Rankin-Cohen algebras and \(\mathfrak{sl}_2\)-algebras
Speaker – Don Zagier
Talk mainly about two algebraic structures and the relations between them. The first is that of a Rankin‑Cohen algebra, which is a graded vector space with infinitely many bilinear operations satisfying the same algebraic identities as those satisfied by the Rankin‑Cohen brackets in the theory of modular forms. The second is that of \(\mathfrak{sl}_2\)‑algebras, meaning an algebra on which the 3‑dimensional Lie algebra \(\mathfrak{sl}_2\) acts by derivations. It turns out that there is a natural isomorphism (actually two, depending on the precise conditions that we impose) between the category of Rankin‑Cohen algebras and the category of commutative and associative \(\mathfrak{sl}_2\)-algebras, both motivated by ideas coming from elliptic modular forms and Siegel modular forms. There is also a non‑commutative version based on work done many years ago with Yuri Manin and which led to the construction of infinitely many associative multiplications on algebras of modular forms. Finally, although, there is an analogous story relating conformal algebras (a variant of the better known notions of vertex algebras and vertex operator algebras from conformal field theory) to \(\mathfrak{sl}_2\)-algebras, but this time to \(\mathfrak{sl}_2\)-Lie algebras rather than to associative ones.
Sheared Witt vectors and Barsotti‑Tate groups
Speaker – Vladimir Drinfeld
I will discuss a conjectural description of the \(p\)-adic completion of the stack of Barsotti‑Tate groups (a.k.a. \(p\)-divisible groups). The description is in the spirit of the classical Dieudonné theory, but the ring scheme of Witt vectors is replaced by a certain ring space, which is called the space of sheared Witt vectors. In some sense, the ring space and the conjectural description go back to the works of Thomas Zink.
Algebraic flat connections and o‑minimality
Speaker – Hélène Esnault
We prove that an algebraic flat connection has \( \mathbb{R}_{\text{an,exp}}\)-definable flat sections if and only if it is regular singular with unitary monodromy eigenvalues at infinity, refining previous work of Bakker–Mullane. This provides e.g. an o‑minimal characterization of classical properties of the Gauss–Manin connection (joint work with Moritz Kerz).
Semi‑infinite Hodge structure and primitive forms for hyperbolic root systems of rank 2
Speaker – Kyoji Saito
Semi‑infinite Hodge structure equipped with primitive forms is constructed for hyperbolic root systems of rank 2. As the consequences, we determine (1) the flat structure and the Frobenius manifold structure and (2) the “virtual” period maps for the primitive forms.
Hyperbolic root systems are the first case of root systems which do not have geometric origin as vanishing cycles (e.g. the monodromy actions are not quasi-unipotent and, hence, the Coxeter number and the exponents are pure imaginary numbers), several new phenomena appear. First, the primitive derivation, whose inverse action determines the Hodge filtration and which itself gives the unit element in the Frobenius algebra, needs to be defined anew by using an equation on the polarization of the Hodge structure and solved by a suitable hypergeometric equation.
After fixing these structures, the constructions of the semi-infinite Hodge structure, primitive forms and the virtual period maps are achieved parallel to the classical period map theory except that the defining domain of the period map is no-longer the Frobenius manifold of the flat coordinates but a wrapping quotient space of its universal covering space. Finally, we describe the inversion maps to the virtual period maps, answering to the classical inversion problem, in case of virtual CY-dimension is equal to 2.
Hodge theory for non‑Archimedean analytic spaces
Speaker – Vladimir Berkovich
By Deligne’s Hodge theory, the integral cohomology groups \( H^n(\mathcal{X}^h,\mathbf{Z}) \) of the \(\mathbf{C}\)-analytification \( \mathcal{X}^h \) of a separated scheme \( \mathcal{X} \) of finite type over \(\mathbf{C}\) are provided with a mixed Hodge structure, functorial in \(\mathcal{X}\).
Given a non-Archimedean field \(K\) isomorphic to the field of Laurent power series \(\mathbf{C}((z))\), there is a functor \( \mathcal{X} \mapsto \mathcal{X}_K^{\text{an}}\) that takes \(\mathcal{X} \) to the non-Archimedean \(K\)-analytification of \( \mathcal{X}K = \mathcal{X} \otimes_{\mathbf{C}} K \). I'll talk about the extension of Deligne's Hodge theory via the above functor to a full subcategory of the category of \(K\)-analytic spaces that contains the \(K\)-analytifications of schemes of finite type over \(K\), proper \(K\)-analytic spaces, and the analytic subdomains of both.
Adelic Percolation
Speaker – Matilde Marcolli
Models of long‑range percolations on lattices and on hierarchical lattices appear at first to represent very different random geometries. However, both can be reduced to building blocks of a similar nature through an adelic perspective suggested by Manin’s “reflections on arithmetical physics”.
Indeed these two types of percolation models can be related through the use of three intermediate geometries: a 1-parameter deformation based on the power mean function, relating lattice percolation to a percolation model governed by the toric volume form; the adelic product formula for a function field, relating the hierarchical lattice model to an adelic percolation model; and the adelic product formula for a number field that relates the toric percolation model on the lattice given by its ring of integers in the Minkowski embedding to another adelic percolation model.
Q‑Zeta and Elliptic Hall Polynomials
Speaker – Ivan Cherednik
The fundamental property of zeta functions and \(L\)-functions is that their meromorphic continuations provide a lot of information about the corresponding objects. Complex values of \(s\) occur as a technical tool, with little arithmetic‑geometric meaning. In the refined theory, \(1/n^s\) are replaced by certain \(q,t,a\)-series, which are invariants of Lens Spaces \(L(n,1)\) directly related to Elliptic Hall Polynomials.
Superduality is one of their key features: under \( q \leftrightarrow 1 / t, a \mapsto a \). As \( t \to 0\), they become Rogers-Ramanujan series (certain string functions and those from the Nahm Conjecture), the limit to Hall Polynomials is \(q \to 0\) etc. We will begin with the \(q\)-deformation of Riemann's zeta: the case of \(A_1\) when \(t = q^{s - 1/2}, a = t^2\).
Jacobians and intermediate Jacobians with additional symmetries
Speaker – Yuriy Zarkhin
We study principally polarized complex abelian varieties \((X,\lambda)\) of positive dimension \(g\) that admit an automorphism \(\delta\) of prime order \(p>2\), whose set of fixed points \(X^{\delta}\) is finite. Such triples \((X,\lambda,\delta)\) exist if and only if \((p-1)\) divides \(2g\).
By functorality, \(\delta\) acts on the \(g\)-dimensional complex vector space \(\Omega^1 (X)\) of dfferentials of the first kind on \(X\) as a diagonalizable linear operator, whose spectrum (the set of eigenvalues) consists of primitive \(p\)th roots of unity with certain multiplicities. Let \(\mathbf{a}_X\) be the corresponding multiplicity function on the set \( \mu^*_p\) of all primitive \(p\)th roots of unity. (In particular \(\mathbf{a}_X(\zeta) = 0\) if and only if \(\zeta\) is not an eigenvalue of \(\delta : \Omega^1(X) \rightarrow \Omega^1(X) \).) It is known that \(\mathbf{a}_X\) is well rounded, i. e., \[ \mathbf{a}_X(\zeta) + \mathbf{a}_X(1/\zeta) = 2g / (p-1) \quad \] \[\forall \zeta \in \mu^*_p\] We describe explicitly all functions on \(\mu^*_p\) that can be realized as the multiplicity functions in the case when \((X, \lambda)\) are canonically polarized jacobians of smooth projective curves of genus \(g\). It turns out that not all well rounded nonnegative integer-valued functions occur as multiplicity functions that arise from jacobians. As an application, we sketch another proof of the (already known) fact that intermediate jacobians of certain cubic threefolds are not isomorphic (as principally polarized abelian varieties) to jacobians of curves. As another application, we prove that certain Prym varieties are not isomorphic to jacobians of curves.
The Brauer group of an abelian variety
Speaker – Alexei Skorobogatov
The importance of the Brauer group for arithmetic geometry was highlighted by Manin in his celebrated 1970 ICM address. In this talk I will discuss the structure of the Brauer group of an abelian variety \(A\) over an algebraically closed field of characteristic \(p\) focusing on the \(p\)-primary torsion, the key part of which is a certain quasi-algebraic unipotent group \(U_A\). I will present results on the dimension and the \(p\)-exponent of \(U_A\) based on the classical Manin-Dieudonné theory, leading to the determination of \(U_A\) up to isogeny for abelian varieties \(A\) of small dimension. This is joint work with Livia Grammatica and Yuan Yang.
Bruhat operads
Speaker – Vadim Schechtman
Higher Bruhat orders have been introduced in our paper with Manin around 1986. They generalize the weak Bruhat orders on the symmetric groups. In this talk, which presents a joint work with Gleb Koshevoy, I will sketch a construction which allows to build some planar operads from higher Bruhat orders. Furthermore I will explain that these operads are operads with multiplication in the sense of McClure‑Smith. This implies that they give rise to some complexes having all formal properties of the Hochschild complex for an associative algebra.
Unsuspected depth of the First Concepts (not streamed via Zoom)
Speaker – Mariusz Wodzicki
We treat the First Concepts that form the foundation of modern Mathematics as a set of primitives devoid of content and carrying little interest on their own. During the century since they were postulated Mathematics witnessed momentous development that was often dramatically changing the landscape of whole areas. One thing that remains constant is the First Concepts. Unsuspected depth of the First Concepts is what I will be talking about.
Effective integration of Lie type algebras
Speaker – Bruno Vallette
I will cover the integration theory with effective formulas of algebraic structures stronger or weaker (higher) than Lie algebras. This will be the occasion to survey some of the recent developments of the operadic calculus and to present solutions to open problems in Algebra, Geometry, and Topology.
Mysterious Triality
Speaker – Alexander Voronov
The striking connection between del Pezzo surfaces and the exceptional \(E\)series of root systems was uncovered by Manin in his seminal 1972 monograph Cubic Forms. There, he identified these root systems within the Picard groups of del Pezzo surfaces, leveraging this structure to systematically investigate their geometry and arithmetic. Decades later, in 2001, the mathematical physicists Iqbal, Neitzke, and Vafa observed an analogous appearance of \(E\)-type root systems in the study of \(\frac{1}{2}\)-BPS branes in M-theory compactifications, such as type IIA string theory.At the time, the link to del Pezzo surfaces remained obscure, prompting them to dub this correspondence Mysterious Duality .
Twenty years on, Hisham Sati and I discovered a similar root system pattern in the context of toroidifications \( \mathcal{T}^k S^4 := \text{Map}(T^k, S^4) // T^k\) of the four-sphere \(S^4\) for \( 0 \le k \le 8\), where \(T^4 = (S^1)^k\) is the \(k\)-torus.
We also showed that these spaces serve as universal target spaces for compactified M-theory and as classifying spaces for supergravity fields. This reveals a new duality---no less enigmatic---between del Pezzo surfaces and toroidifications of \(S^4\), culminating in Mysterious Triality, which intertwines Geometry, Topology, and Physics.
In this talk, I will present our findings on the root systems associated with toroidifications and describe an action of a maximal parabolic subgroup of a Lie group \(G_k\) of type \(E_k\) via rational self-equivalences of \mathcal{T}^k S^4. This action echoes Serganova and Skorobogatov's work linking del Pezzo surfaces to homogeneous spaces \(G_k / P_k\).
Gauss‑Manin connection and noncommutative calculus, revisited
Speaker – Boris Tsygan
We review and revise the constructions of noncommutative calculus and of the Gauss‑Manin superconnection in noncommutative geometry. In particular, we provide explicit formulas that are intriguing in various respects. Namely, they seem to have good convergence properties, both \(p\)-adically and in Archimedean metrics; and they look very much like something from microlocal analysis and mathematical physics (where the role of the Planck constant is played by the formal parameter \(u\)$ from cyclic homology theory).
Geometric aspects of representation theory of supergroups
Speaker – Vera Serganova
The representation theory of Lie superalgebras began in the end of 1970‑s following Kac’s classification of simple Lie superalgebras. In the early 1980‑s, Manin and his students Penkov, Skornyakov and Voronov studied geometry of flag supervarieties $G/B$. In particular, several important results towards super Borel–Weil–Bott theorem were obtained by Penkov. A natural next step would be to generalize the Beilinson–Bernstein localization theorem. However, a satisfactory approach is still unknown.
Even the cohomology groups of an arbitrary \(G\)-equivariant line bundle on \(G/B\) remain largely unknown for a classical supergroup \(G\).
In this talk I will focus on the category \(\text{Rep}G\) of representations of an algebraic supergroup \(G\)with reductive underlying group \(G_0\). Although this category is not as complicated as the category \(\mathcal{O}\), it seems to capture the essential difficulties in passing from \(G_0\) to \(G\)and is still challenging to study.
The tensor category \(\text{Rep}G\) is Frobenius. By this reason one can apply certain techniques developed in the modular representation theory of finite groups. We generalize the notions of \(p\)-subgroups and Sylow subgroups, as well as support and rank varieties.
We also formulate a conjecture about the Balmer spectrum of the stable category \(\text{St } G\), verified in several cases.
Finally, we discuss new results on homogeneous supervarieties and odd vector fields, which play a crucial role in this approach.
Refining Weil groups
Speaker – Dustin Clausen
The theory of Galois cohomology of number fields is very beautiful and powerful, but there are indications coming from several directions that it is not quite “correct”. This is partly fixed by passing to Weil groups, but even there problems remain. I will describe how to go “beyond'' Weil groups to get a more satisfactory theory.
Equivariant birational geometry
Speaker - Yuri Tschinkel
I will discuss new results and constructions in equivariant birational geometry.
Some questions about singular support of étale sheaves
Speaker – Alexander Beilinson
I will discuss some as yet unsolved problems in the theory of singular support of étale sheaves on varieties in finite characteristic.