Peter Philip authored notes on functional analysis beginning with topological vector spaces, seminorms and locally convex spaces. They develop foundational tools including Hahn-Banach separation. The progression explains how geometric separation and continuity remain available in function spaces where a single norm may not describe the natural topology.^[ref]

Peter Petersen authored an exposition of smooth manifolds, transversality and de Rham cohomology. Among its structural results is Ehresmann fibration theorem, which turns a proper submersion into a locally trivial fiber bundle. This connects a differential condition on a map with a uniform local description of all its fibers.^[ref]

Evgueni Tevelev authored a study of projective dual varieties, which encode the tangent hyperplanes of a projective variety. His account develops reflexivity, contact loci and links with discriminants and secant varieties. The viewpoint turns tangency into a new geometric object, allowing properties of the original variety to be studied through its family of tangent hyperplanes.^[ref]

Percy Deift developed an exposition of nonlinear steepest descent for Riemann-Hilbert problems. His lectures explain how an analytic matrix problem with a prescribed jump across a contour can be transformed to reveal asymptotic behavior. Applications include orthogonal polynomials, integrable systems and random matrices, connecting complex analysis with large-parameter limits.^[ref]

Charles Weibel authored an exposition of bilinear forms over a field. It develops diagonalization and the classification of symmetric bilinear forms, including the real signature described by Sylvester's law of inertia. These classifications explain why changing a basis can simplify a form without removing invariants such as its numbers of positive and negative squares.^[ref]

Emmy Noether proved the fundamental connection between continuous symmetries of a variational problem and conservation laws. Her 1918 paper also gives identities among the Euler-Lagrange equations for symmetries depending on arbitrary functions, a foundation for understanding gauge symmetry. The distinction between finite-parameter and function-dependent transformations is essential to the two theorems.^[ref]

Gerard Orriols coauthored notes on elliptic partial differential equations with André Guerra. They develop variational methods, maximum principles and Schauder estimates, including how regularity estimates support existence proofs. This brings together energy minimization and pointwise comparison as complementary tools for analyzing solutions.^[ref]

André Guerra coauthored notes on elliptic partial differential equations with Gerard Orriols. Their exposition develops harmonic functions, maximum principles, Schauder estimates and regularity theory. The treatment connects a priori bounds with existence arguments, explaining why controlling derivatives can help prove that a boundary-value problem has a solution.^[ref]

M. A. Tavel translated Emmy Noether's 1918 paper on invariant variational problems into English; the translation appeared in 1971. It presents the relation between continuous symmetries, conservation laws and identities among the Euler-Lagrange equations. The surviving byline establishes the translator's contribution without requiring an inferred full name or biography.^[ref]

P. D. Seymour studied square-lattice bond percolation with D. J. A. Welsh. Their work constructs crossing bounds for rectangles from square-crossing information, using positive correlation and geometric combinations of crossing events. Such scale comparisons are central to Russo-Seymour-Welsh theorem and planar critical connectivity.^[ref]

D. J. A. Welsh coauthored a study of square-lattice bond percolation with P. D. Seymour. Their crossing-probability arguments relate crossings of squares to crossings of rectangles. These estimates help control connectivity across different scales and form part of Russo-Seymour-Welsh theorem.^[ref]

Michael Aizenman coauthored tree-diagram inequalities for percolation with Charles M. Newman. Their work controls multipoint connectivity and gives a condition for the percolation susceptibility exponent to take its mean-field value. The method relates large-scale critical behavior to convergence properties of diagrams built from pairwise connection probabilities.^[ref]

Charles M. Newman studied critical behavior in percolation with Michael Aizenman. Their tree-diagram inequalities bound multipoint connection probabilities using two-point functions. A diagrammatic convergence criterion gives mean-field behavior for the percolation susceptibility exponent, connecting the geometry of large clusters with quantitative critical scaling.^[ref]

Hugo Duminil-Copin coauthored a proof of the sharp bond percolation transition with Vincent Tassion. Their nearest-neighbor lattice argument establishes exponential decay below the critical probability and a positive lower bound for the infinite-cluster probability above it. The result explains why the transition separates quantitatively different connectivity behavior.^[ref]

Vincent Tassion coauthored a proof of sharpness of the bond percolation transition with Hugo Duminil-Copin. Below the critical probability, connection to distance $n$ decays exponentially; above it, the infinite-cluster probability obeys a quantitative lower bound. The argument distinguishes two regimes on either side of the threshold rather than merely proving that a threshold exists.^[ref]

C. M. Fortuin coauthored the FKG inequality with P. W. Kasteleyn and J. Ginibre. The theorem makes a precise sufficient condition for increasing observables to be positively correlated. Its applications include Ising models and the random-cluster model, connecting ordered probability spaces with statistical mechanics.^[ref]

P. W. Kasteleyn coauthored the FKG inequality with C. M. Fortuin and J. Ginibre. Their theorem gives positive association for increasing functions on a finite distributive lattice when the probability weights satisfy the required lattice condition. It provides a rigorous correlation tool for Ising models and the random-cluster model.^[ref]

R. G. Edwards, identified as Robert G. Edwards in the publisher byline, coauthored the Edwards-Sokal coupling with A. D. Sokal. The construction combines spin and bond variables in one joint probability law, connecting the Potts model and the random-cluster model. This supplies the probabilistic basis for changing entire clusters in a Monte Carlo method.^[ref]

A. D. Sokal, identified as Alan D. Sokal in the publisher byline, coauthored the Edwards-Sokal coupling with R. G. Edwards. Their joint spin-and-bond representation links the Potts model to the random-cluster model and supports cluster algorithms for Monte Carlo simulation. It allows the same joint probability law to be viewed through either spin or bond variables.^[ref]

Ramon van Handel authored notes on probability theory in high dimensions connecting probability theory, analysis and geometry. The notes develop concentration of measure, Gaussian processes and related inequalities through common mechanisms such as tensorization of entropy. This viewpoint explains why a function of many independent inputs can be tightly concentrated even when the full input space has very high dimension.^[ref]