Martin Hairer has won the 2021 Breakthrough Prize in Mathematics. The announcement is here. The official citation is: *For transformative contributions to the theory of stochastic analysis, particularly the theory of regularity structures in stochastic partial differential equations*. His long paper in *Inventiones Mathematicae *is a remarkable and thorough exposition of his work on stochastic PDEs, which has applications in many areas of both pure and applied mathematics. Our review of this work is at the end of this post.

The announcement of the Breakthrough Prize is accompanied by the announcements of some affiliated prizes.

### The New Horizons in Mathematics Prize

For 2021, there are three winners:

**Bhargav Bhatt**(University of Michigan)

Citation: For outstanding work in commutative algebra and arithmetic algebraic geometry, particularly on the development of p-adic cohomology theories.**Aleksandr Logunov**(Princeton University)

Citation: For novel techniques to study solutions to elliptic equations, and their application to long-standing problems in nodal geometry.**Song Sun**(University of California, Berkeley)

Citation: For many groundbreaking contributions to complex differential geometry, including existence results for Kahler-Einstein metrics and connections with moduli questions and singularities.

### Maryam Mirzakhani New Frontiers Prize

This is the first year for the Maryam Mirzakhani New Frontiers Prize, which had three winners. One of the winners, Piccirillo, was written up in Quanta Magazine (and several other places) recently for her amazing result in knot theory.

**Nina Holden**, ETH Zurich (PhD MIT 2018)

Citation: For work in random geometry, particularly on Liouville Quantum Gravity as a scaling limit of random triangulations.**Urmila Mahadev**, Caltech (PhD University of California, Berkeley 2018)

Citation: For work that addresses the fundamental question of verifying the output of a quantum computation.**Lisa Piccirillo**, Massachusetts Institute of Technology (PhD University of Texas at Austin 2019)

Citation: For resolving the classic problem that the Conway knot is not smoothly slice.

**MR3274562**

Hairer, M.(4-WARW)

A theory of regularity structures. (English summary)

*Invent. Math.* 198 (2014), no. 2, 269–504.

60H15 (35R60 60H40 81S20 82C28)

The theory of regularity structures has its roots in the theory of rough paths for SPDEs and represents its generalization to higher dimensions. It is a novel theory developed by the author, for which he was awarded the Fields Medal in 2014. This paper represents a complete self-contained survey on the theory, on its background, motivations, applications and interconnection with other theories. Written in a lucid style with perfect exposition and well-structured into ten sections, it may serve as a book as well.

The purpose of regularity structures is to build a framework that allows one to solve and study the solutions to semilinear SPDEs with highly irregular random input. It results in an algebraic framework that allows one to describe functions and generalized functions by a kind of Taylor expansion around each point in space-time. The main novel idea is to replace the classical polynomial model by arbitrary models that involve other functions and that are purpose-built for the problem at hand. A regularity structure is thus a triplet $(A,T, G)$ where $A\subset \bf R$ denotes an appropriate index set, $T$ denotes the model space which is a graded vector space of the form $T=\bigoplus_{\alpha\in A}T_\alpha$ whose elements describe the local expansion of a function at any given point, and $G$ denotes a group of linear operators acting on $T$ which help to translate the coefficient of a local expansion around a given point into coefficients of an expansion around a different point.

The next step is to build a calculus incorporating the operations of multiplication, composition with smooth functions and integration against singular kernels. Concerning the problem of multiplication of generalized functions, the author provides deep insights, as well as a connection to alternative theories: Colombeau’s algebras of generalized functions, Hida’s white noise theory, Bony’s paraproduct and Lyons’ rough path theory are reflected within the ideas of regularity structures.

In order to solve SPDEs, one has to find an appropriate regularity structure that is rich enough to allow the formulation of a fixed point problem associated to the SPDE. By solving this fixed point problem one arrives at an abstract solution map to the original equation which can also be interpreted as a limit of classical solutions to regularized problems, usually modified by the addition of diverging counterterms that arise through the action of a renormalization group.

Thus, the theory of regularity structures makes it possible to give a pathwise meaning to ill-posed SPDEs that appear when describing the macroscopic behaviour of physical models near a critical point. One of the reasons why the theory is efficient at providing precise small-scale features of solutions to semilinear SPDEs is that it comes with very sharp Schauder estimates.

Some of the examples that illustrate the theory include the continuous parabolic Anderson model, the KPZ-type equations, the Navier-Stokes equation with singular forcing, stochastic quantization equations, etc. As a novel application the author solves the long-standing problem of building a natural Markov process that is symmetric with respect to the finite volume measure describing the $\Phi_3^4$ Euclidean quantum field theory.

Reviewed by Dora Seleši

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