, with k an algebraically closed field of characteristic p >0 . In [3] we proved that a necessary condition for M~ to be a composition factor of V X is that 2 is strongly linked to )6 We proved more generally that the same is true for all cohomology groups of the line bundles on G/B, B a Borel subgroup, induced by )~ and its conjugates under the Weyl group W. Even though Very little is known about the higher cohomology groups of line bundles on G/B (in particular it is not known exactly when they are indecomposable as G-modules) the existence of certain canonical maps between them enables us to deduce some properties of their composition factors and of the structure of Weyl modules. In this paper we shall explore the method from [3] somewhat further. First we obtain an estimate on the set of weights in HI(D, i>0, ;g a character o r b (as in [3] we write Hi00 short for Hi(G/B, L00)). We prove that if )~ is dominant then Z is the unique highest weight in H~(W)(w.z) for all w~W, and it occurs with multiplicity 1. Here ~(w) denotes the length of w and so in characteristic zero, Ht('~)(w.z) is the unique non-vanishing cohomology group [7]. Next we prove that when 2 and X are dominant and 2 is maximal among the weights strongly linked to ;g then M~ occurs exactly once as a composition factor of V z. For p-regular weights this result was proved by Jantzen [9, Satz 10] (see also Carter's and Lusztig's results for groups of type A, [5, Theorem 4.1]). The above result on the highest weight in Ht(W)(w.x) gives a canonical homomorphism Ht{W)(w.z)-~H~ In this way we obtain for each reduced expression for w0, the longest element in W, a filtration of H~ (and of V.~). Using these filtrations we are able to prove the conjecture of Carter and Lusztig [5, p. 239] about intertwining homomorphisms between certain Weyl modules in most cases. We conclude the paper with a general formula for the character of the image of intertwining homomorphisms in the direction of a simple root. As illustrations we give details in the 2 simplest non-trivial cases: type A 2 and B2, Z in the bottom p<alcove.

Floer theory for Lagrangian cobordisms was developed by Biran and Cornea in a series of papers [Lagrangian cobordism. I, J. Amer. Math. Soc. 26 (2013) 295–340; Lagrangian cobordism and Fukaya categories, Geom. Funct. Anal. 24 (2014) 1731–1830; Cone-decompositions of Lagrangian cobordisms in Lefschetz fibrations, Selecta Math. 23 (2017) 2635–2704] to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study Lagrangian cobordisms in Liouville manifolds and the associated exact triangles in the derived wrapped Fukaya category. Furthermore, we compute the cobordism groups of non-compact Riemann surfaces of finite type.

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric Residue Mirror Conjecture, claims that the generating functions of intersection numbers of divisors on a special sequence of simplicial toric varieties are power series expansions of some rational functions obtained as toric residues. We expect that this conjecture holds true for all Gorenstein toric Fano varieties associated with reflexive polytopes and give some evidences for that. The proposed conjecture suggests a simple method for computing Yukawa couplings for toric mirror Calabi-Yau hypersurfaces without solving systems of differential equations. We make several explicit computations for Calabi-Yau hypersurfaces in weighted projective spaces and in products of projective spaces.

We classify smooth toric Fano varieties of dimension $n\\geq 3$ containing a toric divisor isomorphic to the $(n-1)$-dimensional projective space. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\\geq 3$ with a fixed point $x\\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to the $n$-dimensional projective space or to the blow-up of the $n$-dimensional projective space along an invariant linear codimension two subspace. As expected, such results are proved\nusing toric Mori theory due to Reid.

Bivariance, Grothendieck duality and Hochschild homology, II: The fundamental class of a flat scheme-map

Modular representation theory of BIB designs