I - 34151 Trieste Italy

*(+39) 040 2240 111*

*pio@ictp.it*

- Joint ICTP/SISSA PhD Programme in Physics and Mathematics
- Joint PhD Programme, Earth Science and Fluid Mechanics
- Physics PhD Program
- Joint Masters in Physics
- Joint ICTP/Collegio Carlo Alberto Program in Economics
- International Master, Physics of Complex Systems
- Master of Advanced Studies in Medical Physics
- Masters in High Performance Computing

Contacts:

Since 1986 the Mathematics section at ICTP has played an important role in fostering mathematics research and education in developing countries. Research is carried out in various fields of Mathematics by the permanent staff, postdocs, and graduate students, as well as by scientific visitors from all over the world.

Typically, the section organizes from 5 to 10 focused activities a year involving an average of 100 participants. These activities are the core of the section's activities and are crucial for disseminating current mathematics knowledge of the highest level as widely as possible.

In addition the Mathematics section, like all the other sections at ICTP, participates in the Diploma program. Since 2011 Diploma students can apply to stay on to work on a PhD in Mathematics in a joint program with SISSA.

The Mathematics section also offers opportunities for postdocs and research fellows; click here for latest announcements.

Once a month, the section organises The Basic Notions Seminar Series to broaden the understanding of some mathematical concepts.

1 Jun 2018

James Yorke, father of chaos, on the usefulness of defining disorder

31 Jan 2018

Two recipients honored for dedication to Abdus Salam’s vision

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Europe/Rome
2018-06-19 16:00:00
2018-06-19 17:00:00
SPECIALIZED SEMINAR: Exceptional splitting of reductions of abelian surfaces with real multiplication
ICTP
ICTP
pio@ictp.it
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19 Jun 2018

» SPECIALIZED SEMINAR: Exceptional splitting of reductions of abelian surfaces with real multiplication

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Europe/Rome
2018-07-03 16:00:00
2018-07-03 17:00:00
Regularity of spherical maximal functions
Abstract:
Fractional integrals and the classical fractional maximal function are smoothing operators, in the sense that they map Lebesgue spaces into first order Sobolev spaces. We show that this phenomenon continues to hold for the fractional spherical maximal function when the dimension of the ambient space is greater than or equal to 5. A key element in the proof is a local smoothing estimate for the wave equation.
This is joint work with Joao P. G. Ramos and Olli Saari.
ICTP
ICTP
pio@ictp.it
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3 Jul 2018

» Regularity of spherical maximal functions

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Europe/Rome
2018-08-02 14:30:00
2018-08-02 15:30:00
Smooth invariant manifolds of renormalization
Abstract: Topological classes are often invariant manifolds of renormalization. Unfortunately, renormalization is not a differentiable dynamical system. The stable manifold theorem for hyperbolic dynamics can not be applied. The talk will discuss a technique to construct smooth invariant manifolds in such a non-differentiable context.
ICTP
ICTP
pio@ictp.it
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2 Aug 2018

» Smooth invariant manifolds of renormalization

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Europe/Rome
2018-06-18 08:00:00
2018-06-22 22:00:00
Summer School on Geometry of Moduli Spaces of Curves | (smr 3215)
Moduli spaces of stable pointed curves play an important role in algebraic geometry. This School will have one course on vector bundles of coinvariants and on conformal blocks and another one on their cohomology classes in relation with those of moduli of abelian varieties. The cohomology of moduli spaces of curves and abelian varieties carries several natural classes. We focus on the tautological classes and the cohomology classes related to spaces of modular forms. The problem of determining relationships between the tautological classes turns out to be particularly interesting. Moduli spaces of curves carry vector bundles of coinvariants and conformal blocks; they are invariants of a curve C attached to a Lie group G that are canonically isomorphic to global sections of an ample line bundle on the moduli stack of certain G-bundles on C. These are generalized theta functions in case C is smooth. In case g=0, the bundles of co-invariants are globally generated, and their first Chern classes are semi-ample line bundles on the moduli of curves, and shed light on its birational geometry. We can also use the moduli space of curves to learn about generalized theta functions.
TOPICS:
Cohomology classes on moduli of curves and abelian varieties
moduli spaces of curves and abelian varieties
properties of the moduli spaces and their compactifications
natural cohomology classes and their relations
tautological classes and cohomology classes related to spaces of modular forms
Vector bundles of coinvariants and conformal blocks
introduction to moduli spaces of curves
open problems and F-conjecture
vector bundles of coinvariants and conformal blocks
case g=0: global generation and semi-ample divisors
any genus: nef divisors
Chern classes of bundles of coinvariants
Global sections of ample line bundles on Bun_G(C): smooth and nodal case
PARTICIPATION:
Women are particularly encouraged to apply.
Should you come to Trieste with your child(ren), please send an e-mail to smr3215@ictp.it to describe your family needs and we will do our best to meet them.
A limited number of grants are available to support the attendance of selected participants, with priority given to participants from developing countries. There is no registration fee.
To apply, please use the link on the left side of this web page.
The deadline for submitting applications expired on 15 March 2018.
ICTP
ICTP
pio@ictp.it
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18 Jun 2018
- 22 Jun 2018

» Summer School on Geometry of Moduli Spaces of Curves | (smr 3215)

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Europe/Rome
2018-07-09 08:00:00
2018-07-28 22:00:00
EAUMP-ICTP School and Workshop on Homological Methods in Algebra and Geometry II | (smr 3219)
This is a three-week school and workshop on homological methods in algebra and geometry. The first two weeks will be a school for students from East Africa and beyond. International and African researchers will join for a workshop in the third week.
The school begins with two introductory courses on some basic techniques that are widely used in the area. These are then built upon in the second week with slightly more advanced topics. The goal here is to give the participants a glimpse into some ideas used in algebraic geometry and homological algebra with the hope of inspiring them to pursue further research in a related topic.
The school is intended for advanced graduate students (M.Sc. and Ph.D.) and young academic staff members from East Africa and beyond. Senior academic members are also welcome to participate.
In the last week, the activity turns into a research focus-session involving international and African experts. The focus-session will study tensor categories and how they arise naturally in algebraic geometry with the aim of exploiting them using homological algebra.
School Website: eaumpictp2018.weebly.com
School Topics:
Advanced linear algebra
Galois theory
Elementary algebraic geometry
Introduction to homological algebra
Workshop topic:
Tensor categories of coherent sheaves
Organizers:
Tarig Abdelgadir (UNSW Sydney)
Ulrich Krähmer (TU Dresden)
Sylvester Rugeihyamu (Dar Es Salaam)
David Ssevviiri (Makerere)
Balázs Szendrői (Oxford)
Fernando Rodriguez Villegas (ICTP)
School speakers:
Ravi Ramakrishna (Cornell)
David Ssevviiri (Makerere)
Chelsea Walton (Temple)
Paul Wedrich (ANU Canberra)
Michael Wemyss (Glasgow)
Workshop speakers:
Raf Bocklandt (Amsterdam)
John Boiquaye (Accra)
Andre Saint-Eudes Mialebama Bouesso (AIMS-South Africa)
Alexandru Chirvasitu (Boulder)
Joshua Greene (Boston College)
Pinhas Grossman (UNSW Sydney)
Yujiro Kawamata (Tokyo)
Shinnosuke Okawa (Osaka)
Sue Sierra (Edinburgh)
Hermann Sore (Burkina Faso)
Angela Tabiri (Glasgow)
Ralph Twum (Accra)
ICTP Scientific Contact: L. GOETTSCHE (ICTP)
Application is open to all mathematicians and graduate students from developing countries. Applicants from EAUMP member states will be given priority. We encourage participants to secure their own funding for travel and subsistence from their home institution. Limited funds are available for participants from Sub- Saharan African Countries. There is no registration fee.
Deadline: 1st of April 2018
Activity e-mail: smr3219@ictp.it
Dar-es-Salaam - United Republic of Tanzania
ICTP
pio@ictp.it
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9 Jul 2018
- 28 Jul 2018

» EAUMP-ICTP School and Workshop on Homological Methods in Algebra and Geometry II | (smr 3219)

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Europe/Rome
2018-07-09 08:00:00
2018-07-20 22:00:00
ICTP-CIMPA School: AGRA III (Aritmética, Grupos y Analisis III) | (smr 3222)
ICTP-CIMPA School: AGRA III (Aritmetica, Grupos y Analisis III)
from the 9 - 20 July 2018 in Cordoba, Argentina
to held at the Academia Nacional de Ciencias and at FaMAF
Please note: Arrival date should be 8 July 2018 and
Departure date should be 21 July 2018
For Schedule and further information please refer to following link:
http://www.famaf.unc.edu.ar/agra3/
http://www.anc-argentina.org.ar/web/actividades/bienvenida
Deadline for requesting participation: 4 MARCH 2018
The area of the School is number theory, broadly understood - analytic, algebraic, combinatiorial, with links to groups and geometry.
All of the course topics lie at thematic intersections.
The course on Galois representations will focuse on their connections with modular forms and elliptic curves.
The course on arithmetic groups will involve familiarizing students with topics in group theory and modular forms.
Equidistribution in a diophantine context will be the main subject of a course.
Another lecture series will focus on analysis, combinatorics and discrete geometry.
The study of curves over finite fields, and the resulting codes, will allow for an accessible introduction to deep issues in algebraic geometry, with immediate applications.
Finally, the course on primes, parity and analysis will combine classical tools and the use of entropy and independence.
The plan is, then, to introduce advanced students to a variety of fields and tools and to notable recent developments.
Scienfific Committee:
Michael Harris
Harald Helfgott
Roberto Miatello
Nuria Vila Oliva
Fernando Rodriguez Villegas
Local Committee:
Maria Chara
Emilio Lauret
Ariel Pacetti
Ricardo Podesta
Diego Sulca
Invited Speakers:
Miram Abdon, Universidade Federal Fluminense, Brazil
Mikhail Belolipetsky, IMPA - Instit. Nac. de Matematica Pura e Aplicada, Brazil
José Burgos Gil, ICMAT, Spain
Luis Dieulefait, Universitat de Barcelona, Spain
Cicero Fernandes De Carvalho, Universidade Federal de Uberlandia, Brazil
Michael Harris, Université Paris VII, France & Columbia University, U.S.A.
Harald Helfgott, Georg-August Universität Göttingen & CNRS - Centre National
de la Recherche Scientifique & Université Paris VI/VII, France
Marc Hindry, Univesité Paris VII, France
Juan Rivera-Letelier, University of Rochester, U.S.A.
Benjamin Linowitz, Oberlin College, Ohio, U.S.A.
Ricardo Menares, PUCV - Pontificia Universidad Católica de Valparaíso, Chile
Roberto Miatello, Universidad Nacional de Córdoba, Argentina
Ariel Pacetti, Universidad Nacional de Córdoba, Argentina
Daniel Panario, Carleton University, Canada
Marusia Rebolledo, Université Blaise Pascal Clermont-Ferrand 2, France
David Roberts, University of Minnesota, Morris, U.S.A.
Fernando Rodriguez Villegas, ICTP - the Abdus Salam International Centre for
Theoretical Physics, Italy
Adrián Ubis Martínez, Universidad Autonoma de Madrid, Spain
Co-sponsors:
Academia Nacional de Ciencias
Alexander von Humboldt
Centre International de Mathematiques Pures et Appliques (CIMPA)
Facultad de Matematica, Astronomia, Fisica y Computacion (FAMAF)
Foundation Compositio Mathematica
Universidad Nacional de Cordoba (UNC)
Cordoba - Argentina
ICTP
pio@ictp.it
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9 Jul 2018
- 20 Jul 2018

» ICTP-CIMPA School: AGRA III (Aritmética, Grupos y Analisis III) | (smr 3222)

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Europe/Rome
2018-07-16 08:00:00
2018-07-27 22:00:00
Summer School in Dynamics (Introductory and Advanced) | (smr 3226)
PARTICIPANTS WISHING TO TAKE PART ONLY IN THE ADVANCED SCHOOL (SECOND WEEK) SHOULD NOT APPLY FROM THIS PAGE BUT SHOULD APPLY HERE: http://indico.ictp.it/event/8467/
Participants wishing to participate for both weeks can apply through the link available on the sidebar of this page.
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DEADLINE for applicants requesting Financial Support: 2 APRIL 2018
DEADLINE for applicants NOT requesting Financial Support: 25 JUNE 2018
Organizers and Lecturers:
J. Rodriguez-Hertz, SUS Tech., Shenzan, China
C. Ulcigrai, Univ. Bristol, U.K.
A. Wilkinson, Univ. Chicago, U.S.A.
Local Organizer:
S. Luzzatto, ICTP, Trieste, Italy
Week 1
A circle of concepts and methods in dynamics.
Basic concepts in dynamics will be introduced, with many examples, especially in the setting of circle maps. Topics include rotations of the circle, doubling map, Gauss map and continued fractions and an introduction to the basic ideas of symbolic codings and invariant measures. At the end of the week we will discuss some simple examples of structural stability and renormalization.
Week 2
Ergodicity in smooth dynamics (10h, Jana Rodriguez-Hertz and Amie Wilkinson)
The concept of ergodicity is a central hypothesis in statistical mechanics, one whose origins can be traced to Boltzmann's study of ideal gases in the 19th century. Loosely speaking, a dynamical system is ergodic if it does not contain any proper subsystem, where the notion of "proper" is defined using measures. A powerful theorem of Birkhoff from the 1930's states that ergodicity is equivalent to the property that "time averages = space averages:" that is, the average value of a function taken along an orbit is the same as the average value over the entire space. The property of ergodicity is the first stepping stone in a path through the study of statistical properties of dynamical systems, a field known as Ergodic Theory.
We will develop the ergodic theory of smooth dynamical systems, starting with the fundamental, linear examples of rotations and doubling maps on the circle introduced in Week 1. We will develop some tools necessary to establish ergodicity of nonlinear smooth systems, such as those investigated by Boltzmann and Poincaré in the dawn of the subject of Dynamical Systems. Among these tools are distortion estimates, density points, invariant foliations and absolute continuity. Closer to the end of the course, we will focus on the ergodic theory of Anosov diffeomorphisms, an important family of "toy models" of chaotic dynamical systems.
Renormalization in entropy zero systems (5h, Corinna Ulcigrai)
Rotations of the circle are perhaps the most basic examples of low complexity (or "entropy zero") dynamical systems. A key idea to study systems with low complexity is renormalization. The Gauss map and continued fractions can be seen as a tool to renormalize rotations, i.e.study the behaviour of a rotation on finer and finer scales. We will see two more examples of renormalization in action.
The first is the characterization of Sturmian sequences, which arise as symbolic coding of trajectories of rotations (and hint at more recent developments, such as the characterization of cutting sequences for billiards in the regular octagon). The second concerns interval exchange maps (IETs), which are generalizations of rotations. We will introduce the Rauzy-Veech algorithm as a tool to renormalize IETs. As applications, we will give some ideas of how it can be used (in some simplified settings) to study invariant measures and (unique) ergodicity and deviations of ergodic averages for IETs.
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Tutorial and exercise sessions will be held regularly and constitute an essential part of the school.
Tutors: Oliver BUTTERLEY (ICTP), Irene PASQUINELLI (Durham University, UK), Davide RAVOTTI, (University of Bristol, UK), Lucia SIMONELLI (ICTP), Kadim WAR (Ruhr-Universität, Bochum, Germany).
Women in Mathematics: Activities directed to encourage and support women in mathematics, such as panel discussions and small groups mentoring and networking, will be organized during the event.
ICTP
ICTP
pio@ictp.it
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16 Jul 2018
- 27 Jul 2018

» Summer School in Dynamics (Introductory and Advanced) | (smr 3226)

Every year the Mathematics Group offers research opportunities for outstanding mathematicians from developing countries, for short and long-term visits, as well as postdoctoral fellowships, through the Mathematics Research Fellowships programme. A call for postdoctoral fellowships with a starting date of September 2018 is now open. The application deadline is 7 January 7 2018. Other research and education opportunities in mathematics are available through the ICTP's Postgraduate Diploma Programme as well as its Associateship Scheme.

Differential Geometry

Claudio Arezzo studies the geometry of complex algebraic varieties using techniques from analysis and differential geometry.

Algebraic Geometry

Lothar Goettsche is interested in moduli spaces in algebraic geometry and interactions with theoretical physics.

Dynamical Systems

Stefano Luzzatto studies dynamical systems and is particularly concerned with the geometric and statistical properties of chaotic systems as well as the use of rigorous use of numerical techniques to study them.

Number Theory

Fernando Rodriguez Villegas has research interests in representation theory, combinatorics and symmetric functions as well as computational algebra and number theory.

ICTP is governed by UNESCO, IAEA, and Italy, and is a UNESCO Category 1 Institute

http://library.ictp.it/ejds.aspx