Final Papers | 21-317 Functional Inequalities

The final 27-paper research menu, presentation dates, and expository-paper requirements for 21-317, Fall 2026.

21-317 Fall 2026

Overview Syllabus Schedule Final papers Course notes Announcements

21-317 · Fall 2026 · Final papers

Select your final paper Download the October 23 paper menu (PDF) Read the October 23 addendum View presentation dates

Current scheduling details (updated November 2): Round I is complete; Rounds II and III are cancelled. The remaining final research-paper presentations are Round IV. Seven students have 1+2+2+2 individual 25-minute slots: one on November 18, and two each on November 20, 23, and 30. Speaker order is not yet confirmed. The approximately ten-page individual paper is due Friday, December 4, 2026. There is no class on November 25 or 27 (Thanksgiving Break), or on December 2 or 4 (time for writing).

Paper selection is now open: Select your final paper. Enter your name and CMU email; no account or sign-in is required. Selection is first come, first served: a successful confirmation reserves that paper and makes it unavailable to other students. The research-paper descriptions below are unchanged from the supplied October 23 menu; its presentation counts have been updated to the November 2 schedule. The downloadable PDF is the original October 23 handout, whose scheduling details are superseded by the current schedule above.

21-317: Suggested research papers for the final project

Fall 2026 | Paper menu revised October 23, 2026; presentation schedule updated November 2, 2026

This is a list of suggestions, not a required or exhaustive list. You may propose another research paper after consulting the instructor. Please choose a paper because its mathematical question interests you, and try to understand the entire original article to the best of your ability. Your presentation and final expository paper should distinguish the main result, the key argument, and the background results you are quoting.

Paper selection: Review a few possibilities. The first-come, first-served selection form will be available on Tuesday, October 27. You will also receive individual suggestions based on your Round I presentation. Final presentations (updated November 2): one student on November 18 and two each on November 20, 23, and 30, 25 minutes each. Expository paper: Approximately 10 pages, due Friday, December 4. For a very short article, expand the calculations and fill in missing explanations. For a longer article, identify the essential results and organize a readable exposition. Time for writing: No class on December 2 or 4. (November 25 and 27 are Thanksgiving break.) Difficulty of the whole original paper: A: modest extra preparation; B: significant but manageable preparation; C: ambitious, requiring consultation with the instructor. Intermediate ratings reflect background dependence. Length alone is not a measure of difficulty. Research-reading standard: Identify the problem, theorem, hypotheses and proof strategy. You may quote a difficult prerequisite with a correct reference, but should not present it as proved when you have not worked through its proof. An optional Lean formalization exercise is not part of the required assignment.

Suggested starting points if you are undecided: Loomis–Whitney, Hoeffding, Efron–Stein, Dasgupta–Gupta, Hastings, or Rahimi–Recht. The Bauerschmidt–Bodineau spin-system paper is our shared guided reading and is not listed below as an individual project.

1. Functional inequalities, geometry, and analysis

Pure analysis: how inequalities capture geometry, regularity, and the behavior of functions.

Loomis–Whitney (1949). An inequality related to the isoperimetric inequality

A; 2 pp. Bull. Amer. Math. Soc. 55, 961–962. Paper or publisher record.

• Why this paper? A foundational inequality comparing the volume of a set with the volumes of its coordinate projections.
• Mathematical focus: Reconstruct the original discrete counting inequality (Theorem 2), then derive the volume inequality (Theorem 1).
• Preparation and cautions: Finite sums, elementary geometric measure, counting; a natural first paper.

Payne–Weinberger (1960). An optimal Poincaré inequality for convex domains

B/C; 7 pp. Arch. Ration. Mech. Anal. 5, 286–292. Paper or publisher record.

• Why this paper? A sharp diameter-dependent Poincaré inequality on convex domains.
• Mathematical focus: Understand the one-dimensional inequality and trace how convex geometry enters the higher-dimensional argument.
• Preparation and cautions: One-dimensional Poincaré inequalities, calculus and convex geometry. The original proof treats planar convex domains; the higher-dimensional extension requires care.

Beckner (1989). A generalized Poincaré inequality for Gaussian measures

B/C; 4 pp. Proc. Amer. Math. Soc. 105, 397–400. Paper or publisher record.

• Why this paper? Sharp inequalities interpolating between Gaussian Poincaré and logarithmic Sobolev inequalities.
• Mathematical focus: Explain the parameter family, recover both endpoints, and investigate the proof of the interpolation inequality.
• Preparation and cautions: Gaussian integration and semigroup or Hermite-polynomial methods. The four-page original is very compressed.

Talagrand (1996). Transportation cost for Gaussian and other product measures

B/C; 14 pp. Geom. Funct. Anal. 6, 587–600. Paper or publisher record.

• Why this paper? A foundational quadratic transportation-cost inequality for Gaussian and product probability measures.
• Mathematical focus: Understand the one-dimensional Gaussian transport inequality, its entropy bound, and the role of product measures.
• Preparation and cautions: Probability densities, entropy and monotone rearrangement. The full argument is more demanding.

Bobkov (1997). An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space

C; 9 pp. Ann. Probab. 25, 206–214. Paper or publisher record.

• Why this paper? An original functional inequality on the discrete cube leading to Gaussian isoperimetry.
• Mathematical focus: Start with the basic two-point inequality, then show how tensorization produces a discrete isoperimetric statement.
• Preparation and cautions: Gaussian distributions, convexity and a nontrivial limiting argument.

Carlen–Loss (1993). Sharp constant in Nash’s inequality

C; 3 pp. Int. Math. Res. Notices 1993, 213–215. Paper or publisher record.

• Why this paper? Determines the sharp constant in the Nash inequality.
• Mathematical focus: Explain Nash’s inequality and why sharpness matters; reconstruct the optimization and symmetrization steps.
• Preparation and cautions: Sobolev inequalities, variational calculus and rearrangement. Short but not elementary.

Gross (1975). Logarithmic Sobolev inequalities

C; 23 pp. Amer. J. Math. 97, 1061–1083. Paper or publisher record.

• Why this paper? The foundational development of Gaussian logarithmic Sobolev inequalities and hypercontractivity.
• Mathematical focus: Prove a finite-dimensional Gaussian consequence, then explain the equivalence with hypercontractivity and how the paper proceeds beyond it.
• Preparation and cautions: Semigroups, functional analysis and some infinite-dimensional arguments.

Jordan–Kinderlehrer–Otto (1998). The variational formulation of the Fokker–Planck equation

C; 17 pp. SIAM J. Math. Anal. 29, 1–17. Paper or publisher record.

• Why this paper? A foundational connection between free-energy dissipation and optimal transport.
• Mathematical focus: Derive the implicit variational time step in a simple model, and explain its relation to diffusion and entropy.
• Preparation and cautions: Calculus of variations, probability densities, PDE and Wasserstein distance.

2. Concentration and high-dimensional probability

Concentration and variance inequalities for random functions, including high-dimensional examples.

Hoeffding (1963). Probability inequalities for sums of bounded random variables

B; 18 pp. J. Amer. Statist. Assoc. 58, 13–30. Paper or publisher record.

• Why this paper? Classical exponential concentration bounds.
• Mathematical focus: Prove an exponential-moment estimate, derive a bounded-sum tail inequality, then account for the sampling-without-replacement results.
• Preparation and cautions: Independent random variables, exponential moments and convexity.

Efron–Stein (1981). The jackknife estimate of variance

B; 11 pp. Ann. Statist. 9, 586–596. Paper or publisher record.

• Why this paper? A foundational comparison of the jackknife variance estimator with actual variance.
• Mathematical focus: Recover the orthogonal decomposition of variance and derive the principal jackknife estimate.
• Preparation and cautions: Conditional expectation and the variance of jackknife estimates; do not confuse the original paper with only the modern one-line Efron–Stein inequality.

Dasgupta–Gupta (2003). An elementary proof of a theorem of Johnson and Lindenstrauss

A/B; 6 pp. Random Structures Algorithms 22, 60–65. Paper or publisher record.

• Why this paper? An influential, especially readable later proof of the original Johnson–Lindenstrauss dimension-reduction theorem.
• Mathematical focus: Analyze random projections, obtain concentration of lengths, then take a union bound over pairs of points.
• Preparation and cautions: Gaussian probability, elementary exponential-moment estimates and union bounds. This is a later proof, not the original discovery.

3. Entropy and discrete inequalities

Entropy and information as tools for proving combinatorial and analytic inequalities.

Kahn (2002). Entropy, independent sets and antichains: a new approach to Dedekind’s problem

B/C; 8 pp. Proc. Amer. Math. Soc. 130, 371–378. Paper or publisher record.

• Why this paper? Entropy inequalities are a key ingredient in counting independent sets and antichains.
• Mathematical focus: Identify the precise entropy estimate and reconstruct its use in bounding the number of independent sets; relate this to the hard-core partition function.
• Preparation and cautions: Discrete entropy, conditional probability, finite combinatorics; more directly inequality-based than Zhao (2010).

Lovász (1979). On the Shannon capacity of a graph

B/C; 7 pp. IEEE Trans. Inform. Theory 25, 1–7. Paper or publisher record.

• Why this paper? The theta-function method for bounding zero-error communication capacity.
• Mathematical focus: Explain the orthogonal-representation bound and the exact calculation for the pentagon.
• Preparation and cautions: Linear algebra, graph theory and positive semidefinite matrices.

Madiman–Tetali (2010). Information inequalities for joint distributions, with interpretations and applications

C; 15 pp. IEEE Trans. Inform. Theory 56, 2699–2713. Paper or publisher record.

• Why this paper? Refined entropy inequalities using fractional coverings and submodularity.
• Mathematical focus: Prove one fractional-cover inequality, then describe the other applications to joint distributions.
• Preparation and cautions: Conditional entropy, combinatorics of set systems.

Stam (1959). Some inequalities satisfied by the quantities of information of Fisher and Shannon

C; 12 pp. Information and Control 2, 101–112. Paper or publisher record.

• Why this paper? Foundational Fisher-information and entropy-power inequalities.
• Mathematical focus: Isolate the Fisher-information inequality and explain its connection with convolution and Gaussian noise.
• Preparation and cautions: Probability densities, differentiation and convolution; some proofs in the original are not fully elaborated.

4. Markov chains, spectral gaps, and sampling

Convergence to equilibrium, its relation to the spectrum, and exact or approximate sampling.

Hastings (1970). Monte Carlo sampling methods using Markov chains and their applications

B; 13 pp. Biometrika 57, 97–109. Paper or publisher record.

• Why this paper? Generalizes Metropolis sampling to asymmetric proposals and studies Monte Carlo error.
• Mathematical focus: Derive the acceptance rule and detailed balance; discuss why serial correlation affects estimates.
• Preparation and cautions: Reversible chains, conditional probability and variance.

Diaconis–Stroock (1991). Geometric bounds for eigenvalues of Markov chains

B/C; 26 pp. Ann. Appl. Probab. 1, 36–61. Paper or publisher record.

• Why this paper? The canonical-path method relating graph geometry to spectral gaps.
• Mathematical focus: Derive Proposition 1 (the canonical-path Poincaré bound); apply it to the cycle, then explain the other eigenvalue estimates.
• Preparation and cautions: Finite reversible chains, Dirichlet forms, and the Rayleigh quotient. A substantial full-paper project despite the elementary central proof.

Propp–Wilson (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics

C; 30 pp. Random Structures Algorithms 9, 223–252. Paper or publisher record.

• Why this paper? Introduces coupling from the past for exact sampling from stationary laws.
• Mathematical focus: Prove exactness in a finite monotone-chain example and account for the larger applications.
• Preparation and cautions: Coupling, Markov chains and partial orders; this is a long project.

Diaconis–Saloff-Coste (2006). Separation cut-offs for birth and death chains

C; 25 pp. Ann. Appl. Probab. 16, 2098–2122. Paper or publisher record.

• Why this paper? A theorem connecting the cutoff phenomenon to the spectral gap and a spectral time scale for birth-and-death chains.
• Mathematical focus: Begin with the precise cutoff criterion in Theorem 1.1 and examples; identify the role of eigenvalues and explain which spectral and hitting-time results the proof requires.
• Preparation and cautions: Finite-state continuous-time Markov chains, spectral gaps, hitting times. The result concerns separation from an endpoint, not arbitrary chains or all distances.

5. Correlation inequalities and mathematical statistical mechanics

What inequalities imply about correlations, monotonicity, and equilibrium in interacting systems.

Fortuin–Kasteleyn–Ginibre (1971). Correlation inequalities on some partially ordered sets

B/C; 15 pp. Commun. Math. Phys. 22, 89–103. Paper or publisher record.

• Why this paper? The original FKG positive-association theorem for measures on finite distributive lattices.
• Mathematical focus: State Proposition 1 (positive association under the lattice condition), examine the induction, and work out an attractive Ising example.
• Preparation and cautions: Finite probability, monotone functions and lattice operations. The induction needs close reading; the Ising application is more approachable.

Griffiths–Hurst–Sherman (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field

C; 6 pp. J. Math. Phys. 11, 790–795. Paper or publisher record.

• Why this paper? A classical higher-order correlation inequality and magnetization concavity.
• Mathematical focus: Identify the sign inequality and explain how it yields concavity in the applied field.
• Preparation and cautions: Ising systems, magnetization derivatives and correlation inequalities. Particularly demanding in its short format.

Edwards–Sokal (1988). Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm

B/C; 4 pp. Phys. Rev. D 38, 2009–2012. Paper or publisher record.

• Why this paper? An influential joint spin–bond representation connecting Potts models and cluster sampling.
• Mathematical focus: Check the spin–bond coupling on a small graph and derive the conditional spin-cluster law.
• Preparation and cautions: Finite graph probability and Gibbs measures; it is a representation identity, not itself a functional inequality.

Holley–Stroock (1987). Logarithmic Sobolev inequalities and stochastic Ising models

C; 36 pp. J. Stat. Phys. 46, 1159–1194. Paper or publisher record.

• Why this paper? A foundational application of logarithmic Sobolev inequalities to stochastic Ising dynamics.
• Mathematical focus: Identify where logarithmic Sobolev inequalities control the approach to equilibrium for stochastic Ising systems; examine a finite example.
• Preparation and cautions: Markov semigroups, conditional entropy and spin-flip dynamics. A longer, technical project.

6. Statistical learning and information inequalities

Original papers in which a precise probability, entropy, or stability inequality is a principal mathematical result.

McAllester (1998). Some PAC-Bayesian theorems

B/C; 5 pp. Proc. 11th COLT, 230–234. Paper or publisher record.

• Why this paper? An early foundational PAC-Bayesian generalization inequality based on relative entropy.
• Mathematical focus: Work out the exponential change-of-measure inequality between a prior and posterior, then reconstruct a PAC-Bayesian bound.
• Preparation and cautions: Hoeffding-type concentration, relative entropy and conditional probability. The original is only five pages and very compressed.

Rahimi–Recht (2007). Random features for large-scale kernel machines

B; 8 pp. Adv. Neural Inf. Process. Syst. 20, 1177–1184. Paper or publisher record.

• Why this paper? Random Fourier features with quantitative concentration bounds for approximating kernels.
• Mathematical focus: Verify the random-feature expectation identity, the fixed-pair concentration bound, and the passage toward uniform approximation.
• Preparation and cautions: Bounded-variable concentration and basic Fourier analysis; Bochner’s theorem may be quoted.

Hardt–Recht–Singer (2016). Train faster, generalize better: stability of stochastic gradient descent

B; 10 pp. Proc. ICML (PMLR 48), 1225–1234. Paper or publisher record.

• Why this paper? Quantitative stability estimates for stochastic gradient descent and their implications for generalization.
• Mathematical focus: Reconstruct the stability-to-generalization estimate and the contraction bounds for convex gradient updates.
• Preparation and cautions: Lipschitz/convex functions, iterative algorithms, probability. The connection to this course is quantitative stability, not a logarithmic Sobolev inequality.

Lee–Lu–Tan (2022). Convergence for score-based generative modeling with polynomial complexity

C; 13 pp. Adv. Neural Inf. Process. Syst. 35, 22870–22882. Paper or publisher record.

• Why this paper? Quantitative convergence guarantees for score-based generative sampling, using logarithmic Sobolev inequalities to control sampling with an approximate score.
• Mathematical focus: Understand Theorem 2.1 and prove the coupling estimate in Theorem 4.1 (Section 4); explain how the log-Sobolev hypothesis enters the later chi-square divergence estimate.
• Preparation and cautions: Relative entropy, chi-square divergence, Markov chains and Langevin diffusion. The main text is 13 pages, but full proofs are substantially longer; this is an ambitious project requiring instructor guidance.