Algebra and Geometry seminar of Neuchâtel

Program

• 16 September 2024 (exceptionnellement le lundi à 11h): Dhruv Ranganathan (Cambridge) - A story of degenerations in enumerative geometry.
  ▶ Abstract: The basic principle of degeneration has been present in enumerative geometry and intersection theory since their origins in the 19th century. But with the development of Gromov-Witten theory in the 1990s, and more generally the systematic study of moduli, the idea grew into a robust set of techniques that can be used to control the geometry of moduli spaces. I’ll outline the basic ideas, and then highlight some of the successes of the last few years, including connections to tropical geometry, the double ramification cycle, the cohomological study of the moduli space of curves.
• 24 September 2024: Fabio Bernasconi (University of Neuchâtel) - Geometry of 3-dimensional del Pezzo fibrations in positive characteristic.
  ▶ Abstract: A 3-dimensional del Pezzo fibration \(X \to C\) is one of the possible outcomes of the MMP, and one hopes to study the birational geometry of \(X\) in terms of the generic fibre \(X_{k(C)}\) and the base curve \(C\). In positive characteristic, an additional difficulty arises in this study: the generic fibre \(X_{k(C)}\) is defined over an imperfect field, where regularity is a weaker property than smoothness. In recent works with Fanelli-Schneider-Zimmermann and Tanaka, we study these extravagant cases and extend two classical results due to Enriques, Manin, and Iskovskikh: a 3-dimensional del Pezzo fibration over a curve, defined over an algebraically closed field, always admits a section, and the total space is rational if the base curve is rational and the anticanonical degree of a fibre is at least five.
• 01 October 2024 : Thomas Bouchet (Université Côte d'Azur) - Courbes de genre 4: invariants et reconstruction.
  ▶ Abstract: Dans cet exposé, nous allons introduire des invariants algébriques pour les courbes non-hyperelliptiques de genre 4, qui caractérisent les classes d'isomorphismes de ces courbes sur un corps algébriquement clos de caractéristique nulle. Après avoir exposé quelques outils de théorie classique des invariants qui nous ont permis de résoudre ce problème, nous donnerons les idées des preuves des principaux résultats. Enfin, nous expliciterons un algorithme qui, donné une liste d'invariants, reconstruit une courbe non-hyperelliptique de genre 4 qui possède ces invariants. Cet algorithme généralise celui de Mestre pour les formes binaires, et il fonctionne dans un cadre plus général que celui de l'exposé.
• 08 October 2024: Simon Machado (ETH Zurich) - Brunn-Minkowski in \(SO_n(\mathbb{R})\)
  ▶ Abstract: Given a subset \(A\) of a group, the doubling constant is the ratio \(\frac{|A^2|}{|A|}\), where \(A^2\) denotes the set of products of two elements of \(A\) and \(|A|\) is a notion of size appropriate to the setting (e.g. number of points, volume, covering number, etc). The study of doubling, a part of additive combinatorics, has found applications in number theory, dynamics, probability theory, and geometric analysis.
  In this talk, I will focus on doubling in Lie groups with respect to the Haar measure. While doubling is well understood in Euclidean spaces, it becomes more mysterious in other cases. For example, a conjecture by Breuillard and Green predicting a lower bound for doubling in \(SO_n(\mathbb{R})\) remained out of reach until recently. We will discuss a proof of this result and its implications, including a Brunn-Minkowski type inequality and a stability result.

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• 15 October 2024: Ilia Itenberg (Sorbonne Université) - Refined invariants for real curves.
  ▶ Abstract: 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. It turns out that two different rules of signs in the enumeration surprisingly lead to the same collection of refined invariants. The proof of the latter fact uses the tropical counterparts of the invariants under consideration. This is a joint work with Eugenii Shustin.
• 22 October 2024: Giuseppe Melfi (University of Neuchâtel) - On a problem of Erdös about sums of distinct powers of integers.
  ▶ Abstract: This is a joint work with Maximilian F. Hasler. In this talk, I will present a new lower estimate of the counting function of the positive integers that are sums of distinct powers of 3 and 4. This problem was raised by Erdős in 1996. I will also present some possible generalizations for related problems.
• 29 October 2024: No seminar - ColboisFest.
  
• 05 November 2024: Marc Abboud (University of Neuchâtel) - On the dynamics of endomorphisms of affine surfaces.
  ▶ Abstract: Let \(K\) be an algebraically closed field, an affine surface \(S\) over \(K\) is a surface given by polynomial equations in \(K^n\) for some n. An endomorphism \(f\) of \(S\) is a polynomial transformation of \(K^n\) which preserves \(S\). One of the first dynamical invariant that one can define for such transformations is the dynamical degree. For polynomial maps of \(K^2\), it is defined as the n-th power of the degree of the formulas of f^n. Favre and Jonsson in 2007 showed that the dynamical degree of polynomial maps of \(\mathbb C^2\) is an algebraic integer of degree \(\leq 2\). We generalise their result to any affine surface. In this talk, I will give several examples of affine surfaces and explain the main tool of the proof of our result: we study the dynamics of \(f\) on a space of valuations associated to \(S\) and deduce results on the dynamics of \(f\) on \(S\).
• 12 November 2024: Thomas Blomme (University of Neuchâtel) - Correlated Gromov-Witten invariants (joint with Francesca Carocci).
  ▶ Abstract: In this talk we introduce a geometric refinement of Gromov-Witten invariants for \(\mathbb P^1\)-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. We’ll give certain invariance properties of the correlated invariants and provide some computations in the case of \(\mathbb P^1\)-bundles over an elliptic curve. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and \(K3\) surfaces.
• 19 November 2024: Gabriel Dill (University of Neuchâtel) - The modular support problem.
  ▶ Abstract: In 1988, Erdös asked: let \(a\) and \(b\) be positive integers such that for all \(n\), the set of primes dividing \(a^n - 1\) is equal to the set of primes dividing \(b^n - 1\). Is \(a = b\)? In fact, work of Schinzel from 1960 already yields an affirmative answer to this question and even more generally that, if every prime dividing \(a^n - 1\) also divides \(b^n - 1\), then \(b\) is a power of \(a\).
  In joint work that was inspired by the analogy between roots of unity and singular moduli, i.e. \(j\)-invariants of elliptic curves with complex multiplication, Francesco Campagna and I have studied this so-called support problem with the Hilbert class polynomials \(H_D(T)\) instead of the polynomials \(T^n - 1\), both over number fields and over function fields.
• 26 November 2024: Jan Draisma (Bern University) - Subrank of bilinear maps.
  ▶ Abstract: The subrank of a bilinear map \(f:U \times V \rightarrow W\) measures how many independent scalar multiplications can be performed by using \(f\) once. Both subrank and an approximate version called border subrank play a central role in Strassen's work on complexity theory. I will discuss recent work by Derksen-Makam-Zuiddam and Pielasa-Šafránek-Shatsila, which determines the subrank of any sufficiently general \(f\); and joint work with Biaggi-Chang-Rupniewski in which we use a generalisation of the Hilbert-Mumford criterion to determine the asymptotics of the generic border subrank.
• 03 December 2024: Marvin Hahn (Trinity college, Dublin) - Tropical combinatorics of the 2D Toda lattice.
  ▶ Abstract: The study of soliton solutions of the KP hierarchy is a classical topic. In seminal work of Kodama and Williams a tropical geometry approach to their study was introduced. More precisely, it was proved that the phase structure of smooth solutions of the KP hierarchy is described by a tropical curve that naturally arises from combinatorial decompositions of the Grassmannian. The 2D Toda lattice is a generalisation of the KP hierarchy. In this talk, we show that the tropical approach introduced by Kodama and Williams generalises to the 2D Toda lattice answering a question posed by Kodama. This is talk is based on joint work in progress with Betancourt, Posch and Reda.
• 10 December 2024 (at 4pm instead of 1pm, in Room E213): Emanuele Delucchi (SUPSI) - On the \(K(\pi,1)\)-problem for abelian arrangements.
  ▶ Abstract: The study of arrangements of hypersurfaces is a nowadays classical field that was spurred by work of Arnol'd, Brieskorn and Deligne from the Seventies on arrangements of linear hyperplanes, configuration spaces and Artin groups of finite type. A hallmark of the classical theory is the strong interaction between geometry, algebra and combinatorics. This field recently broadened its scope beyond the linear case to include arrangements in the torus, in products of elliptic curves and, more generally, in connected Abelian Lie groups. The classical \(K(\pi,1)\)-problem, i.e. deciding asphericity of an arrangement's complement in combinatorial terms, can be stated in general. In this talk I will review some of the classical history of this problem and present some recent advances in the non-linear case. The talk will report on joint work with Christin Bibby, Alessio D'Alì, Noriane Girard, Giovanni Paolini, Sonja Riedel.
• 17 December 2024: No seminar - Conférence Géométrie des groupes et groupes de Cremona.
• 18 February 2025: Olivier Benoist (École Normale Supérieure) - On the rationality of real conic bundles.
  ▶ Abstract: Deciding whether a given algebraic variety is rational (birational to projective space) is an important problem in algebraic geometry. Over the field of real numbers, this problem is particularly interesting for varieties that are known to be rational over the complex numbers, as it then has an arithmetic flavour. In this talk, I will review the techniques that are available, and focus on concrete examples of real conic bundles for which I will provide new positive and negative results. This is joint work with Alena Pirutka.
• March 4th 2025: Gebhard Martin (Bonn) - Classification of non-F-split del Pezzo surfaces
  ▶ Abstract: Among all smooth cubic surfaces in \(\mathbb P^3\), there is a unique one without three lines forming a triangle: The Fermat cubic in characteristic \(p = 2\). Coincidentally, this is also the unique non-F-split smooth cubic surface. By work of Hara, non-F-split del Pezzo surfaces exist only in degrees 3, 2, and 1. After describing the classification of non-F-split del Pezzo surfaces of degree 2 due to Saito, I will report on joint work with Réka Wagener in which we give a geometric characterization of non-F-split del Pezzo surfaces of degree 1.
• 27 May 2025: Ronan Terpereau (University of Lille)
  ▶ Abstract: