Let
(
M
,
g
)
(M,g)
be a compact (in general, nonorientable) surface with boundary
∂
M
\partial M
and let
Γ
0
\Gamma _0
, …,
Γ
m
−
1
\Gamma _{m-1}
be connected components of
∂
M
\partial M
. Let
u
=
u
f
(
x
)
u=u^{f}(x)
be a solution to the problem
Δ
g
u
=
0
\Delta _{g}u=0
in
M
M
,
u
|
Γ
0
=
f
u\big |_{\Gamma _0}=f
,
u
|
Γ
j
=
0
u\big |_{\Gamma _j}=0
,
j
=
1
j=1
, …,
m
′
m’
,
∂
ν
u
|
Γ
j
=
0
\partial _{\nu }u\big |_{\Gamma _j}=0
,
j
=
m
′
+
1
j=m’+1
, …,
m
−
1
m-1
, where
ν
\nu
is the outward normal. With this problem, one associates the DN map
Λ
:
f
↦
∂
ν
u
f
|
Γ
0
\Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0}
. The purpose is to determine
M
M
from
Λ
\Lambda
.
To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra
A
\mathfrak {A}
of functions holomorphic on the appropriate orientable double cover of
M
M
. It is proved that
A
\mathfrak {A}
is determined by
Λ
\Lambda
up to isometric isomorphism. The spectrum of the algebra
A
\mathfrak {A}
provides a relevant copy
M
′
M’
of
M
M
. This copy is conformally equivalent to
M
M
while its DN map coincides with
Λ
\Lambda
.