The basic principle of degeneration has been present in enumerative geometry since the 19th century. With Gromov-Witten theory in the 1990s, the idea grew into a robust set of techniques. I'll outline the basic ideas and highlight connections to tropical geometry, the double ramification cycle, and the cohomological study of the moduli space of curves.
24 Sep. 2024Fabio BernasconiUniversité de Neuchâtel
Geometry of 3-dimensional del Pezzo fibrations in positive characteristic
A 3-dimensional del Pezzo fibration X → C is one of the possible outcomes of the MMP. In positive characteristic, the generic fibre is defined over an imperfect field, where regularity is weaker than smoothness. We extend classical results due to Enriques, Manin, and Iskovskikh.
01 Oct. 2024Thomas BouchetUniversité Côte d'Azur
Courbes de genre 4 : invariants et reconstruction
Nous introduisons des invariants algébriques pour les courbes non-hyperelliptiques de genre 4, caractérisant les classes d'isomorphismes sur un corps algébriquement clos de caractéristique nulle, et un algorithme de reconstruction généralisant celui de Mestre.
08 Oct. 2024Simon MachadoETH Zurich
Brunn-Minkowski in SOn(ℝ)
We prove a conjecture by Breuillard and Green predicting a lower bound for doubling in SO_n(ℝ), establishing a Brunn-Minkowski type inequality and a stability result.
15 Oct. 2024Ilia ItenbergSorbonne Université
Refined invariants for real curves
New invariants of the projective plane arising from signed enumeration of real algebraic curves of genus 1 and 2, admitting a refinement similar to Mikhalkin's genus zero case. Two different rules of signs surprisingly lead to the same invariants. Joint work with Eugenii Shustin.
22 Oct. 2024Giuseppe MelfiUniversité de Neuchâtel
On a problem of Erdős about sums of distinct powers of integers
Joint work with Maximilian F. Hasler. A new lower estimate of the counting function of positive integers that are sums of distinct powers of 3 and 4, a problem raised by Erdős in 1996.
On the dynamics of endomorphisms of affine surfaces
For polynomial maps of K², Favre and Jonsson (2007) showed the dynamical degree is an algebraic integer of degree ≤ 2. We generalise this to any affine surface by studying the dynamics on a space of valuations associated to S.
12 Nov. 2024Thomas BlommeUniversité de Neuchâtel
Correlated Gromov-Witten invariants (joint with Francesca Carocci)
A geometric refinement of Gromov-Witten invariants for ℙ¹-bundles relative to the fiberwise boundary, with computations over elliptic curves. Expected to play a role in degeneration formulas for abelian and K3 surfaces.
19 Nov. 2024Gabriel DillUniversité de Neuchâtel
The modular support problem
Inspired by an Erdős problem and the analogy between roots of unity and singular moduli, Francesco Campagna and I studied the support problem with Hilbert class polynomials H_D(T) instead of T^n−1.
26 Nov. 2024Jan DraismaUniversité de Bern
Subrank of bilinear maps
The subrank of f: U × V → W measures independent scalar multiplications using f once. We discuss recent work determining the subrank of any sufficiently general f and asymptotics of generic border subrank.
03 Déc. 2024Marvin HahnTrinity College, Dublin
Tropical combinatorics of the 2D Toda lattice
The tropical approach of Kodama-Williams for the KP hierarchy generalises to the 2D Toda lattice, answering a question of Kodama. Joint work in progress with Betancourt, Posch and Reda.
10 Déc. 2024Emanuele DelucchiSUPSI
On the K(π,1)-problem for abelian arrangements
Recent advances on the K(π,1)-problem for arrangements in tori, products of elliptic curves, and connected Abelian Lie groups. Joint work with Bibby, D'Alì, Girard, Paolini, Riedel. Note: séance à 16h, salle E213.