Speakers & Abstracts

Understanding asymptotic complex geometry close to the boundary of moduli spaces requires the development of a hybrid multi-scale geometry that combines both complex and tropical geometries. The framework in dimension one has been now largely developed in the theory of hybrid curves and their moduli spaces. It allows to provide a fairly complete description of the asymptotic geometry of families of Riemann surfaces. Hybrid curves are multi-scale geometric objects with an algebraic geometry analogous to the one for algebraic curves; classical theorems such as Riemann-Roch, Abel-Jacobi, Poincaré-Lelong hold in the hybrid setting. Their moduli space provides a new compactification of the moduli space of curves, and several geometric and analytic constructions on Riemann surfaces extend continuously to the boundary of this compactification. This picture has been extended partially to higher dimensions, for families of higher dimensional varieties. The asymptotic results often hold in a purely topological setting, meaning that the analyticity assumption on the family can be replaced by a weaker topological condition. In this talk, I will provide an introduction to this multi-scale geometry highlighting its connections to combinatorics and Hodge theory, and present some applications in complex geometry and combinatorics.

This talk concerns Connes and Consani’s Gamma-set approach to absolute algebra and geometry. I will first review the theory and describe collaborative work with Nakamura showing that classical axiomatic projective geometries can be embedded fully faithfully in Gamma-sets. In particular, the group of collineations of an axiomatic projective geometry can be recovered as the automorphism group of its associated Gamma-set. Then I will describe preliminary work in developing a derived theory of absolute algebra.

In this talk I will survey some recent results around generalizations of the Beilinson-Bloch-Esnault-Kerz fiber square, relating the algebraic K-theory of a scheme to its topological cyclic homology. In particular, I will discuss how log geometry naturally enters the picture in the case of bad reduction, and how one can control the obstruction to lifting K-theory classes in the semistable case using the "Hyodo-Kato Chern character", extending the work of Antieau-Mathew-Morrow-Nikolaus. The methods involve some reduction from the logarithmic to the non logarithmic case, using some results of motivic origin in the non-A^1-invariant setting.

I’ll report on scheme and stack theory over commutative semirings, especially over the natural numbers, the non-negative reals, and the Boolean numbers. I'll show that the language of scheme theory, in its standard category-theoretic form, works perfectly well here and that it gives a formally satisfying way of bringing positivity into the foundations of algebraic geometry, much as textbook scheme theory did with p-adic integrality in the 1950s. I'll touch on the following specific topics: line bundles and vector bundles, the fpqc topology, the narrow class group, the Picard stack and its algebraicity, positive models of moduli spaces, and the "total" Grassmannian. I'll also mention open questions, many of which appear to be tractable. This is mostly joint work with Jaiung Jun, but also with Robert Culling, Darij Grinberg, A. Johan de Jong, and Ivan Zelich.

Let k be a finite field, and X a normal geometrically connected variety over k. One says that a bounded complex of l-adic constructible sheaves (l different from the characteristic of k) is algebraic if its local L-factors have algebraic coefficients and one says that two bounded complexes of l-adic and l'-adic constructible sheaves are compatible if they are algebraic and their local L-factors coincide. One expects two such complexes to be the incarnation of the same motive and, in particular, to satisfy stronger compatibility conditions. For instance, if X is smooth over k and one restricts to lisse sheaves, a celebrated theorem Serre, Chin and Drinfeld asserts that the neutral component and group of connected components of the monodromy depend only on its class of compatibility. If X is an abelian variety over k, one can also associate to a perverse sheaf on X a monodromy group which, as in the case of l-adic sheaf, if an algebraic group over the algebraic closure of Q_l. We will explain that, in this setting, again, the neutral component and group of connected components of the monodromy depends only on the class of compatibility of the perverse sheaf. Our argument follows the strategy of Drinfeld, relying on the companion conjecture, a reconstruction theorem of Kazhdan, Larsen and Varshavsky and an explicit description of the group of connected components.

Grothendieck conjectured that, for a field K finitely generated over ℚ, there exists a class of "anabelian varieties": varieties whose étale fundamental groups over K are rich enough to completely determine their geometry. He expected that the smooth anabelian varieties of dimension one should coincide with hyperbolic curves, and that "elementary anabelian varieties", varieties built from hyperbolic curves by smooth proper fibrations, should also be anabelian. The fact that hyperbolic curves are anabelian was subsequently proven by Mochizuki. In this talk, we explain our proof that elementary anabelian varieties are anabelian, and speculate about which further higher-dimensional varieties might be anabelian.

In this talk, I will describe recent joint work with Katia Consani on a new geometric structure (Spec Z)_F1, based on our understanding of F1-geometry in terms of Segal's Gamma-rings. The first step is to recognize that the adele class space, which provides the spectral realization of the zeros of the Riemann zeta function, is in fact the Picard monoid of the compactified arithmetic curve. This gives a generalization of the Abel-Jacobi map as a direct relation between Spec Z and the arithmetic site,  endowed with a structure sheaf over F1. Pulling back this structure sheaf defines the absolute geometry (Spec Z)_F1. Evaluating the points of this geometry over an algebraically closed perfectoid field of residue characteristic p provides a rigorous geometric realization of P. Scholze's foundational paradigm from absolute geometry, allowing us to canonically reconstruct the Fargues--Fontaine curve.  Evaluating the points of this geometry over the field of complex numbers produces at each prime p the Tate elliptic curve of modulus 1/p, and its canonical splitting as the product of the adelic periodic orbit C_p by the p-independent analogue of the FF-curve.

This is joint work with Lothar Goettsche. For any proper complex scheme S, there is a natural class in A_0(S) whose degree is the Euler characteristic of the analytification of S; it is the dimension zero part of the Chern-Schwarz-MacPherson class c_SM(S) of S. When S is smooth, then c_SM(S)=c(TS)∩[S]. Let X be a smooth complex projective variety. We compute the generating function for the Chern-Schwarz-Macpherson classes of the symmetric powers of X. We extend the result to the pushforward to the symmetric powers of the CSM classes of the Quot scheme of points Quot^n(E) of a vector bundle E on X. When dim(X)=3 we propose a refined version of the dimension zero MNOP formula for Quot that computes the pushforward of the virtual fundamental class of Quot^n(E) under the Quot to Chow morphism, and prove it in some special case. In a future paper we will lift this refined formula to the Chow ring of Quot.

The usual Witt vector construction can be understood as arising via isogenies of the multiplicative group \mathbb{G}_m. It is natural to ask whether other kinds of isogenies give rise to analogous Witt vector theories. The answer is yes: C. Rezk constructed such a theory for height-h p-divisible groups. In this talk, I will describe the case of isogenies of elliptic curves, the resulting elliptic Witt vectors, and elliptic \delta-rings. This is joint work with J. Borger.

The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate signed enumeration of real algebraic curves of genus 1 and 2. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the genus zero case. We also suggest an extension of the invariants under consideration to a non-toric setting. This is a joint work with Eugenii Shustin.

Developing a suitable notion of vector bundles in tropical geometry has recently attracted considerable interest. In this talk, I will provide an overview of several approaches, focusing in particular on the current approach via the combinatorial structures known as matroids. If time permits, I will also discuss how categorical tools can be used to study tropical vector bundles defined in terms of matroids. This is joint work with Alex Sistko and Cameron Wright.

Degeneration is a basic strategy in algebraic geometry: a smooth variety is degenerated to a more tractable simple normal crossing scheme, and you hope that the structure you care about survives the voyage. For coherent sheaves, together with their moduli spaces and derived categories, the journey is treacherous. This talk introduces the category of logarithmic coherent sheaves which steadies the ship: it offers moduli spaces with good properties parallel to the smooth case and a derived category with perfect diagonal (the categorical signature of smoothness). A theme is that tropical geometry provides the bridge which lets us work with the theory: after crossing it, every construction and computation involving log coherent sheaves becomes a conventional algebro-geometric one. Based on joint work with Dell, Hu, Manali Rahul, and Schimpf.

In his Esquisse d’un programme, Grothendieck called for the development of a “tame” topology, free from the pathologies of general topology. Such a framework was developed in the 1980s and 1990s by logicians under the name o-minimal geometry, based on the simple principle that, in dimension one, the only sets considered are finite unions of intervals. Remarkably, this setting has recently led to striking applications in complex algebraic geometry, particularly in Hodge theory, as well as in number theory, notably in Diophantine geometry. This talk will offer an elementary introduction to these ideas.

Around a hundred years ago Veblen and Young proved what is now known as the Fundamental Theorem of Projective Geometry. One way to interpret this theorem is as a topological characterization of linear geometry. I will report on a new non-linear generalization of this result: a topological characterization of algebraic geometry. This is joint work with Kollár Olsson and Sawin.

Two cubic fourfolds are Fourier Mukai partners if there is an equivalence between their Kuznetsov components. It has been suggested by Huybrechts that a pair of Fourier Mukai cubic fourfold partners are birational, however examples of such phenomena are rare. In an attempt to investigate this phenomena, we develop an algorithm for counting the number of Fourier Mukai partners of a given cubic fourfold, and run it on cubics with automorphisms. We prove that a cubic fourfold admitting a symplectic involution has 1120 non-trivial Fourier-Mukai partners, all birational but non-isomorphic. We’ll explain the 1120 in terms of root systems and non-syzygetic pairs of cubic scrolls.

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. Developing algebraic foundations of tropical geometry will allow us to explore the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we will describe the `coordinate semiring' of a tropical variety, what information it carries, its integral closure, and some examples of normalization in tropical geometry.

In this talk, we will review the general properties of Irreducible Holomorphic Symplectic manifolds and we will focus on their dreamlike birational geometry, with special attention also to singular symplectic varieties.

The differential equation underlying the operator named in the title goes back to classical mathematical physics, where it arose through separation of variables in prolate spheroidal coordinates for the wave (or Helmholtz) equation. The operator gained considerable prominence in the 1960s through the fundamental work of Slepian, Landau, and Pollak on signal concentration theory, which gave an astonishing second life to the operator and its associated prolate wave functions. A third incarnation of the prolate wave ensemble emerged in the late 1990s through a cutoff mechanism in the trace-formula framework developed by Connes in noncommutative geometry, recovering the Riemann-Weil explicit formula in number theory. The same ensemble later played a significant role in the Connes-Consani refinement (2021) of Weil positivity at the Archimedean place. New eigenfunctions for the extension of the prolate operator to the entire real line were discovered by Connes and myself (2022), revealing a striking spectral analogy with the zeros of the Riemann zeta function. Building on recent work of Ramis, Richard-Jung, and Thomann (2025) concerning the behavior of the prolate wave equation in the complex plane, this talk aims to uncover some of the hidden geometric structure underlying - and perhaps reinforcing the mysterious bond between the prolate wave operator and Riemann's zeta function.

Spaces of geometric objects associated with a variety, such as points, maps, or sheaves, often admit several natural compactifications. I will discuss a common structure underlying the compactification problems for these moduli spaces, and explain some fundamental consequences for their geometries. Examples will include Hilbert schemes of points, Hurwitz spaces of covers, and Gieseker spaces of stable sheaves.

Geometric structures are typically encoded by suitable sheaves, which satisfy the condition that sections over a cover can be reconstructed by appropriately gluing sections defined on the pieces of the cover. Such gluing conditions can be expressed via the universal property of a limit, and their precise form depends on the “complexity” of the target category of the sheaf. In particular, for presheaves valued in a higher category, an appropriate notion of limit is required. In this talk, we discuss how to formulate a notion of limits for functors taking values in a higher category. This is joint work in progress with Moser and Rasekh.

When Grothendieck outlined his vision on anabelian geometry, the main object of study were étale fundamental groups and their arithmetic and geometry. We will start with a survey of basic techniques that are used in anabelian geometry to extract geometry from étale fundamental groups. Then we report on a shift of paradigm that replaces the étale fundamental group with the étale homotopy type to expand the scope of anabelian geometry. The latter is joint work with Tim Holzschuh and Alexander Schmidt.

We investigate generic derivations in expansions of topological fields from the perspective of definable geometry. After introducing the notion of a generic derivation and discussing its existence for the theory of algebraically bounded structures, we study the model-theoretic properties of the resulting expansions. A central theme of the talk is the interaction between generic operators and the underlying tame geometry of the field. In particular, we focus on the notion of open core: the open core of a topological structure is the reduct generated by the sets which are both open and definable. We prove that, under very mild assumptions, adding a generic derivation does not create new open definable sets. Moreover, we show that exponential fields do not admit a generic derivation in the absence of compatibility conditions between the derivation and the exponential map. This is joint work with Antongiulio Fornasiero.

In homotopical algebra, we widen the objects of study from classical rings to their higher analogues. These admit homotopy groups, which vanish for classical rings everywhere outside degree 0. On the other extreme are even periodic rings, whose even-degree homotopy groups all coincide and odd-degree ones vanish. We will discuss what effect the restriction to such periodic rings as affines has on algebraic geometry, and the connections this has to formal groups and prismatization.

Methods of linear algebra are ubiquitous throughout mathematics. Over combinatorialised versions of fields, for instance, bands and hyperfields, defining methods that mimic standard processes in linear algebra is significantly more challenging. However, defining such methods would be important for many aspects of tropical geometry and general matroid theory - for instance, it would enable more rigorous tropical representation theory and would help uncovering the mysteries of tropical vector bundles. In this talk, I discuss a framework to perform linear algebra on matroids. We discuss which combinatorial analogues of fields are amenable to linear algebra, and which have structural obstructions. Further, we construct combinatorial analogues of matrices and discuss their multiplication, and the properties of the arising monoids. Joint work in progress with Yifan Guo.

Given a smooth curve $C$, the set of line bundles of fixed degree is parameterized by an abelian variety $J_C$ known as the Jacobian of the curve. When $C$ has nodal singularities, the Jacobian can still be constructed but in this case fails to be compact. The problem of finding compactifications which are again moduli spaces for geometric objects associated with $C$ has a long history, going back to the work of Mumford. A class of solutions to this problem was provided by the work of Oda and Seshadri, in which the combinatorics of the dual graph was used alongside geometric invariant theory to produce a family of such compactifications $J_C(\varphi)$. The resulting compactifications are projective varieties, but in general many of their invariants remain mysterious. By studying the structure of a certain natural group action on the compactifications, we find a bridge to the theory of oriented matroids. This bridge then allows us to utilize techniques from matroid theory and provide a description compactifications in the Grothendieck ring of varieties.

We will explore the algebraic structures over the "field with one element" in the sense of Connes and Consani, where ordinary operations become hyper-operations with multiple possible outcomes. We then discuss its geometric implications and formulate a base change mechanism from F1 to Z.