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Register a new userGalois groups as quotients of Polish groups
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We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_sigma$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $emptyset$. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Andand Pillay, which says that for any strong type defined on a single complete type over $emptyset$, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the aforementioned paper about bounded quotients of type-definable subgroups of definable groups.
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We revisit Kolchins results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in CODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.
We give a detailed proof of Kolchins results on differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. We closely follow former works due to Pillay and his co-authors which were written under the assumption that the field of constant is algebraically closed. In the present setting, which encompasses the cases of ordered or p-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in the theory of closed ordered fields, we establish a relative Galois correspondence for definable subgroups of the group of differential order automorphisms.
Let $n, k geq 3$. In this paper, we analyse the quotient group $B_n/Gamma_k(P_n)$ of the Artin braid group $B_n$ by the subgroup $Gamma_k(P_n)$ belonging to the lower central series of the Artin pure braid group $P_n$. We prove that it is an almost-crystallographic group. We then focus more specifically on the case $k=3$. If $n geq 5$, and if $tau in N$ is such that $gcd(tau, 6) = 1$, we show that $B_n/Gamma_3 (P_n)$ possesses torsion $tau$ if and only if $S_n$ does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order $tau$ in $B_n/Gamma_3 (P_n)$ with those of elements of order $tau$ in the symmetric group $S_n$. We also exhibit a presentation for the almost-crystallographic group $B_n/Gamma_3 (P_n)$. Finally, we obtain some $4$-dimensional almost-Bieberbach subgroups of $B_3/Gamma_3 (P_3)$, we explain how to obtain almost-Bieberbach subgroups of $B_4/Gamma_3(P_4)$ and $B_3/Gamma_4(P_3)$, and we exhibit explicit elements of order $5$ in $B_5/Gamma_3 (P_5)$.
The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives information on its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations.
For a finite-index $mathrm{II}_1$ subfactor $N subset M$, we prove the existence of a universal Hopf $ast$-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call this Hopf $ast$-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.
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