मठ
Mathas
The monastic seat a family owes allegiance to. Where it applies it divides families more sharply than region or surname does: Vaishnava households of the Kashi and Partagali seats and Smarta households of Kavale may share a village and a surname and still keep separate ritual. Plenty of families name no matha at all.
4 recorded in the archive

Shri Kashi Math Samsthan
501 peopleVaranasi, with branches at Walkeshwar (Mumbai), Haldipur, Basrur and Cochin · vaishnava · Dvaita/Vaishnava lineage established under Shrimad Yadavendra Tirtha. Followers are often called Kashi Math Samsthan GSBs.
Photograph Sntshkumar750, CC BY-SA 4.0

Shri Chitrapur Math
372 peopleShirali, Uttara Kannada, Karnataka · smarta · Guru parampara begins with Shrimat Parijnanashram Swami I. The spiritual home of the Chitrapur Saraswats (Bhanaps).
Photograph Nkodikal, CC BY 3.0

Shri Gokarn Partagali Jeevottam Math
361 peoplePartagali, Canacona, Goa · vaishnava · Founded by Shrimad Narayana Teertha. A principal Vaishnava seat of the Goud Saraswat community.
Photograph Pranavhegde006, CC BY-SA 4.0

Shri Saunsthan Gaudapadacharya, Kavale
295 peopleKavale, Ponda, Goa · smarta · Traditionally regarded as the oldest matha of the Saraswats, tracing to Shri Gaudapadacharya. Smarta / Advaita.
Photograph Ashok Prabhu ( talk ), Public domain