I - 34151 Trieste Italy

*(+39) 040 2240 111*

*pio@ictp.it*

- Conferences, workshops and schools
- Junior Associates
- Regular Associates
- Senior Associates
- Federated Institutes
- The Kuwait Programme at ICTP
- APS-EPS-ICTP Travel Award Fellowship Programme (ATAP)
- CIMPA-ICTP Research in Pairs for mathematicians in developing countries
- Researchers Mobility for Sustainable Development

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.

12 May 2022

ICTP celebrates International Year of Basic Sciences for Sustainable Development

23 Mar 2022

Don Zagier named the first holder of ICTP's Ramanujan International Chair

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Europe/Rome
2022-11-30 11:00:00
2022-11-30 12:00:00
On highly degenerate Alt-Phillips problems
Hybrid seminar
Venue: for in-person attendees (Leonardo da Vinci, Lecture Room D).
For virtual attendees:
https://zoom.us/meeting/register/tJwtceytpjwuEtHlba3sEx-YuHV74KjK6RsW
After registering, you will receive a confirmation email containing information about joining the meeting.
Abstract: In this talk, we study regularity properties for nonvariational singular elliptic equations ruled by highly degenerate second-order operators. We obtain optimal regularity along the free boundary. We also show existence of solutions, nondegeneracy properties, and fine geometric estimates for the free boundary.
This is joint work with Ginaldo Sá - UFPB.
Hybrid seminar
ICTP
pio@ictp.it
`
30 Nov 2022

» On highly degenerate Alt-Phillips problems

`
Europe/Rome
2022-12-05 10:00:00
2022-12-06 15:00:00
A Proof of the P=W Conjecture
This short series of four lectures by T. Hausel (IST, Austria) and A. Mellit (Univ. of Vienna) will discuss the recent proof of the P=W conjecture in the theory of the moduli space of Higgs bundles on a curve.
The following seminars will be given on Monday 5 December and Tuesday 6 December in the IFPU Seminar Room (room number 205) (old SISSA Building):
Monday
10:00 - 11:00 Introduction to P=W and Mirror symmetry - T. Hausel
14:00 - 15:00 P=W via H_2 (part 1) - A. Mellit
Tuesday
10:00 - 11:00 P=W via H_2 (part 2) - A. Mellit
14:00 - 15:00 Mirror Symmetry and big algebras - T. Hausel
You are all welcome to join.
____________________
Abstracts
Tamas Hausel
Introduction to P=W and mirror symmetry
Motivated by topological mirror symmetry for Langlands dual Higgs bundle moduli spaces, one can study the arithmetic of the corresponding character varieties to gain information on the weight filtration on their cohomology. In turn this lead to the P=W conjecture which identifies the weight filtration on the cohomology of the character variety with the perverse filtration of the Hitchin system on the cohomology of the Higgs moduli space.
Mirror symmetry and big algebras
First we discuss the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we explain a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety.
Anton Mellit
P=W via H_2
By H_2 we denote the Lie algebra of polynomial hamiltonian vector fields on the plane. We consider the moduli space of stable twisted Higgs bundles on an algebraic curve of given coprime rank and degree. De Cataldo, Hausel and Migliorini proved in the case of rank 2 and conjectured in arbitrary rank that two natural filtrations on the cohomology of the moduli space coincide. One is the weight filtration W coming from the Betti realization, and the other one is the perverse filtration P induced by the Hitchin map. Motivated by computations of the Khovanov-Rozansky homology of links by Gorsky, Hogancamp and myself, we look for an action of H_2 on the cohomology of the moduli space. We find it in the algebra generated by two kinds of natural operations: on the one hand we have the operations of cup product by tautological classes, and on the other hand we have the Hecke operators acting via certain correspondences. We then show that both P
and W coincide with the filtration canonically associated to the sl_2 subalgebra of H_2.
ICTP
pio@ictp.it
`
5 Dec 2022
- 6 Dec 2022

» A Proof of the P=W Conjecture

`
Europe/Rome
2022-12-05 14:00:00
2022-12-08 16:00:00
CIMPA/ICTP Course on "Introduction to Optimization and Operator Theory"
This is a recurring meeting, therefore please register just once for all 4 sessions:
https://zoom.us/meeting/register/tJUodOihqDMvE9fOe6s-MziaNCKeZeoaVSDt
Venue (for in-person attendees):
Luigi Stasi Seminar Room
Timetable:
Monday 5 December: 14:00 - 16:00
Tuesday 6 December: 14:00 - 16:00
Wednesday 7 December: 14:00 - 16:00
Thursday 8 December: 14:00 - 16:00
<span lang="EN-US" style="font-size:13.0pt;font-family:
"Gill Sans MT";mso-fareast-font-family:"Malgun Gothic"">Abstract: <span lang="EN-US" style="font-size:13.0pt;font-family:
"Gill Sans MT";mso-fareast-font-family:"Malgun Gothic"">The lecture presents and analyzes the theory and numerical methods of nonsmooth optimization problems, and is discussed at the master’s or beginning of Ph.D. study level. The lecture can be viewed as an introduction to Nonsmooth Optimization and Operator Theory based on the books [1, 2, 3]. It is oriented in viewing the special cases of the Optimization Problems which I observed through my research work. Rather than consisting of a series of results, this lecture is intended to be didactic by focusing on the motivation of concepts and presenting examples and counter- examples. If this lecture has awakened the curiosity of the students or has improved his/her understanding of nonsmooth optimization problems, then they have achieved their objective.
<span lang="EN-US" style="font-size:13.0pt;font-family:
"Gill Sans MT";mso-fareast-font-family:"Malgun Gothic"">
<span lang="EN-US" style="font-size:13.0pt;font-family:
"Gill Sans MT";mso-fareast-font-family:"Malgun Gothic"">[1] Bauschke, H.H., Combettes, P.L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces, vol. 408. Springer, New York
[2] Bagirov, A. M., Gaudioso, M., Karmitsa, N., Mkel, M. M., Taheri, S. (Eds.). (2020). Numerical nonsmooth optimization: State of the art algorithms.
[3] Bagirov, A., Karmitsa, N., Mkel, M. M. (2014). Introduction to Nonsmooth Optimization: theory, practice and software. Springer.
Hybrid
ICTP
pio@ictp.it
`
5 Dec 2022
- 8 Dec 2022

» CIMPA/ICTP Course on "Introduction to Optimization and Operator Theory"

`
Europe/Rome
2022-12-09 16:00:00
2022-12-09 17:30:00
Basic Notions Seminar - What is a moduli space?
Venue: for in-person attendees (Leonardo da Vinci, Luigi Stasi Seminar Room).
For virtual attendees:
https://zoom.us/meeting/register/tJ0od-ihrTIjH9F3K0rVN45vAwYwC2diqLtL
After registering, you will receive a confirmation email containing information about joining the meeting.
Abstract: There is a beautiful connection between the prime numbers and the zeros of Riemann zeta-function in the complex plane, and understanding the distribution of these zeros can lead to remarkable consequences in our understanding of the primes. A famous open problem, known as the Riemann Hypothesis, states that the non-real zeros of the Riemann zeta-function lie on a vertical line. I will motivate this conjecture, describe some evidence for it, and describe some of the consequences it would lead to in our understanding of the primes. In particular, I will describe how to bound on the maximum gap between prime numbers assuming the Riemann Hypothesis (which is based on joint work with Carneiro and Soundararajan).
hybrid
ICTP
pio@ictp.it
`
9 Dec 2022

» Basic Notions Seminar - What is a moduli space?

`
Europe/Rome
2023-02-27 07:00:00
2023-03-03 21:00:00
Learning Workshop on BPS States and 3-Manifolds | (smr 3818)
The main scientific theme of the workshop is counting BPS states in physical systems associated to 3-manifolds or links, and interpretation of these counts as old or new topological invariants.
Although the ideas behind such relations appeared already in the late 90s, the recent years have seen some dramatic new developments. The goal of this learning workshop is to bring together physicists and mathematicians who are experts in the subject and also those who would like to enter the field. In particular, the talks will be given both by senior scientists and junior ones, including PhD students who only recently began working on the subject. The talks will be supplemented by dedicated discussion sessions where experts can answer questions that will be submitted by participants to the moderator in advance, or asked directly during the session.
Topics:
"Z-hat" and F_K invariants counting BPS states in 6d superconformal field theory compactified on 3- manifolds, and their relation to known invariants of 3-manifolds and links
Knots-quivers correspondence and its generalization to knot complements
Properties of generating functions counting BPS states, including: modularity, relation to characters of (logarithmic) Vertex Operator Algebras, GW/DT invariants
Speakers:
R. AKHMECHET, Columbia University
*M. CHENG, University of Amsterdam
H-J. CHUNG, Jeju National University
F. COSTANTINO, Toulouse University
T. EKHOLM, Uppsala University
B. HAGHIGHAT, Tsinghua University
S. HARICHURN, Bonn University
J. JANKOWSKI, Wrocław University
P. JOHNSON, Michigan State University
S. KRUSHKAL, University of Virginia
H. LARRAGUIVEL, Jagiellonian University
P. LONGHI, Uppsala University
D. NOSHCHENKO, University of Amsterdam
S. PARK, University of Texas at Austin
D. PASSARO, University of Amsterdam
*D. PEI, University of Southern Denmark
P. RAMADEVI, Indian Institute of Technology, Mumbai
S.J. RI, ICTP
M. STOSIC, Instituto Superior Técnico
P. SULKOWSKI, Warsaw University
J. SVOBODA, University of Miami
*C. VAFA, Harvard University
* to be confirmed
Grants: 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.
ICTP
ICTP
pio@ictp.it
`
27 Feb 2023
- 3 Mar 2023

» Learning Workshop on BPS States and 3-Manifolds | (smr 3818)

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. Other research and education opportunities in mathematics are available through the ICTP's Postgraduate Diploma Programme as well as its Associateship Scheme.

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

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