The work of Wendelin Werner

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Laudatio on the occasion of the Field medal award to Wendelin Werner.
Explosion Group action Diffuser (automotive)
Brownian motion Numerical analysis Planar graph
Mathematics Fields Medal Probability theory Physicalism Mathematical physics Cartesian coordinate system Mathematician Fundamental theorem of algebra Statistical physics Directed graph
Randomization Multiplication sign Mereology Dimensional analysis Mathematical structure Explosion Mathematics Sign (mathematics) Plane (geometry) Lattice (group) Different (Kate Ryan album) Series (mathematics) Physical system Social class Area Potenz <Mathematik> Curve Physicalism 3 (number) Two-dimensional space Riemann surface Surface of revolution Probability theory Connected space Proof theory Category of being Binary tree Arithmetic mean Hexagon Configuration space Summierbarkeit Figurate number Resultant Directed graph Geometry Point (geometry) Connectivity (graph theory) Continuum hypothesis Random matrix Graph coloring Theory Kritisches Phänomen Power (physics) Product (business) Gaussian elimination Frequency Kritischer Punkt <Mathematik> Natural number Set theory Stochastic process Percolation Fields Medal Exponentiation Mathematical analysis Algebraic structure Independence (probability theory) Limit (category theory) Evolute Cartesian coordinate system Numerical analysis Planar graph Diameter Local ring
Point (geometry) Complex (psychology) Functional (mathematics) Scaling (geometry) Cohen's kappa Correspondence (mathematics) Curve Two-dimensional space Parameter (computer programming) Cartesian coordinate system Evolute Time domain Brownian motion Event horizon Derivation (linguistics) Frequency Formal power series Plane (geometry) Spherical cap Algebraic closure Well-formed formula Term (mathematics) Boundary value problem Quantum gravity
Randomization Potenz <Mathematik> Scaling (geometry) Model theory Physicalism Two-dimensional space Dimensional analysis Kritisches Phänomen Power (physics) Category of being Kritischer Exponent Kritischer Punkt <Mathematik> Thermal fluctuations Universe (mathematics) Integer Vapor pressure Resultant Physical system
Randomization Interior (topology) Correspondence (mathematics) Multiplication sign Parameter (computer programming) Open set Dimensional analysis Explosion Inference Plane (geometry) Lattice (group) Different (Kate Ryan album) Cluster sampling Perimeter Physical system Social class Area Potenz <Mathematik> Theory of relativity Random walk Sampling (statistics) Physicalism Two-dimensional space Riemann surface Surface of revolution Measurement Time domain Category of being Proof theory Conformal field theory Resultant Spacetime Geometry Classical physics Functional (mathematics) Observational study Probability theory Parity (mathematics) Connectivity (graph theory) Continuum hypothesis Latent heat Well-formed formula Interface (chemistry) Boundary value problem Symmetric matrix Set theory Addition Focus (optics) Scaling (geometry) Percolation Model theory Mathematical analysis Basis <Mathematik> Limit (category theory) Planar graph Formal power series Loop (music) Network topology Combinatory logic Object (grammar) Intercept theorem
Predictability Group action Potenz <Mathematik> Scaling (geometry) Percolation Variance Physicalism Parameter (computer programming) Limit (category theory) Derivation (linguistics) Proof theory Well-formed formula Interface (chemistry) Boundary value problem Family Resultant
Area Distribution (mathematics) Focus (optics) Percolation Theory of relativity Process (computing) Probability theory Model theory Continuum hypothesis Physicalism Two-dimensional space Calculus Mereology Brownian motion Diameter Formal power series Category of being Mathematics Object (grammar) Analytic continuation Curve fitting
Universe (mathematics) Model theory Continuum hypothesis Mathematical analysis Limit (category theory)
or that he'll for diffuse Medal and deadly by Charles Newman Grimsley Hewitt of Mathematical Sciences USA which groups it is my great
pleasure to briefly report
on some of mandolin burners research accomplishments there a number of aspects of vendors work better my pleasure and then 1 of his trainers a probable receiving his Ph.D. in 1993 under the supervision of and swallowed Garland Paris with the dissertation concerning planar Brownian motion which as we shall see plays a major
role in his later work as well until Laos
probability theory has not been represented among fields mess and so I'm enormously pleased to
be here to witness change in that history indeed probability quite well represented a new word that precisely and I myself was originally trained that probability theory and mathematical physics Mariners work together with his collaborators such as drug Lawler of that charm fast involves applications of probability and formal met in very fundamental problems in statistical physics as we shall discuss a 2nd source of my pleasure is the belief that this together with other work of recent years represents a watershed in the interaction between mathematics and physics general namely mathematicians such as burner and that only providing rigorous
proofs of already existing claim physics literature but more reportedly providing White conceptual understanding of basic phenomena in this case a direct geometric picture of the intrinsically random structure of physical systems at critical points at least in 2 dimensions once 1 simple but important examples population here is a simulation of critical percolation configuration and a portion of the triangular lattice corresponding this figure To uniformly random to coloring the hexagon we'll discuss percolation more later particularly interested in discussing things like scaling limits in which the size of a lattice here the diameter of the individual had guns goes
0 and 1 is interested in various geometric properties of the classes here just is connected blue or connected the yellow components were connection means against the same color sharing sharing an edge there's a small cluster 2 yellow Texaco and surrounded by a larger like a larger number of blacks and will get back to that White permit a somewhat more personal remark as director of the Pratt Institute for the past 4 years we have a scientific viewpoint as their predecessors instituting during namely that important goal should be the elimination of artificial distinctions between the mathematical sciences and their applications and other sciences I believe that if and when work lives up to that philosophy as the other work that is being recognized today the yet a the 3rd source of pleasure concerns the collaborative nature of much of his work beautiful and productive mathematics can be the result of many different personal work style but a highly interactive style which burner together with smaller Schramm Minnesota collaborators leading them learn appeals to many of us simultaneously good for the soul while leading to work stronger than the sum of its parts is a promising sign to see Fields Medal awarded for a new style of your theory of probability theory which most strongly interact with statistical physics and that involving stochastic process these nontrivial special structure this area which also interacts with advanced communications theory theoretical computer science and other fields His combined interesting applications with first-class mathematics recent developments however and raise perceived mathematical status the best work From merely first-class 2 outstanding let me begin by mentioning 2 pieces of murders were from 1998 to 2000 these are not only intrinsic significance but also were precursors the breakthroughs about happened in understanding 2 dimensional critical systems with natural conformal invariant reports other significant precursors such as path approach the scathing limits and canyons work and we braced walks and Domino Thailand's songs some of them were related work jointly already mentioned the 1st of the 2 pieces of work is a 1998 papered over Holland House and burner the motivation was to construct a continuum version of earlier where this true self what this led To make quite beautiful mathematical structure an extended version of mostly unpublished in nearly forgotten construction that almost 20 years earlier by rat but coalescing in reflecting one-dimensional Bermuda has running forward and backward in time filling up all of 2 dimensional space time there is a plane feeling curve within the structure that is analogous to 1 that arises limits of uniformly random any which will mention again later and it was 1 the trams motivations In 2000 paper introducing SLD as as many of you know is an acronym for what was originally called to cast lot more evolution and is now called the Schramm revolution will discuss a little bit more about that shortly before the 2nd piece of work from this period consists of 2 papers with red in 1999 and 2000 involving binary planar Brown and intersection exponents and actually given the mission of from struck late in the 2nd of these shows that the same set of X province must occur providing only at certain locality conformal invariant properties are valid this was a key idea which combined with the introduction of only the analysis of two-dimensional critical phenomena led to a remarkable series of 3 papers in 2001 and 2002 by Lawler Schramm and burner which yielded a whole series of intersection extra 1 example public W 1 Derby 2 etc. The Independent cleaner-burning motion starting from distinct points time chief 0 consider the probability that end the 1st evidence of these emotions of cent of them observed up until time are all this joint that probability decays as he tends to infinity like some power of peace and that our feet and the constant describing the powers called 8 and it is an example of an intersection exponent 1 of their Fareham's
is that these intersection expands it and had given exactly by the formula of foreign squared minus 1 24 this formula the conjecture earlier based on numerical results by by 2018 phone and arrived later by 28 9 rigorously using two-dimensional quantum gravity methods despite the simplicity of the formula prior to the introduction of S solely based methods used by polish and burner its derivation by conventional because countless techniques the beer appeared to be quite out of reach for the period from 2001 or so until now has seen an explosion of interest in applications of the SLD approach to discuss this need 1st briefly described as a leaky faucet assured that domain D in the complex plane with distinct points Indiana
boundaries as parameter kappa has still leave the portal site with parameters but as a certain brand them continuous yes the closure starting starting from 18 going to be a frantic effort is Leslie for facility is a simple task of probability 1 that only touches the boundaries at Indy I will In inward dating back to the 1920 s studied evolution from 8 to be nonrandom curves and there associated formal maps in terms of a real value driving function you see as the lead parameter kappa correspondences driving option you see being a stay one-dimensional Brownian motion I was scale with parameter kappa it's call to prop eased dramatically and cap but when have very them for it was no longer a simple per capita his 8 or more actually Plane named her now back to you as solely based events of the
recent test many of these insurance were motivated by now rigorous results from the school physics literature about two-dimensional critical phenomena critical points of physical systems typically have been very specific values the physical parameters such as where the vapor pressure occurred and liquid yet system physical systems of many remarkable properties such as random fluctuations that normally Her observable only and might we the scales man manifesting themselves
macroscopically related feature Is that many quantities at 4 approaching the critical point of power-law behavior with the powers which typically not 9 integer virus known as critical exponents believed to satisfy universality that is the same exponents but microscopically distinct models in the same spatial dimension should have the same Expo at their respective critical points such a diverse Sally possibly be true about other macroscopic features such as spin women will be discussed later 2 dimensional
critical systems now I have an additional amount of property which is at the heart of both approach predecessors in the visit literature that is conformal invariance a hearing on the matter epic scale does in the case of the burning interception exponents of your spouse many of the SLD based results to the were rigorous proof that's what values that have been derived earlier by 9 rigorous arguments primarily arguments were based on what is known the physics literature is conformal field theories which dates back to the work report gap collaborators
and others in the 1970 the eighties and nineties others our results were brand new discuss a few results in more detail but what is most exciting is that SLD based approach is not just a realization of what already existed but conceptually quite complementary approach that conformal field theory Werner and particularly as emphasized needs understand that supplementary relationship this as an example to focus on the restriction property as in his work on the former lingering measurements warning that work is 1 example of a burgeoning interest in extending the original focus during 1st 2 random loops still with conformal invariant both in the case of percolation the more general that in more general context such as formal Rubin samples such as currently study by that Sheffield England here some more examples of the results obtained in the last 6 years collect WTB of playing very emotional I want to find out what's called Runyon inference cheers of that's the following considers these 1st segment of the emotion between time 0 in time t the compliment a plane that her segment is a parable accountable Union of open sets 1 of which is infinite in and the component of is the frontier from as a consequence of deep relations at planar burning motion have with SLD with parameters said Sherman burner proved proved a celebrated 1982 conjecture enameled wrote about running frontier namely that the house dimension of the frontier is for 3rd again this was a result which 1 would have thought would be attainable by more conventional of sick S covers methods but had not been a different set of results stated In formally in next Arab they concerned loop erased random walks and related random objects on lattices will brace random walks are simple symmetric random walks in which any loops their foreign that sequentially erased she so there is a formal version that consider a safe shutdown domain of the plane and consider the scaling limit limit corresponds to letting Villegas and 0 respectively will breaks random walk and its related objects which are uniformly random spanning tree and the related lattice filling offer those respectively the limit which the letter spacing of 0 as described by gasoline the grammar a continuum treaty based only 2 and the plane going at selling the perimeter a still limits of lettuce models as in the serum are among the most among the most interesting and often most difficult results to do them involves a successful combination concepts and techniques from 3 different areas conformal geometry as in the classical look revolutions with be driving functions nonrandom cassock Analysis a leader driving functions about motion and the probability theory the ladders models limits tried to study the worker burner combines all 3 ingredients get well before closing I'd like to discuss 1 more example demonstrates how these 3 areas and work together and that is spending limits of two-dimensional critical percolation of the physics community knew what rigorous basis the expo value released most of their values and even some geometrical information in the form of specific formulas for scale limits crossing probabilities between boundary settlements of domain these formulas would derived by parity following the conjecture that they should be invariant but there was no understanding for example of these scaling limit geometry of objects like cluster interfaces the hostages faces I simply outer boundaries of classes the outer boundary of his blue clusters this dark black hat goes around Gaborik this doubt boundary of this but cluster interface describing the boundary of the year that this move closer would go around like this remember interested in limit which the letter spacing of 0 but you but 1 in a critical percolation system even as letterspacing 0 1 continues the clusters and
macroscopic scale so How to studied the limits of these interfaces on world 1st breakthrough in Trammer argued that the limit of a particular interface called exploration f should be described in the spending limits by canceling the parameter 6 In fact if there was a crew that if there was enough variance in the limit existed have next for special cases are trying their best to prove that indeed across probably due converged for lingering party formulas also
sketched argument the how that could be convergence of the whole exploration have argued further that 1 should be able to watch them those results to a full-scale describing the family of the interface groups of the but describing the boundaries all the hostages using a machinery approved certain percolation exponents using exploration have earned convergence and while ashram underused full-scale convergence from another X from value which ousted there's a lot of the convergence the 2 types of just mentioned can be approved as recent work Rodrigo myself by using ladders percolation machinery including results obtained by essence derivatives and Jane about crossings results of Eisenman and there were only about the nature of narrow your percolation then than the results about percolation exponents of applied and provide another example hello the 3 ingredients mentioned before can work together as 1 example of that is the following text on derivation proof of a prediction that had already exists in the physics villagers in niece need the
in-house about Expo describing these in diameter fit custody origin critical two-dimensional percolation cluster size diameter distribution as tailed came like our minus 5 48 value by its 1st so I close with some comments about continuum models part of probability theory and their relation to other areas of mathematics which
exemplified by the work of burner a traditionally a major focus of probability theory especially so France has been a continual objects such as Brownian motion and calculus Esterly in related processes are the latest continue objects in the camp and those of us raised a different setting such a successful the Canucks sometimes regard let us smiled as more real physical and is continuing models but but I would say that's a narrow view it's only the continue models which Princess the Heckscher properties like formal variants that relate probability theory other areas of mathematics and such relations have become an increasing importance in recent years and will continue to be so even if 1 is primarily interested in the original letters models is quite we're their properties such as critical
opponents and critical universality and that they understood without deep analysis of the continuum models rise spending limit thanks to the work of the burner is collaborators others I would say that now we old continue we stones by for attention