60: Probability theory,... There is even more movement in the stochastic PDE community with Hairer,s regularity structures. The groups appearing there are character groups of certain Hopf algebras and there is the massive work by Bruned, Hairer and Zambotti to uncover the algebraic framework, leading them to certain Hopf algebras: Algebraic renormalisatiom of regularity structures arxiv:1610.08468.

Also the whole topic of rough paths (Now Msc2020 60Lxx) is quite connected, as rough paths can be viewed as paths again in character groups of certain Hopf algebras. See e.g.

What does the group action of a rough path in a Lie group look like?

This can be found In most modern treatments in the guise of dealing with the tensor algebra and shuffle products. Some modern more algebraic developments are also included in the works of Ebrahimi-Fard, Manchon et al.

22Exx Infinite-dimensional Lie theory, turns out that character groups of Hopf algebras are often infinite-dimensional Lie groups (this provides Lie group structures for many well-known examples, such as the Butcher group from numerical analysis):

• Character groups of Hopf algebras as infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), 66 no. 5 (2016), p. 2101-2155 arXiv:1501.05221
• Lie groups of controlled characters of combinatorial Hopf algebras, arXiv:1609.02044
• The geometry of characters of Hopf algebras, Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13 DOI: 10.1007/978-3-030-01593-0_6, arXiv:1704.01099

60: Probability theory,... There is even more movement in the stochastic PDE community with Hairer,s regularity structures. The groups appearing there are character groups of certain Hopf algebras and there is the massive work by Bruned, Hairer and Zambotti to uncover the algebraic framework, leading them to certain Hopf algebras: Algebraic renormalisatiom of regularity structures arxiv:1610.08468.

Also the whole topic of rough paths (Now Msc2020 60Lxx) is quite connected, as rough paths can be viewed as paths again in character groups of certain Hopf algebras. See e.g.

What does the group action of a rough path in a Lie group look like?

This can be found In most modern treatments in the guise of dealing with the tensor algebra and shuffle products. Some modern more algebraic developments are also included in the works of Ebrahimi-Fard, Manchon et al.

22Exx Infinite-dimensional Lie theory, turns out that character groups of Hopf algebras are often infinite-dimensional Lie groups (this provides Lie group structures for many well-known examples, such as the Butcher group from numerical analysis):

• Character groups of Hopf algebras as infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), 66 no. 5 (2016), p. 2101-2155 arXiv:1501.05221
• Lie groups of controlled characters of combinatorial Hopf algebras, arXiv:1609.02044
• The geometry of characters of Hopf algebras, Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13 DOI: 10.1007/978-3-030-01593-0_6, arXiv:1704.01099

60: Probability theory,... There is even more movement in the stochastic PDE community with Hairer,s regularity structures. The groups appearing there are character groups of certain Hopf algebras and there is the massive work by Bruned, Hairer and Zambotti to uncover the algebraic framework, leading them to certain Hopf algebras: Algebraic renormalisatiom of regularity structures arxiv:1610.08468.

Also the whole topic of rough paths (Now Msc2020 60Lxx) is quite connected, as rough paths can be viewed as paths again in character groups of certain Hopf algebras. See e.g.

What does the group action of a rough path in a Lie group look like?

This can be found In most modern treatments in the guise of dealing with the tensor algebra and shuffle products. Some modern more algebraic developments are also included in the works of Ebrahimi-Fard, Manchon et al.

60: Probability theory,... There is even more movement in the stochastic PDE community with Hairer,s regularity structures. The groups appearing there are character groups of certain Hopf algebras and there is the massive work by Bruned, Hairer and Zambotti to uncover the algebraic framework, leading them to certain Hopf algebras: Algebraic renormalisatiom of regularity structures arxiv:1610.08468.

Also the whole topic of rough paths (Now Msc2020 60Lxx) is quite connected, as rough paths can be viewed as paths again in character groups of certain Hopf algebras. See e.g.

What does the group action of a rough path in a Lie group look like?

This can be found In most modern treatments in the guise of dealing with the tensor algebra and shuffle products. Some modern more algebraic developments are also included in the works of Ebrahimi-Fard, Manchon et al.

22Exx Infinite-dimensional Lie theory, turns out that character groups of Hopf algebras are often infinite-dimensional Lie groups (this provides Lie group structures for many well-known examples, such as the Butcher group from numerical analysis):

• Character groups of Hopf algebras as infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), 66 no. 5 (2016), p. 2101-2155 arXiv:1501.05221
• Lie groups of controlled characters of combinatorial Hopf algebras, arXiv:1609.02044
• The geometry of characters of Hopf algebras, Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13 DOI: 10.1007/978-3-030-01593-0_6, arXiv:1704.01099