There are many mathematical research institutes worldwide. Some are conference centers or simply departments by another name, but most run semester-long periods of concentration in specific areas. They often give support to young researchers and graduate students.
The Fields Institute also has a List of Institutes
* Supports research in pairs or teams. Some institutes require participation of a researcher from the home country.
North America
AIM*, Caltech
BIRS*, Banff, Canada
BIRS-UBCO*, Kelowna, BC, Canada
Brin Mathematics Research Center*, University of Maryland
CIMAT, Guanajuato, Mexico
Clay Mathematics Institute, Cambridge, Massachusetts
CRM, Montreal
CMO, Oaxaca, Mexico
DIMACS, Rutgers. Discrete math and computer science
Fields Institute, Toronto
Institute for Advanced Study, Princeton
ICERM*, Brown University
IMA, University of Minnesota. Mostly for applied work
IMSI, Chicago
IPAM, UCLA
MBI, Ohio State. Mathematical biosciences
NITMB, Northwestern and University of Chicago, Mathematical biology
Norbert Wiener Center for Harmonic Analysis and Applications, University of Maryland
Park City Math Institute, Utah
PIMS, Vancouver and at nearby universities
Simons Center for Geometry and Physics, SUNY Stony Brook
Simons Institute for the Theory of Computing*, Berkeley
SLMath*, (formerly MSRI), Berkeley
Europe
Banach Center, Warsaw
Bernoulli Center, Lausanne, Switzerland
CIRM*, Luminy, France
CIRM*, Trento, Italy
CRM, Barcelona
De Giorgi Center*, Pisa
Erdos Center, Budapest
Erwin Schrodinger Institute*, Vienna
Euler Institute, St. Petersburg
Eurandom), Eindhoven, Netherlands. Stochastics.
FIM, Zurich
Hamilton Institute, Maynooth, Ireland
Hamilton Mathematics Institute, Dublin
Hausdorff Center, Bonn
Hausdorff Institute, Bonn
Heilbronn Institute, Bristol
IACM-FORTH, Crete
ICMAT, Madrid
ICMS*, Edinburgh
IHES, Paris
IMAG, Granada
Institute Henri Poincare*, Paris
IIAS*, Israel Inst. for Adv. Studies, Jerusalem
Lebesgue Center, Rennes, France
Max Planck Institute, Bonn
Mittag-Leffler Institute, Stockholm
Newton Institute, Cambridge, UK
Oberwolfach*, Germany
Rényi Institute/Erdős Center, Budapest
RISM, Varese
Saint Flour summer school in probability. France
Turing Institute, London. Data science
Asia and Australia
AMSI, Distributed, Australia
BISMA, Beijing
CIM, Nankai
CMI, Chennai
IASM], Hangzhou
IBS-CGP*, Pohang, Korea
IMS, Singapore
Indian Statistical Institute, (several branches, do a search)
KIAS*, Seoul
MATRIX, Near Melbourne
Morningside Center, Beijing
NIMS, Daejeon, Korea
RIMS*, Kyoto University
Tata Institute, India (click on Faculties/Centres)
VIASM, Hanoi
YMSC, Tsinghua University
South America
IMPA, Rio