Алексей Бондал, "Pure mathematics in foundations of quantum information theory"

26.11.2025, IPMU Colloquium, Kavli IPMU, Япония.

Heisenberg indeterminacy principle is based on complementarity of coordinate and impulse. Complementary observables in a finite dimensional Hilbert space lead to the notion of Mutually Unbiased Bases (MUB). The classification of MUB attracts a lot of attention in Quantum Information Theory, because they are ubiquitous in Quantum Computers, Quantum Tomography, Quantum Criptography, etc. Still the problem is widely open. The proof of the existence a 4-dimensional family of MUB’s in dimension 6 (joint work with I. Zhdanovskiy) is heavily based on Algebraic Geometry.

We will discuss the relation of the MUB problem with the classification of orthogonal decompositions of Lie algebra sl(n,C) and with Winnie the Pooh problem. We will give an interpretation in terms of representation theory of Temperley-Lieb like algebras of graphs, and outline how this leads to a noncommutative version of perverse sheaves.

We describe the set of MUB’s as the critical locus of a potential, a Laurent polynomial. The Symplectic Geometry behind this description and Mirror Symmetry of the potential will be discussed.

Алексей Бондал, "Категорификация некоммутативных деформаций"

11.08.2025, V Конференция математических центров России, Красноярск, Россия.

Будет рассказано как определять деформации объектов в абелевой категории над некоммутативной базой. В качестве обобщения, будут определены некоммутативные деформации функторов между абелевыми категориями. Обсудим геометрический пример флоппируемых стягиваний.

Alexey Bondal, "Categorical and geometric approach to matrix multiplication complexity"

15.07.2025, International Congress of Basic Science 2025 (ICBS), BIMSA, Пекин, Китай.

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It is known after V. Strassen that the multiplication of 2x2 matrices requires 7 products of numbers, not 8 as one gets from the standard algorithm of matrix multiplication. The minimal number of products necessary for multiplication of bigger size matrices is an open problem. I will describe a new categorical approach to this and similar problems. This allows us to use the techniques of triangulated categories and the geometry of coherent sheaves in the theory of computation complexity.

Alexey Bondal, "Mirror symmetry for minuscule varieties"

06.05.2025, Enumerative and Derived Geometry in Mirror Symmetry Conference, Wuhan University, Wuhan, China.

We will discuss the minuscule varieties, the orbits of highest vectors in minuscule representations of complex reductive Lie groups. Then we will describe the minuscule descent, a procedure of transfer from minuscule variety to a minuscule variety for another reductive group, which has many appearance in mathematics and physics. Then we describe toric degenerations and mirror symmetry for minuscule varieties. This is a joint work with S. Galkin.

Алексей Бондал, "Многообразия вырождений скобок Пуассона"

21.04.2025, семинар "Гамильтоновы системы и статистическая механика", МИАН, Москва.

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Скобки Пуассона на алгебраических многообразиях можно воспринимать как касательные вектора к некоммутативным деформациям этих многообразий. Гипотеза докладчика утверждает, что на многообразиях Фано (например, на проективных пространствах) локусы вырождений скобок Пуассона имеют большую размерность, например, что локус нулей скобки Пуассона - это, по меньшей мере, кривая. Гипотеза доказана для многообразий размерности до 4-х, но в высшей размерности остаётся открытой более 30 лет. Будет рассказано про подход к гипотезе с помощью описания структуры специальных слоений на кэлеровых многообразиях, а так же про свежие результаты, где доказывается, что локус нулей скобки Пуассона на многообразии Фано не пустой в размерности вплоть до 7-ми.

Alexey Bondal, "Categorifying noncommutative deformations"

18.03.2025, The University of Hong Kong, Institute of Mathematical Research Department of Mathematics Colloquium, Гонконг.

We will discuss noncommutative formal deformations of finite families of objects in abelian category. The main feature is that the moduli space of such deformations is naturally noncommutative. Moreover, we will present this moduli space as an abelian category. As a geometric application of the theory, the case of flopping contractions will be described. We will also discuss the generalization of the formalism to the framework of noncommutative deformations of functors between abelian categories.

Alexey Bondal, "Dolbeault superconnections on complex-analytic manifolds"

13.03.2025, School of Mathematics and Statistics, Wuhan University, Ухань, Китай.

We will discuss various versions of the derived category of coherent sheaves on compact complex-analytic manifolds and dg-enhancement of the suitable version of the category via Dolbeault superconnections. This enhancement allows us to define Chern and Bott-Chern classes of objects of the category, in particular of coherent sheaves. Also it allows us to construct the complex-analytic moduli spaces of objects in the category. The theory is generalized to non-compact complex-analytic manifolds, where one should consider the category of restricted objects of the derived category. We will analyze when this notion is homological.

Alexey Bondal, "Poisson brackets and non-commutative deformations of algebraic varieties"

16.10.2024, Poisson Geometry and Artin-Schelter Regular Algebras (workshop), IASM, Ханчжоу, Китай.

I will survey some aspects of the theory of holomorphic Poisson brackets on projective algebraic varieties and related noncommutative deformations. In particular, some conjectures about these objects will be discussed.

Alexey Bondal, "Derived approach to highest weight categories" (Пленарный доклад)

19.07.2024, International Congress of Basic Science 2024 (ICBS), BIMSA, Пекин, Китай.

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Highest weight categories (hw categories) are an abstract version of categories of perverse sheaves on stratified topological spaces with contractible strata. They are popular in Representation Theory, because many categories of representations of various origin are hw categories. I will discuss how one can interpret hw categories and Ringel duality for them in terms of exact categories and their abelian envelopes and in terms of exceptional collections in derived categories. Hw categories have interesting application to Birational Geometry, where they show up as the abelian null-categories of birational morphisms. The techniques of hw categories allow us to identify the derived support of the null-category for any smooth surface contraction as the discrepancy divisor. Conjecturally this holds in higher dimensions too.

Алексей Бондал, "Гипотезы на стыке алгебраической и дифференциальной геометрий"

28.07.2024, XXIII Летняя школа "Современная математика" имени Виталия Арнольда, Дубна, Россия.

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Основная цель лекции — познакомить слушателей с интересными гипотезами о фундаментальных свойствах некоторых дифференциально-геометрических объектов на алгебраических многообразиях. Это гипотеза Зарисского—Липмана о локальной (не)свободе касательного пучка на особом алгебраическом многообразии, гипотеза Бергера о кручении пучка дифференциальных форм на кривой, и моя гипотеза о размерности локусов вырождений скобки Пуассона на многообразиях Фано. Первые две гипотезы имеют локальный характер, а последняя — глобальная и самая молодая, но и ей уже 35 лет.

Алексей Бондал, "Гипотеза Винни-Пуха: развитие сюжета."

26.02.2024, Семинар кафедры Высшей алгебры МГУ, посвященный 95-летию со дня рождения Алексея Ивановича Кострикина, Москва, Россия.

Alexey Bondal, "Nc resolutions of finite dimensional commutative algebras via smooth surface blow-ups"

21.12.2023, McKay correspondence, Tilting theory and related topics, Kavli IPMU, Кашива, Япония.

I will explain how to get noncommutative resolutions of some finite dimensional commutative algebras via birational morphisms of smooth surfaces. The resolution is given by the null category which happens to be highest weight category, whose projective generator is the discrepancy sheaf.

Alexey Bondal, "Derived categories of complex-analytic manifolds"

10.11.2023, Китайско-российская междисциплинарная математическая конференция, Пекинский университет, Пекин, Китай.

We discuss how to define Chern and Bott-Chern classes of coherent sheaves on smooth compact complex-analytic manifolds. To this end we consider the derived categories of dbar-superconnections and state the theorem that the homotopy category of those is equivalent to the derived category of complexes of sheaves with coherent cohomology. We outline how this approach allows to look algebraically on the problem of describing complex-analytic manifolds which are not algebraic varieties.

Alexey Bondal, "Commutative and noncommutative resolutions of singularities"

06.11.2023, ZJU School Of Mathematical Sciences Colloquium, Чжэцзянский университет, Ханчжоу, Китай.

Singularity Theory is considered an important tool in application of geometry to various problems in physics: Singularities of complex and algebraic varieties have been recently studied by a combination of geometric and algebraic methods. The structure of a given singularity can be described by means of its classical commutative resolution. Relatively recently, noncommutative resolutions were proposed and intensively studied. We discuss recent results and conjectures about the derived categories of singularities and their commutative and noncommutative resolutions. These include the null categories of the resolutions, spherical functors, flop-flop functors and Knörrer periodicity for noncommutative Auslander resolutions.

Alexey Bondal, "Complex geometry via twist closed enhancements"

31.08.2023, Sino-Russian Hybrid Conference “Geometry and Physics" , Москва, Россия.

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Alexey Bondal, "Complex geometry via derived categories"

31.07.2023, MIST: A satellite conference to the ICBS, by The Chinese University of Hong Kong, Гонконг, Китай.

Alexey Bondal, "The null-categories of noncommutative resolutions" (пленарный доклад)

21.07.2023, The first International Congress of Basic Science (ICBS) , Пекин, Китай.

Various constructions for noncommutative resolutions and their properties will be discussed. In particular, the Knoerrer periodicity for the null-categories of Auslander resolutions will be explained

Vyacheslav Shokurov, "Complements" (пленарный доклад)

21.07.2023, The first International Congress of Basic Science (ICBS) , Пекин, Китай.

A survey of theory of complements with applications will be presented.

Vyacheslav Shokurov, "Boundedness vs optimal bounds."

14.07.2023, ICBS Satellite Conference on Algebraic and Arithmetic Geometry , Пекин, Китай.

We discuss recently obtained optimal bounds (joint with Jihao Liu) on certain threshold for algebraic surfaces.

Alexey Bondal, "Reconstruction of a normal surface from the category of reflexive sheaves."

13.07.2023, ICBS Satellite Conference on Algebraic and Arithmetic Geometry , Пекин, Китай.

I will describe how to recover the most complete (codim-2 saturated) algebraic surface from the additive category of reflexive sheaves. This is a joint work with A. Bodzenta.

Alexey Bondal, "Некоммутативные деформации"

07.07.2023, Летняя школа "Некоммутативная и производная геометрия матричных факторизаций", Центр Фундаментальной Математики МФТИ

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А. И. Бондал, "Как перемножать ассоциативные алгебры и что из этого можно получить"

01.07.2023, Вторая летняя студенческая математическая школа «Алгебра и теория чисел-2023» , Учебный Центр НИУ ВШЭ в Вороново

А. И. Бондал, "Некоммутативные резольвенты особенностей и их нуль-категории"

05.06.2023, Конференция по алгебре, алгебраической геометрии и теории чисел, посвященная 100-летию И.Р. Шафаревича, Москва, Россия.

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Alexey Bondal, "Некоммутативные резольвенты специального типа" (in Russian)

28.05.2023, Mini-workshop on Algebraic Geometry , Saint Petersburg, Russia.

Video recording.