The subject of Gaussian space is too enormous to be surveyed here; some recommended texts include Janson **[Jan97]** and Bogachev **[Bog98]**, the latter having an extremely thorough bibliography.

The Ornstein–Uhlenbeck semigroup dates back to the work of Uhlenbeck and Ornstein **[UO30]** whose motivation was to refine Einstein’s theory of Brownian motion **[Ein05]** to take into account the inertia of the particle. The relationship between the action of $\mathrm{U}_\rho$ on functions and on Hermite expansions (i.e., Proposition 31) dates back even further, to Mehler **[Meh66]**. Hermite polynomials were first defined by Laplace **[Lap11]**, and then studied by Chebyshev **[Che60]** and Hermite **[Her64]**. See Lebedev **[Leb72]** (Chapter 4.15) for a proof of the pointwise convergence of a piecewise-$\mathcal{C}^1$ function’s Hermite expansion.

As mentioned in the notes on Chapter 9, the Gaussian Hypercontractivity Theorem is originally due to Nelson **[Nel66]** and now has many known proofs. The idea behind the proof we presented — first proving the Boolean hypercontractivity result and then deducing the Gaussian case by the Central Limit Theorem — is due to Gross **[Gro75]** (see also Trotter **[Tro58]**). Gross actually used the idea to prove his Gaussian Log-Sobolev Inequality, and thereby deduced the Gaussian Hypercontractivity Theorem. Direct proofs of the Gaussian Hypercontractivity Theorem have been given by Neveu **[Nev76]** (using stochastic calculus), Brascamp and Lieb **[BL76a]** (using rearrangement **[BL76a]**), and Ledoux **[Led13]** (using a variation on Exercises 26–29); direct proofs of the Gaussian Log-Sobolev Inequality have been given by Adams and Clarke **[AC79]**, by Bakry and Emery **[BE85a]**, and by Ledoux **[Led92]**, the latter two using semigroup techniques. Bakry’s survey **[Bak94]** on these topics is also recommended.

The Gaussian Isoperimetric Inequality was first proved independently by Borell **[Bor75]** and by Sudakov and Tsirel’son **[ST78]**. Both works derived the result by taking the isoperimetric inequality on the sphere (due to Lévy **[Lev22]** and Schmidt **[Sch48]**, see also Figiel, Lindenstrauss, and Milman **[FLM77]**) and then taking “Poincaré’s limit” — i.e., viewing Gaussian space as a projection of the sphere of radius $\sqrt{n}$ in $n$ dimensions, with $n \to \infty$ (see Lévy **[Lev22]**, McKean **[McK73]**, and Diaconis and Freedman **[DF87]**). Ehrhard **[Ehr83]** gave a different proof using a symmetrization argument intrinsic to Gaussian space. This may be compared to the alternate proof of the spherical isoperimetric inequality **[Ben84]** based on the “two-point symmetrization” of Baernstein and Taylor **[BT76]** (analogous to Riesz rearrangement in Euclidean space and to the polarization operation from Exercise 2.45).

To carefully define Gaussian surface area for a broad class of sets requires venturing into the study of geometric measure theory and functions of bounded variation. For a clear and comprehensive development in the Euclidean setting (including the remark in Exercise 15), see the book by Ambrosio, Fusco, and Pallara **[AFP00]**. There’s not much difference between the Euclidean and finite-dimensional Gaussian settings; research on Gaussian perimeter tends to focus on the trickier infinite-dimensional case. For a thorough development of surface area in this latter setting (which of course includes finite-dimensional Gaussian space as a special case) see the work of Ambrosio, Miranda, Maniglia, and Pallara **[AMMP10]**; in particular, Theorem 4.1 in that work gives several additional equivalent definitions for $\text{surf}_\gamma$ besides those in Definition 48. Regarding the fact that $\mathbf{RS}_A’(0^+)$ is an equivalent definition, the Euclidean analogue of this statement was proven in Miranda et al. **[MPPP07]** and the statement itself follows similarly **[Mir13]** using Ambrosio et al. **[AFR13]**. (Our heuristic justification of 11.4.(2) is similar to the one given by Kane **[Kan11a]**.) Additional related results can be found in Hino **[Hin10]** (which includes the remark about convex sets at the end of Definition 48), Ambrosio and Figalli **[AF11]**, Miranda et al. **[MNP12]**, and Ambrosio et al. **[AFR13]**.

The inequality of Theorem 51 is explicit in Ledoux **[Led94]** (see also the excellent survey **[Led96]**); he used it to deduce the Gaussian Isoperimetric Inequality. He also noted that it’s essentially deducible from an earlier inequality of Pisier and Maurey **[Pis86]** (Theorem 2.2). Theorem 43, which expresses the subadditivity of rotation sensitivity, can be viewed as a discretization of the Pisier–Maurey inequality. This theorem appeared in work of Kindler and O’Donnell **[KO12]**, which also made the observations about the volume-$\tfrac{1}{2}$ case of Borell’s Isoperimetric Theorem at the end of Section 3 and in Remark 75.

Bobkov’s Inequality **[Bob97]** in the special case of Gaussian space had already been implicitly established by Ehrhard **[Ehr84]**; the striking novelty of Bobkov’s work (partially inspired by Talagrand **[Tal93]**) was his reduction to the two-point Boolean inequality. The proof of this inequality which we presented is, as mentioned a discretization of the stochastic calculus proof of Barthe and Maurey **[BM00a]**. (In turn, they were extending the stochastic calculus proof of Bobkov’s Inequality in the Gaussian setting due to Capitaine, Hsu, and Ledoux **[CHL97]**.) The idea that it’s enough to show that Claim 54 is “nearly true” by computing two derivatives — as opposed to showing it’s exactly true by computing four derivatives — was communicated to the author by Yuval Peres. Following Bobkov’s paper, Bakry and Ledoux **[BL96b]** established Theorem 55 in very general infinite-dimensional settings including Gaussian space; Ledoux **[Led98]** further pointed out that the Gaussian version of Bobkov’s Inequality has a very short and direct semigroup-based proof. See also Bobkov and Götze **[BG99]** and Tillich and Zémor **[TZ00]** for results similar to Bobkov’s Inequality in other discrete settings. Borell’s Isoperimetric Theorem is from Borell **[Bor85]**. Borell’s proof used “Ehrhard symmetrization” and actually gave much stronger results — e.g., that if $f, g \in L^2({\mathbb R}^n, \gamma)$ are nonnegative and $q \geq 1$, then $\langle (\mathrm{U}_\rho f)^q, g\rangle$ can only increase under simultaneous Ehrhard symmetrization of $f$ and $g$. There are at least four other known proofs of the basic Borell Isoperimetric Theorem. Beckner **[Bec92]** observed that the analogous isoperimetric theorem on the sphere follows from two-point symmetrization; this yields the Gaussian result via Poincaré’s limit (for details, see Carlen and Loss **[CL90]**). (This proof is perhaps the conceptually simplest one, though carrying out all the technical details is a chore.) Mossel and Neeman **[MN12]** gave the proof based on semigroup methods outlined in Exercises 26–29, and later together with De **[DMN12]** gave a “Bobkov-style” Boolean proof (see Exercise 30). Finally, Eldan **[Eld13]** gave a proof using stochastic calculus.

As mentioned in Section 5 there are several known ways to prove the Berry–Esseen Theorem. Aside from the original method (characteristic functions), there is also Stein’s Method **[Ste72,Ste86a]**; see also, e.g., **[Bol84,BH84,CGS11]**. The Replacement Method approach we presented originates in the work of Lindeberg **[Lin22]**. The mollification techniques used (e.g., those in Exercise 40) are standard. The Invariance Principle as presented in Section 6 is from Mossel, O’Donnell, and Oleszkiewicz **[MOO10]**. Further extensions (e.g., Exercise 48) appear in the work of Mossel **[Mos10]**. In fact the Invariance Principle dates back to the 1971 work of Rotar’ **[Rot73,Rot74]**; therein he essentially proved the Invariance Principle for degree-$2$ multilinear polynomials (even employing the term “influence” as we do for the quantity in Definition 63). Earlier work on extending the Central Limit Theorem to higher-degree polynomials had focused on obtaining sufficient conditions for polynomials (especially quadratics) to have a Gaussian limit distribution; this is the subject of *U-statistics*. Rotar’ emphasized the idea of invariance and of allowing any (quadratic) polynomial with low influences. Rotar’ also credited Girko **[Gir73]** with related results in the case of positive definite quadratic forms. In 1975, Rotar’ **[Rot75]** generalized his results to handle multilinear polynomials of any constant degree, and also random vectors (as in Exercise 48). (Rotar’ also gave further refinements in 1979 **[Rot79]**.)

The difference between the results of Rotar’ **[Rot75]** and Mossel et al. **[MOO10]** comes in the treatment of the error bounds. It’s somewhat difficult to extract simple-to-state error bounds from Rotar’ **[Rot75]**, as the error there is presented as a sum over $i \in [n]$ of expressions $\mathop{\bf E}[F({\boldsymbol{x}}) \boldsymbol{1}_{|F({\boldsymbol{x}})| > u_i}]$, where $u_i$ involves $\mathbf{Inf}_i[F]$. (Partly this is so as to generalize the statement of the Lindeberg CLT.) Nevertheless, the work of Rotar’ implies a Lévy distance bound as in Corollary 69, with some inexplicit function $o_\epsilon(1)$ in place of $(1/\rho)^{O(k)} \epsilon^{1/8}$. By contrast, the work of Mossel et al. **[MOO10]** shows that a straightforward combination of the Replacement Method and hypercontractivity yields good, explicit error bounds. Regarding the Carbery–Wright Theorem **[CW01]**, an alternative exposition appears in Nazarov, Sodin, and Vol’berg **[NSV02]**.

Regarding the Majority Is Stablest Theorem (conjectured in Khot, Kindler, Mossel, and O’Donnell **[KKMO04]** and proved originally in Mossel, O’Donnell, and Oleszkiewicz **[MOO05a]**), it can be added that additional motivation for the conjecture came from Kalai **[Kal02]**. The fact that (SDP) is an efficiently computable relaxation for the Max-Cut problem dates back to the 1990 work of Delorme and Poljak **[DP93]**; however, they were unable to give an analysis relating its value to the optimum cut value. In fact, they conjectured that the case of the $5$-cycle from Example 72 had the worst ratio of $\mathrm{Opt}(G)$ to $\mathrm{SDPOpt}(G)$. Goemans and Williamson **[GW94]** were the first to give a sharp analysis of the SDP (Theorem 71), at least for $\theta \geq \theta^*$. Feige and Schechtman **[FS02]** showed an optimal integrality gap for the SDP for all values $\theta \geq \theta^*$ (in particular, showing an integrality gap ratio of $c_{\text{GW}}$); interestingly, their construction essentially involved proving Borell’s Isoperimetric Inequality (though they did it on the sphere rather than in Gaussian space). Both before and after the Khot et al. **[KKMO04]** UG-hardness result for Max-Cut there was a long line of work **[Kar99,Zwi99,AS00,ASZ02,CW04,KV05,FL06,KO06]** devoted to improving the known approximation algorithms and UG-hardness results, in particular for $\theta < \theta^*$. This culminated in the results from O’Donnell and Wu **[OW08]** (mentioned in Remark 74), which showed explicit matching $(\alpha,\beta)$-approximation algorithms, integrality gaps, and UG-hardness results for all $\frac12 < \beta < 1$. The fact that the best integrality gaps matched the best UG-hardness results proved not to be a coincidence; in contemporaneous work, Raghavendra **[Rag08]** showed that for *any* CSP, *any* SDP integrality gap could be turned into a matching Dictator-vs.-No-Notables test. This implies the existence of matching efficient $(\alpha,\beta)$-approximation algorithms and UG-hardness results for every CSP and every $\beta$. See Raghavendra’s thesis **[Rag09]** for full details of his earlier publication **[Rag08]** (including some Invariance Principle extensions building further on Mossel **[Mos10]**); see also Austrin’s work **[Aus07,Aus07a]** for precursors to the Raghavendra theory.

Exercise 31 concerns a problem introduced by Talagrand **[Tal89]**. Talagrand offers a $1000 prize **[Tal06a]** for a solution to the following Boolean version of the problem: Show that for any fixed $0 < \rho < 1$ and for $f : \{-1,1\}^n \to {\mathbb R}^{\geq 0}$ with $\mathop{\bf E}[f] = 1$ it holds that $\mathop{\bf Pr}[\mathrm{T}_\rho f > t] = o(1/t)$ as $t \to \infty$. (The rate of decay may depend on $\rho$ but not, of course, on $n$; in fact, a bound of the form $O(\frac{1}{t \sqrt{\log t}})$ is expected.) The result outlined in Exercise 31 (obtained together with James Lee) is for the very special case of $1$-dimensional Gaussian space; Ball, Barthe, Bednorz, Oleszkiewicz, and Wolff **[BBB+13]** obtained the same result and also showed a bound of $O(\frac{\log \log t}{t \sqrt{\log t}})$ for $d$-dimensional Gaussian space (with the constant in the $O(\cdot)$ depending on $d$).

The Multifunction Invariance Principle (Exercise 48 and its special case Exercise 46) are from Mossel **[Mos10]**; the version for general product spaces (Exercise 49) is from Mossel, O’Donnell, and Oleszkiewicz **[MOO10]**.

Thanks for writing these excellent chapter notes. I wish more math textbooks would attempt this level of historical detail.