The 14-karat gold medal, which is worth about €3,600, was left in his briefcase on top of a table.
The briefcase was stolen and later found without the medal, Rio news agency G1 reported, which later said the thief was identified by surveillance video.
Birkar is an Iranian refugee who has been living in the UK for over a decade. He is a major contributor to modern birational geometry, having previously received various maths awards.
The Fields Medal is presented every four years to up to four mathematicians under the age of 40.
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