Pagina 9

Date | Type | Start | End | Room | Speaker | From | Title |
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30/01/24 | Seminario | 14:30 | 16:00 | 1101 D'Antoni | Luca Battistella | Università di Bologna | Hyperelliptic curves and differentials Strata of differentials have been studied from several perspectives, from dynamics to geometry and topology.
Algebraic geometers need compactifications. One, which is a posteriori logarithmic, was introduced by Bainbridge--Chen--Gendron--Grushevsky--Möller in terms of multiscale (collections of) differentials (on reducible nodal curves). This natural compactification is unfortunately not irreducible, but they were able to single out the main component in terms of a condition on global residues; their proof rests on transcendental methods. I will mention a conjecturally equivalent condition which is purely algebraic. In joint work with Sebastian Bozlee, we give a proof of concept of this conjecture in the case of hyperelliptic differentials. The technical core of our work is a flexible tool for constructing hyperelliptic Gorenstein contractions of reducible nodal curves, given a hyperelliptic tropical differential, i.e. a suitable piecewise-linear function, on the dual graph. |

26/01/24 | Colloquium | 14:30 | 15:30 | 1201 Dal Passo | Kevin Buzzard | Imperial College London | Formalising modern research mathematics
A few years ago, the idea of formalising modern research level mathematics seemed completely out of reach. Since then, more and more examples have appeared. I'll go through several examples (some related to the mathematics of Scholze, Tao and Gowers), and talk about how the process is evolving, enabling multiple people to collaborate in the formalisation of modern research in real time.
Computer-verified proofs: 48 hours in Rome |

25/01/24 | Seminario | 14:30 | 15:30 | 1201 Dal Passo | Gihan Marasingha | University of Exeter | The benefits and challenges of teaching proof with Lean
This presentation will explore the pivotal role of the Lean in enhancing first-year undergraduates' understanding of mathematical proofs. I will share insights from my experiences and initial educational research on teaching a large first-year undergraduate cohort with Lean, focusing on how this tool can significantly impact student perception of proofs. Additionally, I will address the challenges encountered in teaching with Lean and the implications for learning and comprehension.
Computer-verified proofs: 48 hours in Rome |

25/01/24 | Seminario | 12:00 | 13:00 | 1201 Dal Passo | Floris van Doorn | University of Bonn | The internals of Lean
In this talk I will describe what goes on behind the scenes of Lean. I will explain the logic of Lean, called dependent type theory, what Lean tactics are and explain why we can trust proofs that are checked by Lean.
Computer-verified proofs: 48 hours in Rome |

24/01/24 | Seminario | 15:30 | 16:30 | 1201 Dal Passo | Filippo A. E. Nuccio Mortarino Majno di Capriglio | Université Jean Monnet Saint-Étienne | How to enjoy a mathematical discussion with your laptop
In this talk I will illustrate how certain programs, of which Lean is an example, permit to interact with a computer about the logical soundness of mathematical arguments. I will go through the details of well-known proofs trying to understand the feedback provided by the computer and will try to share the fun involved in the process.
Computer-verified proofs: 48 hours in Rome |

23/01/24 | Seminario | 14:30 | 16:00 | 1101 D'Antoni | Michele Pernice | KTH (Stockholm) | A stacky Castelnuovo's contraction theorem In this talk, we are going to discuss a generalization to weighted blowups of the classical Castelnuovo' contraction theorem. Moreover, we will show as a corollary that the moduli stack of n-pointed stable curves of genus 1 is a weighted blowup.
This is a joint work with Arena, Di Lorenzo, Inchiostro, Mathur, Obinna. |

23/01/24 | Seminario | 14:00 | 15:00 | 1201 Dal Passo | Marco Lenci | University of Bologna | Internal-wave billiards in trapezoids and similar tables We call internal-wave billiard the dynamical system of a point particle that moves freely inside a planar domain (the table) and is reflected by its boundary according to this rule: reflections are standard Fresnel reflections but with the pretense that the boundary at any collision point is either horizontal or vertical (relative to a predetermined direction representing gravity). These systems are point particle approximations for the motion of internal gravity waves in closed containers, hence the name. The phenomenon of internal waves in a fluid occurs in many situations and has been intensively studied by physicists. One of the first experiments, which became paradigmatic, was done in a container shaped like a rectangular trapezoid (with some thickness).
For a class of tables including rectangular trapezoids, we prove that the dynamics has only three asymptotic regimes: (1) there exist a global attractor and a global repellor, which are periodic and might coincide; (2) there exists a beam of periodic trajectories, whose boundary (if any) comprises an attractor and a repellor for all the other trajectories; (3) all trajectories are dense (that is, the system is minimal). If time permits, we will also discuss the prominent case where the table is an actual trapezoid, studying the sets in parameter space relative to the three regimes. We prove in particular that the set for (1) has positive measure (giving a rigorous proof of the existence of Arnold tongues for internal-wave billiards), whereas the sets for (2) and (3) are non-empty but have measure zero.
Joint work with C. Bonanno and G. Cristadoro. |

19/01/24 | Seminario | 16:00 | 17:00 | 1201 Dal Passo | Università di Roma | Algebra & Representation Theory Seminar (ARTS) "Homology operations for gravity algebras" In the early nineties Getzler discovered a nice algebraic structure on the equivariant homology of a topological conformal field theory. He called this algebraic structure a "gravity algebra" and he showed that it is governed by an operad which is closely related to the homology of M_{0,n+1} , the moduli space of genus zero Riemann surfaces with n+1 marked points. A gravity algebra can be thought as a generalization of a (dg) Lie algebra, in the sense that other than the Lie bracket we also have higher arity operations which satisfies a "generalized Jacobi identity". In this talk we will first give an introduction to gravity algebras, providing many interesting examples from both algebra (cyclic cohomology of a Frobenius algebra) and topology ( S^{1}-equivariant homology of the free loop space on a manifold). Then I will briefly explain that any class in the S^{1}-equivariant homology of the (unordered) configuration spaces of points in the plane H_{*}^{S1}(C(_{n}RF_{p}) (with coefficients in a field F_{p} of p elements) gives rise to an homology operation for gravity algebras. After that we will see how to compute this equivariant homology and if time permits we will see some applications.
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19/01/24 | Seminario | 14:30 | 15:30 | 1201 Dal Passo | Università de L'Aquila | Algebra & Representation Theory Seminar (ARTS) "Normalizer chain, modular idealizer chain and partitions" In recent joint works with Civino, Gavioli and Scoppola, we studied the conjugacy classes of an elementary Abelian regular subgroup T of the symmetric group on 2^{n} elements. In particular we computed, via GAP software package, a chain of normalizers in a Sylow 2-subgroup of this symmetric group defined iteratively, starting from T. We noticed that the logarithm of the indice of the (i-1)-th normalizer in the i-th normalizer of our chain is equal to the i-th partial sum of the sequence of the numbers of partitions of an integer in at least two distinct parts. In this talk we present some techniques developed in order to prove this result, including the notion of a special family of elements of a Sylow 2-subgroup, called rigid commutators. Finally, some generalizations to Lie algebras are given, considering similar results for an idealizer chain. | |

16/01/24 | Seminario | 14:30 | 16:00 | 1101 D'Antoni | Katharina Müller | Universität der Bundeswehr München | On towers off isogeny graphs with full level structure Let k be a finite field of chracteristic q. Let p,l be primes corpime to q and let N be a positive integer coprime to pql.
In this talk we will define graphs X_l^q(Np^n) whose vertices are tuples (E,P,Q), where E/k is an elliptic curves and P,Q is a basis for E[Np^n]. The edges are given by degree l isogenies.
We will discuss when X_l^q(Np^n)/X_l^q(Np^{n-1}) is Galois and will describe the structure of these graphs as volcanos.
This is joint work with Antonio Lei. |

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