I am interested in how multiplication symbols came about, and where different ones were used. When were $x\cdot y$ and $xy$ first used and by whom?
Apparently, early Babylonian tablets used an ideogram for multiplication (depicted here). But "Babylonian scribes did not use the ‘times’ symbol every time they did a calculation. Instead, they just drew a vertical line on their clay tablet, putting numbers to be multiplied on the left of the line and the answers on the right". Diophantus (c.250 AD) follows suit without even a line, he simply writes the result. What of juxtaposition? Generally he uses it for addition rather than multiplication, but he does write coefficients next to power symbols, and means that the two are multiplied. This convention remains in some 15th century European manuscripts.
The earliest known Indian mathematical work, the so-called Bakhshali manuscript (date estimates range from 300 BC to 400 AD), uses juxtaposition for multiplication. Bhaskara I (c.650 AD) usually also juxtaposes factors, but occasionally inserts a dot between them. There is no explanation of the usage in the text however, and he has an alternative notation that does not function like an operation. In Europe Stifel (1545) used $M$ between factors, Vieta $in$, and Oughtred (1631) the familiar St. Andrews cross $\times$. Modern juxtaposition convention became standard after Descartes's Geometry (1637).
Cajori's History of Mathematical Notations details many such issues. It is in the public domain and is available online.