Talks
Senior Talks
Alessandra Sarti
Automorphisms of irreducible holomorphic symplectic manifolds, Enriques manifolds and the Morrison-Kawamata cone conjecture.
Irreducible holomorphic symplectic (IHS) manifolds can be seen as higher dimensional generalisations of K3 surfaces.
Enriques manifolds are non simply connected manifolds whose universal cover is irreducible holomorphic symplectic and as such they are natural generalizations of Enriques surfaces. In the first part of the talk I will recall basic facts about IHS
manifolds and their automorphisms group, I will then introduce Enriques manifolds and their properties. I will then show the famous Morrison-Kawamanta cone conjecture for certain Enriques manifolds and time permitting I will discuss it for
the known examples of Enriques manifolds. This is a joint work with Gianluca Pacienza.
Cinzia Casagrande
Fano 4-folds with large second Betti number are products of surfaces
We will start with an introduction to (smooth, complex) Fano varieties and their properties. Then we will discuss the following theorem: if X is a Fano 4-fold with second Betti number b_2>12, then X is a product of del Pezzo surfaces. This implies, in particular, that the maximal b_2 of a Fano 4-fold is 18. Finally we will discuss some examples and some ideas/techniques used to study this problem.
Diletta Martinelli
Title: Mathematics activities in the Global South.
Abstract: In the last years I have been involved in a variety of mathematical activities in the Global South, especially in Africa. I have been teaching and supervising students from many different countries and I have organized several workshops and training schools. Lately, I am working on the creation of a new mathematics master program in Kigali, Rwanda. I will describe my experiences and explain what are possible ways to engage with these type of activities.
Alice Pozzi
Modular generating series for real quadratic Heegner objects
Abstract: The theory of elliptic curves with complex multiplication has yielded some striking arithmetic applications, ranging from (cases of) Hilbert’s Twelfth Problem to the Birch and Swinnerton-Dyer Conjecture. These applications rely on the construction of certain “Heegner objects”, arising from imaginary quadratic points on the complex upper half plane; the most famous examples of these are Heegner points.
In recent years, conjectural analogues of these Heegner objects for real quadratic fields have been constructed via p-adic methods. In this talk, I will discuss how Heegner objects for real quadratic fields can be used to obtain modular generating series, that is, formal q-series that are q-expansions of classical modular forms. This is joint work with Judith Ludwig, Isabella Negrini, Sandra Rozensztajn and Hanneke Wiersema.
Min Lee
Title: Murmurations
Abstract: In April 2022, He, Lee, Oliver and Pozdnyakov made an interesting discovery using machine learning – a surprising correlation between the root numbers of elliptic curves and the coefficients of their L-functions. They coined this correlation `murmurations of elliptic curves’. Naturally, one might wonder whether we can identify a common thread of `murmurations’ in other families of L-functions. In this talk, I will introduce joint works with Jonathan Bober, Andrew R. Booker, David Lowry-Duda, Andrei Seymour-Howell and Nina Zubrilina, demonstrating murmurations in holomorphic modular forms and Maass forms in archimedean families.
Federica Pasquotto
Title: Contact topology of hypersurface singularities
Abstract: Given an isolated hypersurface singularity, its link is a smooth manifold which carries a natural contact structure. In the 1960's, Mumford proved that any isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. In higher dimension, the diffeomorphism type of the link can fail to detect the singularity. On the other hand, the contact structure on the link retains much more information.
This talk will be a gentle introduction to contact structures, links and Milnor fibres.
My interest in the topic is motivated by joint work with N. Adaloglou and A. Zanardini concerning symplectic invariants of contact structures on links of isolated hypersurface singularities.
Junior Talks
Geometry
Crislaine Kuster
Foliations on algebraic varieties
Abstract: A holomorphic foliation of dimension r on an algebraic variety 𝑋 consists of a decomposition of 𝑋 into a disjoint union of subvarieties of dimension r, satisfying specific compatibility conditions. In this talk, I will present an introduction to the theory of codimension one holomorphic foliations on algebraic varieties. For this purpose, I will start by providing basic concepts from this theory, highlighting examples and fundamental results. In conclusion, we will be interested in the following problem: considering an embedding of X in a projective space P^n, when a foliation on X is a restriction of a foliation on the ambient space P^n.
Abigail Hollingsworth
A “canonical” finite representation of the figure-eight knot group.
The figure-eight knot complement can be triangulated into two ideal tetrahedra in hyperbolic three-space. We use the developing map to ‘develop’ the gluing information we know from this triangulation by gluing copies of the tetrahedra along faces in hyperbolic three-space. We will choose the shapes of the ideal tetrahedra in this triangulation and find that the representation of these shapes has finite image.
Ana Victoria Martins Quedo
The Kawamata-Morrison Cone Conjecture for Generalized Hyperelliptic Variety
Abstract: A Generalized Hyperelliptic Variety (GHV) is the quotient of an abelian variety by a free action of a finite group which does not contain any translation. These varieties are the natural generalization of the bi-elliptic surfaces. In a collaboration with Martina Monti, we prove the Kawamata-Morrison Cone Conjecture for these manifolds using the analogous results established by Prendergast-Smith for abelian varieties."
Benedetta Facciotti
Introduction to wild Riemann-Hilbert correspondence
Abstract: In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles."
Erroxe Etxabarri Alberdi
1-dimensional K-moduli of Fano 3-folds
Abstract: We give a friendly introduction to K-stability, and the motivation behind it. We will see how to study and completely describe all one-dimensional components of the K-moduli of smooth Fano 3-folds. And we will finish giving some specific examples for family 3.12. This result is in collaboration with Abban, Cheltsov, Denisova, Kaloghiros, Jiao, Martinez-Garcia and Papazachariou.
Girtrude Hamm
Combinatorial Automorphisms of Spherical Varieties
Abstract: Toric varieties are a special class of variety which can be described combinatorialy by lattice fans. Toric morphisms have a neat combinatorial description in terms of lattice homomorphisms. This can be used in classifications to show when the toric varieties of two fans are equivalent. Spherical varieties are a generalisation of toric varieties with a similar combinatorial description. However, the combinatorial description of toric morphisms does not easily generalise to the spherical case. I will give a sketch of how to combinatorialy describe spherical varieties and give examples where the expected notion of morphism breaks down. I will then describe certain lattice automorphisms which are associated to automorphisms of spherical varieties and which are enough to make some classification questions possible."
Ines Chung-Halpern
Derived Categories under Birational Transformations
Abstract: In this talk, I will give an overview of the behaviour of derived categories of coherent sheaves when the underlying variety is subject to birational transformations such as flips and flops. In particular, such transformations can be realised in the Toric case as the result of wall crossings in a GIT problem. I will discuss the semi-orthogonal decompositions of derived categories of Toric varieties induced from these wall crossings, and how this relates to the more general decomposition formulas of Orlov and Bondal.
Ines Garcia Redondo
Computable Stability of Persistence Rank Function Machine Learning
Persistent homology (PH) barcodes and diagrams are a cornerstone of topological data analysis. Widely used in many real data settings, they relate variation in topological information (as measured by cellular homology) with variation in data, however, they are challenging to use in statistical settings due to their complex geometric structure. In this talk, we will revisit persistent rank functions and rank invariants as alternative representations of the PH output, easily integrable in Machine Learning and inferential tasks. Due to their functional nature, these invariants are amenable to Functional Data Analysis (FDA), a well-stablished branch of statistics dealing with data coming in the form of functions. I will present and showcase the effectiveness of such approach through several applications of rank functions and biparameter rank invariants to real and simulated data. After that, stability results for rank functions and rank invariants under $L^p$ metrics, the metric space needed for the applications above, will be discussed. This is joint work with Qiquan Wang, Pierre Faugère, Anthea Monod and Gregory Henselman-Petrusek."
Lucia Tessarollo
An introduction to Sub-Riemannian geometry.
Abstract: Sub-Riemannian geometry has emerged in the last twenty years as an independent research domain, with motivations and ramifications in several parts of pure and applied mathematics.
This presentation aims to introduce the concept of sub-Riemannian manifolds and and to construct a fundamental sub-Riemannian manifold, the Heisemberg group, through a famous problem: the isoperimetric problem.
Additionally, I will provide insights into my research focus, which revolves around surfaces embedded in contact sub-Riemannian manifolds, presenting the concept of characteristic points.
Mehidi Sara
Extending torsors via log schemes.
Abstract: Log geometry was introduced in the late 1980s by Fontaine-Illusie, Deligne-Faltings and K. Kato, among others, mainly to address two fundamental and interrelated problems in algebraic geometry: compactification and degeneration. Roughfly speaking, a log scheme is a scheme which, in addition, keeps track of a ""boundary"".
On the other hand, the problem of extending fppf torsors has been largely encountered in the literature, without giving a satisfactory solution in the classical setting. It turns out that the additional structure that one considers to define a log scheme allows to enlarge the category of fppf torsors by defining log torsors. In particular, this provides a new framework for studying the problem of extending torsors, presenting a fresh perspective on the question. In this talk, we will be discussing about this problem of extending torsors in the log setting."
Sofía Marlasca Aparicio
Ultrasolid homotopical algebra
We introduce ultrasolid modules, a new framework for the purpose of formal geometry. This generalises the solid modules over Q or F_p of Clausen and Scholze, and provides a new theory that works over any field.
Ultrasolid modules can be described as a suitable notion of a complete topological vector space. We can form derived variants of this theory to study deformation theory. This gives a generalisation of the Schlessinger criterion that says, for example, that Galois deformation rings always exist as objects in the ultrasolid category."
Thamarai Valli Venkatachalam
Classification of Fano 3-folds
Abstract: Fano varieties are positively curved algebraic varieties. They serve as fundamental components in the construction of various algebraic varieties. Hence, understanding and classifying them holds a very significant value in understanding algebraic varieties in general.
In this talk, we will begin with a formal introduction to Fano varieties and see some of the previously known classification results about them. Then, we will see some key ideas in classifying Fano 3-folds with mild singularities that occur as hypersurfaces in Picard rank 2 toric varieties.
Number Theory
Anouk Greven
The delta method for possibly skew boxes
Abstract: In 1996, Heath-Brown introduced a new version of the circle method, now known as the delta method, and applied it to quadratic forms: looking at zeroes of the quadratic form that lie in a certain box, how does the number of zeroes grow as the box grows? In this talk I describe the results I obtained when allowing the box to be so-called skew and I will highlight some parts of the method.
Ashleigh Wilcox
A systematic approach to solving diophantine equations
Abstract: Diophantus of Alexandria, who lived in the Third Century, was interested in determining integer solutions to polynomial equations with integer coefficients. Such equations are now known as (polynomial) Diophantine equations.
In 1970, Yuri Matiyasevich [1] proved the negative answer to Hilbert’s tenth problem, showing that there is no general method that can solve all Diophantine equations. Instead, researchers focus on solving restricted classes of equations. This is an active area of research, but, traditionally, it is a matter of researcher’s taste which equations to solve.
In this project, we define the “size” of an equation, and then consider all equations systematically in order of their size, starting with the smallest. We also do the same approach in some restricted families of equations, such as equations with a restricted number of variables, degree, or the number of monomials. We used dozens of existing and new methods to solve many thousands of equations, with computer assistance. The smallest equations for which our methods do not work are listed as open. As we know from the example of Fermat’s Last Theorem, such easy-to-state but difficult-to-solve equations have the potential to stimulate the development of new methods in number theory.
[1] Hilary Putnam Yuri Matiyasevich, Martin Davis. Hilbert’s tenth problem. Foundations of computing. MIT Press, 1993.
Bethany (Beth) Dixon
The Distribution of Close p-adic Numbers
The space of rational numbers while having many desirable properties lacks completeness. Because of this certain analysis cannot take place, so it is desired instead to move to the space of real numbers. This however is not the only option. The complementary approach is to instead consider completing the field by extending to the $p$-adic numbers. These numbers are particularly useful for problems in number theory involving divisibility and congruences.
In this talk I will introduce the idea of $p$-adic numbers and discuss how close algebraic $p$-adic numbers of degree $n$ can be.
Catinca Mujdei
Kloostermania
Abstract: Kloosterman sums were introduced by Hendrik Kloosterman, when he refined the circle method to give an asymptotic expression for the number of integral representations of a large integer by a diagonal quaternary quadratic form. An optimal bound for these exponential sums is due to André Weil, and arises from his proof of the Riemann hypothesis for curves over finite fields. The relevance of Kloosterman sums has increased over the past century due to their ubiquity in many number-theoretic applications; this productive activity has eventually been referred to as “Kloostermania”. In this talk, I'll illustrate some of those places where Kloosterman sums appear, such as the Kuznetsov trace formula."
Chiara Sabadin
The Herbrand-Ribet Theorem
Abstract: In 1850s, Kummer discovered a remarkable connection between class groups and special values of the Riemann Zeta function. Over a century later, Ribet devised a new method of connecting class groups and the Zeta function, using modular forms and Galois representations, which refined Kummer's results and inspired new research.
Diana Mocanu
Variants of the modular method
Abstract: We are going to briefly introduce the modular method for solving Diophantine equations and talk about some classical approaches. Then, we will talk about a higher dimensional variant that originated in Darmon program, where we replace Frey elliptic curves by Jacobians of “Frey hyperelliptic curves”. If time permits, I will present some ongoing work involving computing conductors of these Jacobians using cluster pictures.
Erin Hales
"What is homomorphic encryption and why is it so noisy?"
Abstract: I would give an overview of homomorphic encryption and describe how the addition of noise both gives security and creates difficulties in terms of implementation.
Maryam Nowroozi
Perfect powers in elliptic divisibility sequences
Abstract: Let E be an elliptic curve over the rationals given by an integral Weierstrass model and let P be a rational point of infinite order. The multiple nP has the form (An/B2n,Cn/B3n) where An, Bn, Cn are integers with AnCn and Bn coprime, and Bn positive. The sequence (Bn) is called the elliptic divisibility sequence generated by P. In this talk we answer the question posed in 2007 by Everest, Reynolds and Stevens: does the sequence (Bn) contain only finitely many perfect powers?
Philip Holdridge
Title: Random Diophantine Equations with Prime Solutions
If a Diophantine equation has nontrivial solutions in the rational numbers, then it will also have solutions in the reals and in every p-adic field. If the converse implication holds, then the equation is said to satisfy the Hasse principle. We discuss a modified version of the Hasse principle which concerns whether an equation has solutions in the set of prime numbers. We prove that this "prime Hasse principle" holds for almost all homogeneous equations with sufficiently many variables. Our method mostly involves the geometry of numbers.
Sara Varljen
Exploring Bianchi Groups
Abstract: The modular group PSL2(Z) is a very familiar object when dealing with classical modular forms. If, instead of considering Z, we take the ring of integers Ok of an imaginary quadratic field k, we obtain Bianchi groups: modular groups of the form PSL2(Ok). They act on the hyperbolic 3-space, similarly to how PSL2(Z) acts on the hyperbolic plane, and are fundamental in defining Bianchi modular forms.
In this talk we will focus on giving some geometric and number-theoretic background on Bianchi groups and see how the fundamental domain for the action can be defined from a computational point of view. Time permitting, we will mention some (co)homological aspects of Bianchi groups.
Sophie Maclean
Gap distributions and the Metric Poissonian Property
Abstract: When studying dilated arithmetic sequences modulo 1 (i.e. the fractional parts of arithmetic sequences, multiplied by a constant) there are several pertinent questions to ask: Is the resulting sequence equidistributed? What can we say about the gap size between consecutive elements of the new set? In this talk I will explore the gap distributions in dilated arithmetic sequences modulo 1, including what it means for a sequence to have the metric poissonian property. I will also give an overview of the current progress in answering these questions and what is believed to hold, including a famous conjecture of Rudnik and Sarnak.
Vaidehee Thatte
Title: Generalized Ramification Theory and Its Applications
Abstract: Generalized ramification theory serves the dual purpose of a diagnostic tool as well as treatment by helping us locate, measure, and treat the anomalous behavior of mathematical objects. In the classical setup, the degree of a finite Galois extension of ‘nice’ fields splits up neatly into the product of two well-understood numbers that encode how the base field changes. In the general case, however, a third factor called the defect (or
ramification deficiency) can pop up. The defect is a mysterious phenomenon; it is the main obstruction to several long-standing open problems in many areas of mathematics. The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many adjustments" fails when the defect is present. I will discuss my previous and ongoing work in arithmetic algebraic geometry, number theory, and valuation theory that allows us to understand and treat the defect.