Categorical polynomial entropy

Entropy of an autoequivalence on Calabi-Yau manifolds

We prove that the categorical entropy of the autoequivalence $T_{\mathcal{O}}\circ(-\otimes\mathcal{O}(-1))$ on a Calabi-Yau manifold is the unique positive real number $λ$ satisfying $$ \sum_{k\geq 1}\frac{χ(\mathcal{O}(k))}{e^{kλ}}=e^{(d-1)t}. $$ We then use this result to construct the first counterexamples of a conjecture on categorical entropy by Kikuta and Takahashi.

Categorical vs topological entropy of autoequivalences of surfaces

In this paper, we give an example of an autoequivalence with positive categorical entropy (in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich) for any surface containing a (-2)-curve. Then we show that this equivalence gives another counter-example to a conjecture proposed by Kikuta and Takahashi. In a second part, we study the action on cohomology induced by spherical twists composed with standard autoequivalences on a surface S and show that their spectral radii correspond to the topological entropy of the corresponding automorphisms of S.

稳定流形中的轨道与范畴熵：有限 Albanese 态射簇的应用

Orbits in stability manifolds and categorical entropy: applications to varieties with finite Albanese morphisms

本文证明具有有限 Albanese 态射的光滑射影簇满足 Gromov--Yomdin 性质，通过轨道逃逸原理和有限平展覆盖归约，并构造正范畴熵自等价，其存在性等价于非一般型。

关于 $\mathbb{P}$-twists 的熵

On entropy of $\mathbb{P}$-twists

本文通过计算范畴熵函数，证明光滑射影簇上 $\mathbb{P}$-对象诱导的 $\mathbb{P}$-twist 不共轭于标准自等价，并在附加条件下确定其与 spherical twist 的范畴多项式熵。

阿贝尔簇的范畴动力学

Categorical Dynamics of Abelian Varieties

本文研究了阿贝尔簇导出范畴自等价的动力学，证明了 Gromov--Yomdin 等式、熵上界、平移数刻画及约化平移数有限性条件，并否证了范畴熵函数的代数性问题。

On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

In topological dynamics, the Gromov--Yomdin theorem states that the topological entropy of a holomorphic automorphism $f$ of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map $f^*$. In order to establish a categorical analogue of the Gromov--Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov--Yomdin type theorem holds for them.

A note on entropy of auto-equivalences: lower bound and the case of orbifold projective lines

Entropy of categorical dynamics is defined by Dmitrov-Haiden-Katzarkov-Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov-Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

Wreath Macdonald polynomials and categorical McKay correspondence

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of ${\mathbb A}^{2n}$ by the symmetric group $S_n$. A short proof of a similar derived …

Geometric Complexity Theory V: Efficient algorithms for Noether Normalization

We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show that: (1) The categorical quotient for any finite dimensio…

Entropy of the composition of two spherical twists

Given a categorical dynamical system, i.e. a triangulated category together with an endofunctor, one can try to understand the complexity of the system by computing the entropy of the endofunctor. Computing the entropy of the composition of two endofunctors is hard, and in general the result doesn't have to be related to the entropy of the single pieces. In this paper we compute the entropy of the composition of two spherical twists around spherical objects, showing that it depends on the dimension of the graded vector space of morphisms between them. As a consequence of these computations we …

Categorical Complexity

We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest, such as finite sets, Boolean functions, topological spaces, vector spaces, semi-linear and semi-algebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings clas…

Hochschild-Serre 代数的群作用与范畴重构

Group action of Hochschild-Serre algebra and categorical reconstruction

本文研究 Serre 函子在 Hochschild-Serre 代数上的作用，并应用于证明加权射影空间中超曲面的 Kuznetsov 分量及孤立奇点拟齐次多项式的范畴 Torelli 定理，实现范畴重构。

Symmetries of categorical representations and the quantum Ngô action

We observe that all classical Hamiltonian systems coming from the invariant polynomials on a reductive Lie algebra g can be integrated in a universal way. This is a consequence of Ngô's action of the group scheme J of regular centralizers in G on all centralizers: the Hamiltonian flows associated to invariant polynomials integrate to an action of J as commutative symplectic groupoid. We quantize the Ngô action, providing a universal integration for all quantum Hamiltonian systems coming from the center Z=Z(Ug) of the enveloping algebra. Namely we extend Kostant's Whittaker description of Z to …

Entropy of radical ideal of a tropical prevariety

The entropy of a tropical ideal is introduced. The radical of a tropical ideal consists of all tropical polynomials vanishing on the tropical prevariety determined by the ideal. We prove that the entropy of the radical of a tropical bivariate polynomial with vanishing coefficients equals zero. Also we prove that the entropy of a zero-dimensional tropical prevariety vanishes. An example of a non-radical tropical ideal having a positive entropy is exhibited.

Signomial and Polynomial Optimization via Relative Entropy and Partial Dualization

We describe a generalization of the Sums-of-AM/GM Exponential (SAGE) relaxation methodology for obtaining bounds on constrained signomial and polynomial optimization problems. Our approach leverages the fact that relative entropy based SAGE certificates conveniently and transparently blend with convex duality, in a manner that Sums-of-Squares certificates do not. This more general approach not only retains key properties of ordinary SAGE relaxations (e.g. sparsity preservation), but also inspires a novel perspective-based method of solution recovery. We illustrate the utility of our methodolog…