Meeting of the Catalan, Spanish, Swedish Math Societies (CAT‐SP‐SW‐MATH)

(1)

Meeting of the Catalan, Spanish, Swedish

Math Societies (CAT‐SP‐SW‐MATH)

A joint Meeting of the Catalan, Spanish, Swedish Math Societies (CAT-SP-SW-MATH) will be held in Umeå (Sweden) from 12th to 15th June 2017.

The meeting is a symposium devoted to mathematics at large.

The conference is thought as a meeting point between the different areas of mathematics and its applications.

The programme will consist of several plenary lectures, covering a wide range of areas of mathematics, and special sessions devoted to a single topic or area of mathematics. The venue of the conference will be the Department of Mathematics and Mathematical Statistics of Umeå University.

Welcome!

Milagros Izquierdo (Svenska matematikersamfundet) Xavier Jarque (Societat Catalana de Matemàtiques)

Francisco José Marcellán (Real Sociedad Matemática Española)

(2)

Committees

Scientific Committee

 Mats Andersson (Chalmers/Göteborgs universitet)  María Ángeles Gil (Universidad Oviedo)

 Gemma Huguet (Universitat Politècnica de Catalunya)  Ignasi Mundet (Universitat Barcelona)

 Joaquín Pérez (Universidad Granada)  Sandra di Rocco (KTH) Chairperson

 Xavier Tolsa (Universitat Autonoma Barcelona)  Tatyana Turova (Lunds universitet)

 Juan Luis Vázquez (Universidad Autónoma Madrid)

Organizing Committee

 Klas Markström (Umeå universitet)

(3)

Plenary Speakers

 Tomás Alarcón, Centre de Recerca Matemàtica, Bellaterra (Barcelona)  Yacin Ameur, Lund University

 Viviane Baladi, CNRS and University Pierre et Marie Curie, Paris  Fabrizio Catanese, University of Bayreuth

 Rosa Donat, University of Valencia

 Maria J. Esteban, CNRS & University Paris-Dauphine  Luis Guijarro, UAM, Madrid

 Kathryn Hess, EPFL, Lausanne, EMS Distinguished Lecturer  Kurt Johansson, KTH, Stockholm

 Jonatan Lennells, KTH, Stockholm.

 Maria Teresa Lozano, University of Zaragoza  Joaquim Ortega Cerdà, University of Barcelona  Marta Sanz-Solé, University of Barcelona

(4)

Special Sessions

(5)

(6)

Participants

First name Surname Organization

Francesc Aguiló-Gost Dept. Matemàtiques, UPC

Tomas Alarcon ICREA-CRM

Seidon Alsaody Université Lyon 1

Josep Alvarez Montaner Universitat Politècnica de Catalunya

Yacin Ameur Lund University

Johan Andersson Högskolan i Skövde

Rikard Anton Umeå University

Pere Ara Universitat Autonoma de Barcelona

Paco Arandiga Universitat de València

Paul Baginski Fairfield University

Viviane Baladi CNRS and UPMC, Paris

Adson Banda LiU

Francisco (Kiko) Belchi-Guillamon University of Southampton

Alexander Berglund Stockholm University

Göran Bergqvist LiU

Anders Björn LiU

Jana Björn LiU

Chiara Boiti University of Ferrara

Maria Bras-Amorós Universitat Rovira i Virgili

Lance Bryant Shippensburg University

Åke Brännström Umeå University

Emilio Bujalance UNED

Xing Shi Cai Uppsala University

Federico Cantero Morón Universitat de Barcelona

(7)

Natalia Castellana Universitat Autònoma de Barcelona

Angel Castro ICMAT

Fabrizio Catanese Bayreuth University

Javier Cirre UNED

David Cohen Umeå University

Antonio F. Costa UNED

Laura Costa Universitat de Barcelona

Cristina Costoya UDC

Veronica Crispin Uppsala universitet

Jose Antonio Cuenca Mira Universidad de Malaga

Ferran Dachs Cadefau Martin-Luther-Universität Halle-Wittenberg Konstantinos Dareiotis Uppsala University

Shagnik Das Freie Universität Berlin

Alessandro De Stefani KTH

Felix Del Teso NTNU

Manuel Delgado University of Porto

Raj Narayan Dhara NTNU

Sandra Di Rocco KTH

Raquel Diaz Universidad Complutense Madrid

Ernst Dieterich Uppsala universitet

Kien Do Van Hanoi Pedagogical University No.2

Rosa Donat Universitat de València

Blas Echebarria Universitat Politècnica de Catalunya

Mats Ehrnström NTNU

Alberto Elduque Universidad de Zaragoza

Shalom Eliahou Université du Littoral Côte d'Opale Anne-Maria Ernvall-Hytönen Åbo Akademi University

Maria J. Esteban CNRS & University Paris-Dauphine José Javier Etayo Gordejuela Universidad Complutense Madrid

Victor Falgas-Ravry Umeå universitet

Jose Ignacio Farrán Martin Universidad de Valladolid

(8)

Alberto Fernández Boix Ben-Gurion University of the Negev

Francesc Fité Universitat Politècnica de Catalunya/BGSmath

Raóon Flores Universidad de Sevilla

Stiofáin Fordham University College Dublin (UCD)

Jens Forsgård Texas A&M University

Elías Fuentes Guillén USAL/UAEMéxico

Antonio Galbis Universitat de València

Francisco Gancedo Universidad de Sevilla

Evelia Rosa Garcia Barroso Universidad de La Laguna Juan Ignacio García García Universidad de Cádiz Ignacio Garcia-Marco Aix-Marseille University

Olav Geil Aalborg University

Samia Ghersheen LiU

Arpan Ghosh LiU

Konstantina-Stavroula Giannopoulou NTNU

Federica Giardina Stockholm University

Jesús Gómez Ayala Universidad del País Vasco

Ramón González Rodríguez Vigo University

Victor González-Alonso Leibniz University, Hannover

Joana Grah University of Cambridge

Jordi Guàrdia Universitat Polit ècnica de Catalunya

Luis Guijarro UAM

Xavier Guitart Universitat de Barcelona

Gustav Hammarhjelm Uppsala University

Eskil Hansen Lund University

Dolors Herbera Universitat Autónoma de Barcelona

Kathryn Hess EPFL

Anders Holst Lund University

(9)

Rym Jaroudi LiU

Kurt Johansson KTH

Christian Johansson University of Cambridge

Johan Jonasson Chalmers/GU

David Jornet Casanova Universitat Politècnica de València

Philipp Korell TU Kaiserslautern

Mihaly Kovacs Chalmers University of Technology and

University of Gothenburg

Ewa Kozlowska-Walania University of Gdansk, Institute of Mathematics Thomas Kragh Uppsala universitet - matematiska institutionen

Julia Kroos BCAM - Basque Center for Applied Mathematics

Kaie Kubjas Aalto University

Pär Kurlberg KTH

Annika Lang Chalmers & University of Gothenburg

Joan-C. Lario UPC, Barcelona

Stig Larsson Chalmers/University of Gothenburg

Joel Larsson Umeå Univerisy

Abid Ali Lashari Stockhom University

Jonatan Lenells KTH

Karl-Olof Lindahl Linnæus University

Erik Lindgren KTH

Svante Linusson KTH

Evgeniy Lokharu LiU

Santiago López de Medrano Instituto de Matemáticass, UNAM

María Teresa Lozano Universidad de Zaragoza

Franz Luef Norwegian University for Science and Technology

Per Håkan Lundow Umeå universitet

Niklas Lundström Umeå University

Michael Lönne Universität Bayreuth

Jana Madjarova Chalmers University of Technology

Francisco Marcellan Universidad Carlos III de madrid

(10)

Consuelo Martinez Lopez Oviedo University

Marc Masdeu Universitat Autónoma de Barcelona

Olivier Mathieu UdL (Lyon)

Naoyuki Matsuoka Meiji University

Leif Melkersson LIU

Francisca Miguel Universidad de Malaga

Salvador Moll University of Valencia

Fernando Montaner Universidad de Zaragoza

M. Angeles Moreno Frías Universidad de Cádiz Julio-José Moyano-Fernández Universitat Jaume I Juan Carlos Naranjo University of Barcelona

Enric Nart UAB

Luis Narváez Macarro University of Sevilla

Sergey Natanzon National Research University

Higher School of Economics

Jonas Nordqvist Linnæus University

Eulalia Nualart University Pompeu Fabra

Alessandro Oliaro University of Torino

Anna Oneto University of Genova

Joaquim Ortega Cerdà Universitat de Barcelona

Long Pei KTH

Guillem Perarnau University of Birmingham

Fabio Perroni University of Trieste

Tomas Persson Lund University

Andreas Petersson Chalmers and The University of Gothenburg

Lan Anh Pham Umeå University

Arturo Pianzola University of Alberta and CAECE

Ana M. Porto UNED

Eva Primo Tàrraga Universitat de València

(11)

Anita Rojas Universidad de Chile Martha Judith Romero Rojas Universidad del Cauca

Marcel Rubió KU Leuven

Albert Ruiz Universitat Autònoma de Barcelona

Carles Sáez Calvo BGSMath - UB - CRM

Marta Sanz-Solé University of Barcelona

Denys Shcherbak Umeå University

Matas Sileikis Charles University Prague

Mercedes Siles Molina Universidad de Malaga

Olof Sisask KTH

Tord Sjödin Umeå University

Fiona Skerman Bristol University

Anna Somoza Henares Universitat Politècnica de Catalunya / Universiteit Leiden

Dario Spirito Universitá di Roma Tre

Dumitru Stamate ICUB/University of Bucharest

Britt-Marie Stocke Umeå University

Theresa Stocks Stockholm University

Diana Stoeva Austrian Academy of Sciences

Klara Stokes Högskolan i Skövde

Alexander Stolin Chalmers/University of Göteborg

Francesco Strazzanti University of Seville

Martin Strömqvist Uppsala universitet

Olof Svensson LiU

Blazej Szepietowski Gdansk University

Grazia Tamone University of Genova

Erik Thörnblad Uppsala University

Vladimir Tkachev LiU

Joachim Toft Linnæus University

Laura Tozzo Universitá di Genova

Chau Tran Do Minh Thai Nguyen University of Education

(12)

Andrew Treglown University of Birmingham

Peter Turbek Purdue University Northwest

Tatyana Turova Lund University

Ville Turunen Aalto University

Ewa Tyszkowska University of Gdansk

Alberto Vigneron-Tenorio Universidad de Cádiz

Antonio Viruel Universidad de Málaga

Patrik Wahlberg Linnæus University

Erik Wahlén Lund University

Yuexun Wang NTNU

Jonas Wickman Umeå University

Felix Wierstra Stockholms universitet

Frank Wikström Lund University

Jens Wittsten Lund university

Santiago Zarzuela University of Barcelona

Lai Zhang Umeå University

Yi Zhao Georgia State University

Lars-Daniel Öhman Umeå University

(13)

Abstracts

Plenary Speakers

 Tomás Alarcón, Centre de Recerca Matemàtica, Bellaterra (Barcelona)  Yacin Ameur, Lund University

 Viviane Baladi, CNRS and University Pierre et Marie Curie, Paris  Fabrizio Catanese, University of Bayreuth

 Rosa Donat, University of Valencia

 Maria J. Esteban, CNRS & University Paris-Dauphine  Luis Guijarro, UAM, Madrid

 Kathryn Hess, EPFL, Lausanne, EMS Distinguished Lecturer  Kurt Johansson, KTH, Stockholm

 Jonatan Lennells, KTH, Stockholm.

 Maria Teresa Lozano, University of Zaragoza  Joaquim Ortega Cerdà, University of Barcelona  Marta Sanz-Solé, University of Barcelona

(14)

Abstracts and Presentations

Special Sessions

1. Mathematical Biology

2. Algebraic Geometry and Commutative Algebra: Book of Abstracts 3. Nonlinear PDEs

4. SPDEs: From Theory to Simulation 5. Numerical Semigroups and Applications

6. Numbers in Number Theory: Book of Abstracts

7. Loci of Riemann and Klein Surfaces with Automorphisms 8. Time-Frequency Analysis and Pseudo-Differential Operators 9. Graphs, Hypergraphs and Set Systems

10. Non-Commutative Algebras 11. Homotopy Theory

(15)

Meeting of the Catalan, Spanish and Swedish Math Societies

Mathematics of cancer: multi-scale modelling of

the tumour growth dynamics

Tomás Alarcón

ICREA,Pg. Lluís Companys 23, 08010 Barcelona, Spain.

Centre de Recerca Matemàtica. Edifici C, Campus de Bellaterra, 08193 Bellaterra (Barcelona), Spain.

Abstract

In recent years multi-scale modelling of tumour growth has become an emergent area in Mathematical Biology. Multi-scale models seek to integrate different mathematical descriptions of phenomena characterised by a widely varying time and length scales, whereby the global behaviour of the tumour is an emergent property of the non-linear coupling between scales. In this talk, I will present a review of multi-scale models of tumour growth as well as recent advancements in coarse-grained models and hy-brid simulation techniques that allow to simulate this very complex models more efficiently.

(16)

The two-dimensional Coulomb plasma

Yacin Ameur

Lund University. yacin.ameur@math.lu.se

Abstract

The Coulomb gas model could be regarded as a statistical-mechanical bridge between classical potential theory and modern, physical field the-ories. Each level (classical, statistical, field-theoretical) gives rise to non-trivial questions, and depending on the choice of dimension, the analy-sis gets specific traits which makes it unique. My talk will concern the twodimensional case, which is tied to logarithmic potential theory, and to conformal field theory. The area is currently quite active and comprises several fundamental questions, for example, pertaining to the Hall effect and crystallization.

(17)

New analytical tools for dynamics with

singularities, including Sinai billiards

Viviane Baladi

Sorbonne Université, UPMC Univ. Paris 6, CNRS, Institut de Mathématiques de Jussieu (IMJ-PRG), Paris, France, viviane.baladi@imj-prg.fr

Abstract

In the past 15 years, tools from analysis, in particular new Banach spaces of anisotropic distributions on manifolds, have allowed substantial progress in dynamical systems.

After briefly explaining how a spectral gap for a transfer operator furnishes ergodic information, we shall focus on Sinai billiards maps and flows. These natural but technically challenging systems are uniformly hyperbolic and volume preserving — however grazing orbits give rise to singularities. New analytic tools recently allowed us to obtain exponential mixing for finite horizon Sinai billiard flows (with M. Demers and C. Liverani), and the natural volume. We shall finish by discussing ongoing work (with M. Demers) on other Gibbs states (including the measure of maximal entropy).

(18)

Topological methods in moduli theory

Fabrizio Catanese

Universität Bayreuth, Fabrizio.Catanese@uni-bayreuth.de

Abstract

Complex algebraic curves are Riemann surfaces: the interplay of moduli and topology goes back to this well known truth.

The topological approach, via Riemann?s existence theorem, is particu-larly useful when analyzing the moduli spaces of curves with symmetries. In turn, the study of curves with certain groups G of automorphisms, is crucial for the construction of higher dimensional varieties, and for the description of their moduli spaces.

In the talk, after illustrating basic examples of projective classifying spaces, and by now classical constructions (Hirzebruch-Kummer covers associated to line configurations) I shall review several results, obtained jointly with Ingrid Bauer, Michael Lönne and Fabio Perroni.

Finally, I shall concentrate on quite recent results and work in progress concerning rigid complex manifolds and projective classifying spaces, es-pecially in the crucial complex dimension 2.

(19)

Nonlinear approximation and nonlinear

subdivision

Rosa Donat

University of Valencia, Rosa.M.Donat@uv.es

Abstract

Data-dependent and adaptive reconstruction techniques are used in numerical analysis in order to improve accuracy in the presence of dis-continuities, or to comply with shape preservation properties. In this talk we shall review the use of various nonlinear interpolatory tools in the de-sign of nonlinear subdivision schemes, which inherit the non-oscillatory or shape preserving properties of the underlying reconstruction. Their analysis, however, requires a different approach than in the linear case. When a nonlinear scheme can be written as a nonlinear perturbation of a convergent linear scheme, convergence and stability can be studied in a rather systematic way that can be applied also to other nonlinear subdivi-sion schemes (not necessarily related to an interpolatory reconstruction). These observations are useful in order to analyze a simple (stationary) nonlinear interpolatory subdivision scheme capable of reproducing conic sections.

(20)

Rigidity, nonlinear flows and optimal symmetry

for extremals of functional inequalities

Maria J. Esteban

CNRS & University Paris-Dauphine , esteban@ceremade.dauphine.fr

Abstract

The analysis of optimality and symmetry properties of extremals in functional inequalities has been performed recently by introducing non-linear flows into the picture. These results solve conjectures about sym-metry and symsym-metry breaking in functional inequalities which play an important role in various areas of analysis. Also, as a consequence we have obtained optimal estimates for the principal eigenvalues of linear operators and rigidity results of solutions of nonlinear elliptic PDEs for compact and noncompact in Riemaniann manifolds.

This work has been done in collaboration with J. Dolbeault and M. Loss

(21)

Symmetries in metric spaces

Luis Guijarro

Universidad Autónoma de Madrid. luis.guijarro@uam.es

Abstract

Understanding a metric space is easier if it has a good amount of symmetries; compare for instance the amount of statements about homo-geneous spaces to what can be said of arbitrary Riemannian manifolds. In this talk, aimed to a general audience, we will examine the symmetry groups of some metric spaces that have acquired big popularity during the last twenty years: Alexandrov spaces, and a far reaching refinement of Lott-Otto-Villani spaces.

(22)

Configuration spaces of products

Kathryn Hess

EPFL, kathryn.hess@epfl.ch

Abstract

For any topological space X, the configuration space Confn(X) of n

points in X is the subspace of the iterated product X×n consisting of

n-tuples of distinct points in X. Configuration spaces play an

impor-tant role in low-dimensional topology and homotopy theory. For exam-ple, the fundamental group of Confn(R2) is the pure n-stranded braid

group, while the orbit space of the natural action of the symmetric group, Confn(R2)/Σn, has fundamental group isomorphic to the entire n-stranded

braid group. Moreover, Confn(R2)/Σn is homeomorphic to the space of

complex monic polynomials of degree n with exactly n roots.

In this talk I will provide a brief overview of the theory of configu-ration spaces, then describe the connection between configuconfigu-ration spaces and little disks operads, which encode operations and relations among op-erations in iterated loop spaces. To conclude I will explain a new method for computing homotopy invariants of the configuration space of a prod-uct of two closed manifolds in terms of the configuration spaces of each factor separately that exploits this relationship with operads.

(23)

Edge fluctuations of limit shapes

Kurt Johansson

KTH-Royal Institute of Technology, kurtj@kth.se

Abstract

I will survey results on the fluctuations of the boundaries between re-gions with different phases in some random tiling or dimer models. These fluctuations are of a universal nature and the same type of limit laws ap-pear also in other contexts, in particular in local random growth models, directed polymers and random matrix theory. Behind the fact that cer-tain models can be analyzed in detail is the fact that the tiles or dimers form determinantal point processes.

(24)

The Wavemaker Problem

Jonatan Lenells

KTH Royal Institute of Technology, 100 44 Stockholm, Sweden E-mail: jlenells@kth.se

Abstract

I will discuss the problem of determining the wave profile generated by a periodically moving wavemaker mounted at one end of a wave tank. Mathematically, the problem can be expressed as an initial-boundary value problem for a nonlinear integrable equation with time-periodic forc-ing. I will discuss a new technique for analyzing problems of this type and describe how it can be used to at least partially answer some questions raised by experiments 35 years ago.

(25)

Thurston’s geometries and cone-manifolds

María Teresa Lozano

Departamento de Matemáticas e Instituto Universitario de Matemáticas y Aplicaciones Universidad de Zaragoza, tlozano@unizar.es

Abstract

The Geometrization Theorem states that every closed 3-manifold can be decomposed in a canonical way into pieces each having one of the eight Thurston’s geometries. In the first part of this talk we discuss some tran-sitions (degenerations, regenerations and limits) between the Thurston’s geometries.

A cone-manifold (M, K, α) is a manifold M with a geometric structure which is singular along the knot K where the angle around K is α instead of 2π. If α = 2π/n, (M, K, 2π/n) is a geometric orbifold.

In the second part of the talk I will describe some applications of the previous transitions to cone-manifold structures in a manifold M with singularity along a link L. There are some numerical invariants associated to the pair (M, L), as limit of hyperbolicity and/or limit of sphericity. The continuous family of cone-manifold structures in (M, L) contains all the orbifold geometric structures in M with singularity along L as a discrete subset. Therefore, the family of cone-manifold structures in (M, L) is the correct framework to compute geometric magnitudes for these orbifolds, as volume and Chern-Simons invariant, using type Schäfli differential forms. Then, the values of volume and Cher-Simons invariant of the manifolds obtained as covering of M branched over L can be computed.

(26)

Algebraic designs

Joaquim Ortega Cerdà

Universitat de Barcelona

Abstract

I will lecture on a generalization of the classical Bernstein inequality for polynomials on algebraic manifolds and see some of its applications, namely the construction of algebraic designs, building upon a similar re-sult of Bondarenko, Radchenko and Viazovska on spherical designs.

(27)

Wong and Zakai theorems for stochastic partial

differential equations

Marta Sanz-Solé

University of Barcelona

Abstract

The classical Wong and Zakai theorem refers to approximations of Itô stochastic differential equations when the Brownian motion is replaced by a sequence of stochastic processes with smooth sample paths. For stochastic partial differential equations, the analysis of a similar natural question brings up some difficult problems related to the infinite dimen-sional character of the noise and the subsequent identification of the

cor-rection term. Starting with an introduction addressed to non specialists,

we will go through results on the stochastic heat and wave equations, with special attention to the nonlinear stochastic wave equation in dimension three. We will also show how the approximations provide a characteriza-tion of the support of the law of the solucharacteriza-tion in spaces of Hölder continuous functions.

(28)

Meeting of the Swedish, Spanish and Catalan Mathematical Societies Ume˚a, June 12 – June 15, 2017

Special session: Mathematical Biology Preliminary Program

Wedenesday June 14

Time Speaker Talk title

14.00–14.45 Blas Echebarria Mechanisms underlying electro-mechanical cardiac alternans 14.45–15.30 Arpan Ghosh, A one dimensional asymptotic model of blood flow through a curved,

elastic blood vessel

15.30–16.00 COFFEE BREAK

16.00–16.45 Joana Grah Mathematical Imaging Methods for Mitosis Analysis in Live-Cell Phase Contrast Microscopy

16.45–17.30 Rym Jaroudi Source Localization of Brain Tumors via Reaction-Diffusion Mod-els

Jonas Wickman Determining selection across heterogeneous landscapes: a perturbation-based method and its application to modeling evolution in space

Thursday June 15

Time Speaker Talk title

14.00–14.45 ˚Ake Br¨annstr¨om On the convergence of the Escalator Boxcar Train

14.45–15.30 Federica Giardina Statistical methods to estimate the size of the undiagnosed HIV-1 infected population

15.30–16.00 COFFEE BREAK

16.00–16.45 Abid Ali Lashari Malaria model with asymptomatic class and superinfection 16.45–17.30

(29)

Abstracts for Wednesday June 14

Mechanisms underlying electro-mechanical cardiac alternans Blas Echebarria

Polytechnic University of Catalonia, Spain, blas.echebarria@upc.edu

Cardiac excitation starts at the sino atrial node as a periodic change in the myocytes trans-membrane potential, that then propagates along the atria and ventricles, inducing the contraction of the heart and the pumping of blood throughout the body. Cardiac alternans is a disturbance in the normal rhythm of the heart that can be described as beatto- beat oscillations in the duration of the excited phase of the transmembrane potential, i.e., in the action potential duration (APD) and in the concentration of cytosolic calcium, which is the messenger that initiates contraction in car-diac myocytes. In particular, electromechanical carcar-diac alternans consists in beat-to-beat changes in the strength of cardiac contraction. Despite its important role in cardiac arrhythmogenesis, its molecular origin is not well understood. In this talk I will review recent results by our group on the different mechanisms that can give rise to calcium alternans, as well as the differences in calcium response between ventricular and atrial cells.

A one dimensional asymptotic model of blood flow through a curved, elastic blood vessel

Arpan Ghosh

Link¨oping University, Sweden, arpan.ghosh@liu.se

We derive a one dimensional model of blood flow through an arbitrarily curved blood vessel having anisotropic, laminar and elastic wall structure. The blood vessel is assumed to have a circular cross section of varying radius along its length while having a general curvature and torsion for its given centre line. We formulate a suitable moving frame of reference in order to better suit the geometry in consideration to have simpler expressions. For modelling the wall, constitutive relations of elasticity and Newtons second law are used to obtain the partial differential equation system for the displacement of the wall material. A dynamic boundary condition and a kinematic no-slip boundary condition are assumed on the inner surface of the wall which help in coupling the equations with the Navier-Stokes equation governing the blood flow within the vessel. We assume the thickness of the wall of the vessel to be very small compared to the radius, while the radius is also taken to be small compared to the length of the vessel. Under such assumptions, we perform dimension reduction using asymptotic expansions to first obtain a two dimensional model for the wall. Using this as the boundary condition for the Navier-Stokes system, we perform dimension reduction again to obtain a one dimensional model for the blood flow.

Mathematical Imaging Methods for Mitosis Analysis in Live-Cell Phase Contrast Microscopy

Joana Grah

University of Cambridge, United Kingdom, jg704@cam.ac.uk

Mathematical image processing has recently become enormously important in the biomedical sciences. In particular, in the context of microscopy imaging, performance of technical equipment

(30)

is constantly improving and huge amounts of data can be acquired in very short periods of time. Manual analysis of this data is tedious and expensive, if not completely impossible. Hence, there is an urgent need for reliable, reasonably fast and fully automated image analysis tools.

In cancer research, observation of cell cultures in time-lapse live-cell imaging experiments is a key process in research and development of chemotherapy drugs. These so-called antimitotic drugs specifically target cells in the stage of division, i.e. mitosis. Since cancer cells undergo mitosis much more often and apoptosis, the process of programmed cell death helping to maintain a balanced number of cells in the body, much less often than healthy cells, abnormal mitotic behaviour can be a crucial indicator of successful treatment in drug studies.

Choosing the correct microscopy imaging modality to ensure that the cells behave as naturally as possible is essential. Therefore, we are focussing on phase contrast microscopy imaging, since it avoids staining as well as transgenic expression of fluorescent markers, both causing phototoxicity. However, due to the common halo- and shade-off-effect, phase contrast microscopy highly impedes image processing and standard algorithms cannot be applied straightforwardly.

In this talk, we introduce mathematical methods to tackle the challenges associated with mitosis image analysis and present an automated framework making them easily usable and applicable to real data.

Source Localization of Brain Tumors via Reaction-Diffusion Models Rym Jaroudi

Link/”oping University, Sweden, rym.jaroudi@liu.se

We present a well-established model of reaction-diffusion type for brain tumor growth, and give full 3-dimensional simulations of the tumor in time on two types of data, the 3d Shepp-Logan phantom and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repos-itory (IBSR). These simulations are obtained using standard finite difference discretisation of the space and time-derivatives, generating a simplistic approach that performs well. Moreover, we also discuss a mathematical method for the inverse problem of locating the brain tumor source (origin) based on the reaction-diffusion model. Our approach consists in recovering the initial spa-tial distribution of the tumor cells starting from a later state, which can be given by a medical image. We use a regularization method posing the inverse problem as a sequence of well-posed forward problems. Simulations with synthetic images show the accuracy of our approach for lo-cating brain tumor sources. This work is a joint collaboration with George Baravdish (Linkping University), B. Tomas Johansson (Linkping University) and Freddie strm (Heidelberg University).

Determining selection across heterogeneous landscapes: a perturbation-based method and its application to modeling evolution in space

Jonas Wickman

Ume/aa University, Sweden, jonas.wickman@umu.se

(31)

symmetrical dispersal between patches. For directional selection on a quantitative trait, this yields a way to integrate local directional selection across space and determine whether the trait value will increase or decrease. The robustness of this prediction is validated against quantitative genetics. For stabilizing/ disruptive selection, we show that spatial heterogeneity always contributes to dis-ruptive selection and hence always promotes evolutionary branching. The expression for directional selection is numerically very efficient, and hence lends itself to simulation studies of evolutionary community assembly.

The presented work is a joint collaboration with Sebastian Diehl, Bernd Blasius (Carl-v- Ossiet-zky University Oldenburg, Germany), Christopher Klausmeier (Michigan State University, USA), Alexey B. Ryabov (International Institute for Applied Systems Analysis (IIASA), Austria) and ˚Ake Br¨annstr¨om (Ume˚a University).

Abstracts for Thursday June 15

On the convergence of the Escalator Boxcar Train ˚

Ake Br¨annstr¨om

Ume/aa University, Sweden, ake.brannstrom@umu.se

The Escalator Boxcar Train (EBT) is a numerical method used in theoretical biology to inves-tigate the dynamics of physiologically structured population models. It works by discretizing the population into a finite number of cohorts whose dynamics are described by ordinary differential equations. The method was developed more than two decades ago, but had long resisted attempts to give a formal proof of convergence. Using a modern framework of measure-valued solutions, we investigate the EBT method and show that the sequence of approximating solution measures gen-erated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate.

This is joint work with Linus Carlsson and Daniel Simpson.

Statistical methods to estimate the size of the undiagnosed HIV-1 infected population Federica Giardina

Stockholm University, Sweden, federica@math.su.se

Knowledge of the size of the undiagnosed HIV-1 infected population is highly relevant for public health and HIV-1 prevention. In 2014, the Joint United Nations Programme on HIV and AIDS launched an international target that aims at reaching 90% of all people living with HIV diagnosed by 2020. In this talk, I will discuss the main challenges involved in the estimation of the undiagnosed HIV-1 population at the end of 2015 in Sweden, where HIV has been relatively stable over the past few years. I will also describe a stochastic SIR (Susceptible-Infected-Removed) model allowing for the presence of undiagnosed infections and discuss parameter identifiability at the end of an outbreak based on data availability (e.g. temporal data or sequence data).

(32)

Malaria model with asymptomatic class and superinfection Abid Ali Lashari

Stockholm University, Sweden, abid@math.su.se

In this talk, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If successfully treated, the symptomatic individuals recover and move to the susceptible class. The basic reproduction number, R0, is computed using the next generation approach. The system has a disease-free equilibrium (DFE) which is locally asymptomatically stable when R0 < 1, and may have up to four endemic equilibria. The model exhibits backward bifurcation generated by two mechanisms; standard in-cidence and superinfection. If the model does not allow for superinfection or deaths due to the disease, then DFE is globally stable which suggests that backward bifurcation is no longer possible. We also study optimal control strategies applied to bed-net use and treatment as main tools for reducing the total number of symptomatic and asymptomatic individuals.

Stochastic activation in a genetic switch model Alvaro Correales Fern´andez

ICMAT and UAM, Spain, alvaro.correales@icmat.es

Proteins are amazing biological entities. Among other important functions within cells, they can regulate the creation of other proteins and even of themselves. In the absence of noise, this autoregulation process can lead to bistability, but noise can induce transitions between the two states. We present a model in which a path integral formulation lead us to find the mean first jump time between this states, in total agreement with numerical simulations. In addition, the model exhibit bursting behaviour in the limit of short mRNA lifetime. This work has been done in collaboration with Joanna Tyrcha (Stockholm University) and John Hertz (Nordita).

(33)

Meeting of the Catalan, Spanish, and Swedish

Mathematical Societies (12–15 June, 2017, ˚

Umea,

Sweden)

Special Session on Algebraic Geometry and

Commutative Algebra

Editors: Alberto F. Boix and Julio–Jos´

e Moyano Fern´

andez

May 1, 2017

(34)

(35)

Foreword

This special session on Algebraic Geometry and Commutative Algebra aims to present the more recent developments on those subjects within the framework of a broader mathematical community on the ocassion of the joint meeting of the Catalan, Spanish, and Swedish Mathematical Societies. A selection of both young and experienced researchers has been chosen in order to give short talks concerning the following topics:

Algebraic curves and surfaces: plane curves, K3 and ruled surfaces Derived categories, D-modules and differential methods

Group-based phylogenetic models

Local and graded algebras: local cohomology, Hilbert functions, free resolu-tions, multiplier ideals

Maximal Cohen-Macaulay modules

Noetherian and Artinian modules and algebras Positive characteristic methods

Singularity theory

We would like to express our gratitude to the Scientific Committee for letting us the opportunity to organize this session, and to Milagros Izquierdo for all the support she gave us.

We wish to all participants a successful conference and a very pleasant stay in the city of ˚Umea.

Alberto F. Boix and Julio-Jos´e Moyano Fern´andez, ˚

Umea, June 2017

(36)

(37)

Kubjas) . . . 6 Finiteness properties of local cohomology modules (Leif Melkersson) . . . . 6 On the Xiao conjecture for families of plane curves (Joan–Carles Naranjo) 7 Hasse-Schmidt derivations versus classical derivations (Luis Narv´aez Macarro) 7 Spectral sequences in local cohomology (Santiago Zarzuela) . . . 8

(38)

(39)

Schedule

Tuesday Wednesday Thursday

2pm-2:30pm Kubjas Garc´ıa Marco De Stefani

2:30pm-3:00pm Fordham Homs Naranjo

3pm-3:30pm Costa Dachs Gonz´alez Alonso

3:30pm-4:00pm Coffee break Coffee break Coffee break 4pm-4:30pm Forsg˚ard Herbera

4:30pm-5:00pm Narv´aez Alvarez Montaner` 5pm-5:30pm Casta˜no Dom´ınguez Melkersson

5:30pm-6:00pm Banda Zarzuela

(40)

(41)

Abstracts

D–modules, Bernstein–Sato polynomials, and F –invariants

of direct summands

Josep `Alvarez Montaner

Universitat Polit`ecnica de Catalunya

In this work we study D-module structures over a ring that is a direct summand of the polynomial or the formal power series ring with coefficients over a field. We prove that localizations and local cohomology modules have finite length and we will show the existence of a Bernstein-Sato polynomial in this non-regular framework. Time permiting, we will turn our attention to some invariants of singularities in positive characteristic. We prove the discreteness and rationality of F-jumping numbers and we also extend some relations between F-thresholds and roots of the Bernstein-Sato polynomial.

This is joint work with C. Huneke and L. Núñez-Betancourt [ ÀMHNB].

Characterization of Half Exact Coherent Functors over

Principal Ideal Domains and Dedekind domains

Adson Banda

Link¨opings Universitet

We give necessary and sufficient conditions for a coherent functor over a principal ideal domain (PID) and over a Dedekind domain to be half exact. We show that every half exact coherent functor over a PID and more generally over a Dedekind domain arises from a complex of projective modules.

Under the assumption that A is noetherian commutative ring, we consider the exact sequence

F (A) ⊗ − α //F //F0 //0 (1)

and show that αM : F (A) ⊗ M → F (M ) is injective for any A–module M with

projective dimension at most 1. We then show that if F is a half exact coherent functor over a PID, then F0 in (1) is left exact. In this case F0 ∼= HomA(N, −) for

some finitely generated module N . We further show that over a PID, the sequence (1) splits, that is, F ∼= (F (A) ⊗ −) ⊕ F0. This shows that F arises from a complex

since both F (A) ⊗ − and Hom(N, −) arise from a complex of projective modules.

(42)

Irregular Hodge filtration of hypergeometric D-modules

Alberto Casta˜no Dom´ınguez

Technische Universit¨at Chemnitz

Mixed Hodge modules provide the correct framework to incorporate singularities, which arise usually in geometry, to the differential equations induced by variations of Hodge structures. Nevertheless, they are not the most general context since those singularities are always regular. Sabbah and Mochizuki have constructed an impressive extension of this setting, becoming the so-called theory of mixed twistor modules, allowing to introduce irregular singularities.

Recently, Sabbah (based on joint work with Esnault and Yu and ideas of Deligne and Katzarkov, Kontsevich and Pantev) introduced in [Sab] the so-called irregular Hodge filtration, allowing one to assign numerical invariants (i.e., irregular Hodge numbers) to certain mixed twistor modules. He has shown that all rigid irreducible D-modules on the projective line admit a unique irregular Hodge filtration provided that the formal local monodromies are unitary. Rigid D-modules are particularly interesting since they can be algorithmically constructed from simple objects by an algorithm due to Arinkin and Katz. Among the first and best understood examples of such rigid D-modules are the classical hypergeometric D-modules. In the regular case, Fedorov has recently given in [Fed] a closed formula for the Hodge numbers using the work of Dettweiler and Sabbah.

In this talk, we will report on how one can determine irregular Hodge numbers for a general irregular hypergeometric D-module. For a specific class formed by those modules with a purely irregular singularity, a direct calculation is possible and yields numbers related to other Hodge theoretic invariants such as Hodge spectra of non-degenerate Laurent polynomials. If time permits, we will also explain the possible strategies to achieve the general case.

The content of this talk is based on joint work [CDS] with Christian Sevenheck.

Ulrich bundles on ruled surfaces

Laura Costa

An Ulrich bundle on a smooth projective variety is a vector bundle that admits a completely linear resolution as a sheaf on the projective space. They appeared in commutative algebra, being associated to maximal Cohen Macaulay graded modules with maximal number of generators. In my talk, I’ll focus the attention on the existence of special rank two Ulrich bundles on ruled surfaces. This is based on joint work with Marian Aprodu and Rosa Maria Mir´o–Roig [ACMR].

(43)

Computing jumping numbers in higher dimensions

Ferran Dachs–Cadefau

Martin-Luther-Universit¨at Halle-Wittenberg

Multiplier ideals and jumping numbers are invariants that encode relevant informa-tion about the structure of the ideal to which they are associated. A first part of this talk will be devoted to introduce some basics about multiplier ideals in the case of 2-dimensional local rings.

In the second part of the talk, we will introduce some results for the multi-plier ideals in the higher-dimensional case. For this, we introduce the notion of π-antieffective divisors, a generalization of antinef divisors to higher dimensions. Using these divisors, we present a way to find a small subset of the classical candi-date jumping numbers of an ideal, containing all the jumping numbers. Moreover, many of these numbers are automatically jumping numbers, and in many other cases, it can be easily checked.

The presented results are part of a joint work with Hans Baumers [BDC].

Globalizing F-invariants

Alessandro De Stefani

KTH Royal Institute of Technology

The Hilbert-Kunz multiplicity and the F-signature are two important numerical invariants, defined for local rings of prime characteristic. They are subtly connected with the theory of singularities, and they often provide a good measure of how ill-behaved a ring can be. We will survey some classical results on the Hilbert-Kunz multiplicity, and we will discuss how to extend this notion to rings that are not necessarily local, in a way that still detects relevant information. This gives a possible way to meaningfully extend these concepts to more geometric objects, such as algebraic varieties. Time permitting, we will discuss analogous results for the F-signature. The talk is based on joint work with Thomas Polstra and Yongwei Yao [DSPY].

On the height of the formal group associated to a smooth

projective hypersurface

Stiof´ain Fordham

University College Dublin

Given a smooth projective hypersurface over a finite field, we can associate to the defining polynomial a descending chain of ideals after `Alvarez-Montaner–Blickle– Lyubeznik [AMBL05]. Then, a result of Boix–De Stefani–Vanzo [BDSV15] related the point at which the chain stabilizes to the height of the associated formal group in the case of an elliptic curve. I will describe the interpretation of their result using Frobenius splitting methods, and work in progress in extending their result to the case of Calabi-Yau hypersurfaces.

(44)

Singularities of Discriminants

Jens Forsg˚ard

Texas A&M University

We will discuss singularities of A-discriminants in terms of the Horn–Kapranov uniformization. Applications to the study of dual defect toric varieties will be given. This is joint work with J. Maurice Rojas.

Noether resolutions in dimension 2

Ignacio Garc´ıa-Marco

Aix-Marseille Universit´e

Let R := K[x1, . . . , xn] be a polynomial ring over an infinite field K, and let I ⊂

R be a homogeneous ideal with respect to a weight vector ω = (ω1, . . . , ωn) ∈

(Z+)n such that dim(R/I) = d. In this work we study the minimal graded free resolution of R/I as A-module, that we call the Noether resolution of R/I, whenever A := K[xn−d+1, . . . , xn] is a Noether normalization of R/I. When d = 2 and I

is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gr¨obner basis of I with respect to the weighted degree reverse lexicographic order. In the particular case when R/I is a 2-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semigroup. Whenever we have the Noether resolution of R/I or its multigraded version, we obtain formulas for the corresponding Hilbert series of R/I, and when I is homogeneous, we obtain a formula for the Castelnuovo-Mumford regularity of R/I.

As an application of the results for 2-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution and the Macaulayfication of either the coordinate ring of a projective monomial curve C ⊆ PnK associated to an arithmetic sequence or the coordinate ring of any

canonical projection πr(C) of C to Pn−1K .

All the results that will be presented in this talk are based on joint work with Isabel Bermejo, Eva Garc´ıa Llorente, and Marcel Morales [BGLGMM].

(45)

Entropy of automorphisms of supersingular K3 surfaces

V´ıctor Gonz´alez–Alonso

Leibniz Universit¨at Hannover

It is known that the topological entropy of an automorphism of an algebraic surface is either zero or the logarithm of a Salem number, but not much is known about which Salem numbers can actually be realized. In this talk I will present a joint work with Simon Brandhorst [BGA] in the case of supersingular K3 surfaces, giving a method to construct an automorphism with given entropy and showing in particular which minimal Salem numbers occur in this case.

A construction of big Maximal Cohen Macaulay modules

Dolors Herbera

Universitat Aut`onoma de Barcelona

A module is pure projective if it is isomorphic to a direct summand of a direct sum of finitely presented modules. There is a big amount of work done on the study of finitely generated (hence, finitely presented) modules over commutative noetherian rings, but not so much is known on direct summands of infinite direct sums of such modules. With our work we want to give some insight on this problem.

Let R be a local commutative noetherian ring. If R is complete then all pure projective modules are direct sums of finitely generated modules, but we give plenty of examples showing that this is far from true in the non-complete case. However there is still a close relation between pure-projective R-modules and pure projective

ˆ

R-modules, as we can prove that two pure-projective modules P and Q are isomor-phic as R-modules if and only if P ⊗RR ∼ˆ = Q ⊗RR as ˆˆ R-modules. The proof of

such result is modeled on Pˇr´ıhoda’s one that two arbitrary projective modules are isomorphic if and only if they are isomorphic modulo the Jacobson radical [Pˇ07].

A rather difficult question is to determine which pure-projective ˆR-modules are extended from R-modules. We will present an answer to this problem when R is a one dimensional domain and for direct summands of an arbitrary direct sum of copies of a single finitely generated R-module. Our techniques to do that use heavily the results in [HP10] and one of our main source of examples is [Wie01].

The results that will be presented are part of an ongoing long joint project with Pavel Pˇr´ıhoda and Roger Wiegand.

(46)

On low Gorenstein colength of Artin local rings

Roser Homs

I will introduce the notion of Gorenstein colength of an Artin local k-algebras and how to compute it. I will give a complete characterization of k-algebras of colengths 0, 1 and 2 in terms of its Macaulay inverse system and discuss the problem for higher colengths. This is a joint work with Joan Elias [EH].

Maximum likelihood geometry for group-based phylogenetic

models

Kaie Kubjas

Aalto–yliopisto

Based on Matsen’s work on inequalities for group-based phylogenetic models in Fourier coordinates, we study polynomial equations and inequalities that cut out group-based models in original coordinates. We apply this knowledge to the study of boundaries and maximum likelihood estimation on group-based models. In par-ticular, we use the degree of an algebraic variety and numerical algebraic geometry to obtain the maximum likelihood estimate exactly for small group-based models. This talk is based on joint work with Dimitra Kosta [KK].

Finiteness properties of local cohomology modules

Leif Melkersson

Link¨opings Universitet

I will give a survey of finiteness properties of local cohomology modules and discuss some topics related to this as finiteness of Ext and Tor-modules. I will also discuss some open problems in the area. If time permits I will prove some new results of mine.

(47)

On the Xiao conjecture for families of plane curves

Joan–Carles Naranjo

Xiao’s conjecture deals with the relation between the natural invariants present on a fibred surface f : S → B: the irregularity q of S, the genus b of the base curve B and of the genus g of the fibre of f . In a paper with M.A. Barja and V. Gonz´ alez-Alonso [BGAN] we have proved the inequality q − b ≤ g − c, where c is the Clifford index of the generic fibre. This gives in particular a proof of the (modified) Xiao’s conjecture, q − b ≤ g/2 + 1, for fibrations whose general fibres have maximal Clifford index. In this talk we we will report on recent progress on the Xiao’s conjecture for fibrations whose generic fibre is a plane curve. More precisely we will focus on a joint work with F. Favale and G. P. Pirola [FNP] where we have proved the conjecture for quintic plane curves, we also have proved the inequality q − b ≤ g − c − 1 for plane cuves of any degree greater than or equal to 5.

Hasse-Schmidt derivations versus classical derivations

Luis Narv´aez Macarro

Universidad de Sevilla

In this talk we will review on the notion of Hasse–Schmidt derivation of a commu-tative algebra, and we will see how these natural objects allow us to define good differential smoothness properties and to describe rings of differential operators in any characteristic (equal or unequal), and modules over them. On the other hand, the notion of Hasse–Schmidt derivation gives rise to a notion of integrability for (usual) derivations. This property always hold in characteristic zero. So, one can expect that integrability should play a role in the different behavior of singularities in characteristic zero and in positive characteristic.

The content of this talk is based on [NM12], [NM09], [FLNM05] and [FLNM03].

(48)

Spectral sequences in local cohomology

Santiago Zarzuela

Spectral sequences are often used to compute local cohomology functors. In this talk I’m going to review how to use them in order to calculate local cohomology from the primary decomposition of an ideal I in a commutative Noetherian ring R. In the homological case, we shall deal with the computation of several general-ized local cohomology functors supported on I. In the cohomological case we shall mainly be concerned with the computation of the local cohomology of R = I. The construction of these spectral sequences is done by means of the computation of the left and right derived functors of the direct and inverse limits in terms of the homol-ogy (or cohomolhomol-ogy) of a particular explicit complex, that we call homological (or cohomological) Roos complex. In each case, one can also give sufficient conditions in order to guarantee the degeneration of the corresponding spectral sequence. As a guiding cases we have in mind the results obtained by `Alvarez-Garc´ıa- Zarzuela [ `AMGLZA03] and G. Lyubeznik [Lyu07] in the homological case.

The content of this talk is based on a joint work in progress with Josep ` Alvarez-Montaner and Alberto F. Boix [ `AMBZ].

(49)

Bibliography

[ACMR] M. Aprodu, L. Costa, and R-M. Mir´o-Roig. Ulrich bundles on ruled surfaces. Available at https://arxiv.org/pdf/1609.08340.pdf.

[AMBL05] J. Alvarez-Montaner, M. Blickle, and G. Lyubeznik. Generators of D-modules in positive characteristic. Math. Res. Lett., 12(4):459–473, 2005.

[ `AMBZ] J. `Alvarez Montaner, A. F. Boix, and S. Zarzuela. On some local cohomology spectral sequences. In preparation.

[ ÀMGLZA03] J. Àlvarez Montaner, R. Garc´ıa López, and S. Zarzuela Armengou. Local cohomology, arrangements of subspaces and monomial ideals. Adv. Math., 174(1):35–56, 2003.

[ ÀMHNB] J. Àlvarez Montaner, C. Huneke, and L. Núñez-Betancourt. D-modules, Bernstein-Sato polynomials and F-invariants of direct sum-mands. Available at https://arxiv.org/pdf/1611.04412.pdf.

[BDC] H. Baumers and F. Dachs-Cadefau. Computing jumping numbers in higher dimensions. Available at https://arxiv.org/pdf/1603. 00787.pdf.

[BDSV15] A. F. Boix, A. De Stefani, and D. Vanzo. An algorithm for construct-ing certain differential operators in positive characteristic. Matem-atiche (Catania), 70(1):239–271, 2015.

[BGA] S. Brandhorst and V. Gonz´alez-Alonso. Automorphisms of minimal entropy on supersingular K3 surfaces. Available at https://arxiv. org/pdf/1609.02716.pdf.

[BGAN] M. A. Barja, V. Gonz´alez-Alonso, and J. C. Naranjo. Xiao’s conjec-ture for general fibred surfaces. To appear in J. reine angew. Math. Available at https://arxiv.org/pdf/1401.7502.pdf.

[BGLGMM] I. Bermejo, E. Garc´ıa-Llorente, I. Garc´ıa-Marco, and M. Morales. Noether resolutions in dimension 2. To appear in J. of Algebra, available at https://arxiv.org/pdf/1704.01777.pdf.

[CDS] A. Casta˜no Dom´ınguez and C. Sevenheck. Irregular Hodge filtration of confluent hypergeometric systems. In preparation.

(50)

[DSPY] A. De Stefani, T. Polstra, and Y. Yao. Globalizing F-invariants. Available at https://arxiv.org/pdf/1608.08580.pdf.

[EH] J. Elias and R. Homs. On the Gorenstein colength. In preparation.

[Fed] R. Fedorov. Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles. Available at https://arxiv.org/pdf/1505.01704.

[FLNM03] M. Fernández-Lebrón and L. Narváez-Macarro. Hasse-Schmidt derivations and coefficient fields in positive characteristics. J. Al-gebra, 265(1):200–210, 2003.

[FLNM05] M. Fernández-Lebrón and L. Narváez-Macarro. Coefficient fields and scalar extension in positive characteristic. J. Algebra, 285(2):819–834, 2005.

[FNP] F. F. Favale, J. C. Naranjo, and G. P. Pirola. On the Xiao conjec-ture for plane curves. Available at https://arxiv.org/pdf/1703. 07173.pdf.

[HP10] D. Herbera and P. Pˇr´ıhoda. Big projective modules over noetherian semilocal rings. J. Reine Angew. Math., 648:111–148, 2010.

[KK] D. Kosta and K. Kubjas. Maximum likelihood geometry for group-based phylogenetic models. In preparation.

[Lyu07] G. Lyubeznik. On some local cohomology modules. Adv. Math., 213(2):621–643, 2007.

[NM09] L. Narv´aez Macarro. Hasse-Schmidt derivations, divided powers and differential smoothness. Ann. Inst. Fourier (Grenoble), 59(7):2979– 3014, 2009.

[NM12] L. Narv´aez Macarro. On the modules of m-integrable derivations in non-zero characteristic. Adv. Math., 229(5):2712–2740, 2012.

[Pˇ07] P. Pˇr´ıhoda. Projective modules are determined by their radical fac-tors. J. Pure Appl. Algebra, 210(3):827–835, 2007.

[Sab] C. Sabbah. Exponential-Hodge theory. Available at https://arxiv. org/pdf/1511.00176.

[Wie01] R. Wiegand. Direct-sum decompositions over local rings. J. Algebra, 240(1):83–97, 2001.

(51)

On the height of a smooth projective hypersurface

Stiof´ain Fordham

University College Dublin

(52)

Set-up

k perfect field of characteristic p > 2. R := k[x0, . . . , xn].

(53)

Level of X

Following Boix–di Stefani–Vanzo: for integer e ≥ 0 let Ie = Ie(f ) := smallest ideal J ⊂ R s.t. (fp

e₋₁

) ⊂ J[pe], where J[pe]:= {xpe

: x ∈ J}.

Obtain descending chain of ideals:

R = I0 ⊃ I1⊃ · · · ⊃ In⊃ . . .

that stabilises rigidly. Definition

Thelevelof X denoted `(X ) is the least integer n > 0 such that I_n−1 = In.

(54)

Level of X

e₋₁

: x ∈ J}. Obtain descending chain of ideals:

R = I0 ⊃ I1⊃ · · · ⊃ In⊃ . . .

that stabilises rigidly.

Definition

(55)

Level of X

e₋₁

: x ∈ J}. Obtain descending chain of ideals:

R = I0 ⊃ I1⊃ · · · ⊃ In⊃ . . .

that stabilises rigidly. Definition

(56)

Geometric interpretation of level

An elliptic curve X isordinaryif X [p] ∼= Z/pZ and supersingular otherwise.

Theorem (Boix–di Stefani–Vanzo, 2015) If X is an elliptic curve then

`(X ) = (

1 if X is ordinary, 2 if X is supersingular.

Proof.

Delicate check of possible generators of I1 and I2.

(57)

Geometric interpretation of level

`(X ) = (

1 if X is ordinary, 2 if X is supersingular. Proof.

Delicate check of possible generators of I1 and I2.

Objective: go beyond the case of elliptic curves.

(58)

Geometric interpretation of level

`(X ) = (

1 if X is ordinary, 2 if X is supersingular.

(59)

Level and differential operators

For a ring S let DS denote thering of differential operatorson S ,

e.g. for S = k[x0, . . . , xn] then DS is free S -algebra on generators

Di ,t = 1 t! ∂t ∂xt i , i = 0, 1, . . . , n, t = 1, 2, . . . ,

(what is 1/t! when t ≥ p? Formally

Di ,t(xis) := ( _s tx s−t i t ≤ s, 0 otherwise. So it ‘acts like’ _t!1_∂x∂tt i . Note that ∂t ∂xt i

≡ 0 for t ≥ p but the Di ,t are

never identically zero.)

(60)

Level and differential operators

Di ,t = 1 t! ∂t ∂xt i , i = 0, 1, . . . , n, t = 1, 2, . . . , (what is 1/t! when t ≥ p? Formally Di ,t(xis) := ( _s tx s−t i t ≤ s, 0 otherwise. So it ‘acts like’ _t!1_∂x∂tt i . Note that ∂t ∂xt i

(61)

Level and differential operators

Di ,t = 1 t! ∂t ∂xt i , i = 0, 1, . . . , n, t = 1, 2, . . . , (what is 1/t! when t ≥ p? Formally

Di ,t(xis) := ( _s tx s−t i t ≤ s, 0 otherwise. So it ‘acts like’ _t!1_∂x∂tt i . Note that _∂x∂tt i

never identically zero.)

(62)

Level and differential operators

Di ,t = 1 t! ∂t ∂xt i , i = 0, 1, . . . , n, t = 1, 2, . . . , (what is 1/t! when t ≥ p? Formally

Di ,t(xis) := ( _s tx s−t i t ≤ s, 0 otherwise.

(63)

Level and differential operators

Filtration: D_S(e) = h{Di ,t}i =1,...,n: t ≤ pe− 1iS for e = 1, 2, . . . .

Proposition (`Alvarez-Montaner–Blickle–Lyubeznik, 2005) Have Ie−1= Ie iff there exists δ ∈ D_S(e) with δ(fp

e₋₁

) = fpe−p. So:

level 1 ⇔ ∃ δ ∈ D_R(1) with δ(fp−1) = 1 and level 2 ⇔ ∃ δ0 ∈ D_R(2) with δ0(fp2−1) = fp2−p. Example

Let f = x0+ x1 then a choice of above operators δ = D0,p−1 and

δ0 = (p − 1)!(p

2_{− p)!}

(p2_{− 1)!} D0,p−1,

(symmetry in f means could swap D1,p−1 for D0,p−1 in δ, δ0).

(64)

Level and differential operators

Filtration: D_S(e) = h{Di ,t}i =1,...,n: t ≤ pe− 1iS for e = 1, 2, . . . . Proposition (`Alvarez-Montaner–Blickle–Lyubeznik, 2005) Have Ie−1= Ie iff there exists δ ∈ D_S(e) with δ(fp

e₋₁

) = fpe−p.

So:

level 1 ⇔ ∃ δ ∈ D_R(1) with δ(fp−1) = 1 and level 2 ⇔ ∃ δ0 ∈ D_R(2) with δ0(fp2−1) = fp2−p. Example

δ0 = (p − 1)!(p

2_{− p)!}

(p2_{− 1)!} D0,p−1,

(65)

Level and differential operators

e₋₁

) = fpe−p. So:

level 1 ⇔ ∃ δ ∈ D_R(1) with δ(fp−1_{) = 1 and}

level 2 ⇔ ∃ δ0 ∈ D_R(2) with δ0(fp2−1) = fp2−p.

Example

δ0 = (p − 1)!(p

2_{− p)!}

(p2_{− 1)!} D0,p−1,

(symmetry in f means could swap D1,p−1 for D0,p−1 in δ, δ0).

(66)

Level and differential operators

e₋₁

) = fpe−p. So:

level 1 ⇔ ∃ δ ∈ D_R(1) with δ(fp−1_{) = 1 and}

level 2 ⇔ ∃ δ0 ∈ D_R(2) with δ0(fp2−1) = fp2−p. Example

(67)

Connection with Frobenius splitting

Let F : R → R be the Frobenius morphism. Write F∗R for R with R-module structure

r1· r2= r1pr2,

so F : R → F∗R is an R-module homomorphism.

(68)

Frobenius splitting

Seminal paper of Mehta–Ramanathan, 1985. Definition

Say that R isFrobenius-splitor F -split if F : R → F∗R is split i.e.

there exists R-module homomorphism φ : F∗R → R with

φ ◦ F = id.

Globalise: Definition

(69)

Frobenius splitting

Seminal paper of Mehta–Ramanathan, 1985. Definition

Say that R isFrobenius-splitor F -split if F : R → F∗R is split i.e.

there exists R-module homomorphism φ : F∗R → R with

φ ◦ F = id. Globalise: Definition

A variety X isF -split if F : OX → F∗OX is split.

(70)

Frobenius splitting

Theorem

Let X = V (f ) ⊂ Pn be smooth and deg f = n + 1. Then `(X ) = 1 iff X is ordinary.

Remark

Ordinary: F acts non-trivially on Hn(X , OX). Corollary

A hyperelliptic curve of genus > 1 never has level 1.

If d = deg f < n + 1 and p > n then there exists δ ∈ D_R(2) with δ(fp2−1) = fp2−p and there is an algorithm to construct δ with complexity O(dp2).

(71)

Frobenius splitting

Theorem

Remark

Ordinary: F acts non-trivially on Hn(X , OX).

Corollary

(72)

Frobenius splitting

Theorem

Remark

(73)

Frobenius splitting

Theorem

Remark

Meeting of the Catalan, Spanish, Swedish Math Societies (CAT‐SP‐SW‐MATH)