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1. Lecture room 5402: 2(15:55-18:20); 5(14:00-15:35).

Lecture 1 Introduction; Hamonic functions and Laplacian on graphs.

Lecture 2 Courant's minimax principle; Diameter and the second eigenvalue.

Lecture 3 Alon-Boppana; Examples: cycles and discrete tori; discrete/continous time heat equation.

Lecture 4 Random walks: reversibility and convergence; Spectral decomposition; Heat kernels; Normalized heat diffusion.

Lecture 5 Nica's theorem; Laplacian and heat equation on locally finite infinite graphs .

Lecture 6 Maximum principle; Dirichlet heat kernel.

Lecture 7 Exhaustion by finite connected graphs. Stochastic completeness: an introduction.

Lecture 8 Stochastic incompleteness: characterizations.

Lecture 9 \lambda-harmonic and subharmonic functions; Bakry-Emery curvature lower bound estimates; Ricci flat graphs.

Lecture 10 Bakry-Emery curvature as an eigenvalue problem; Examples and open problems.

Lecture 11 Gradient estimate; Stochastic completeness revisited; Chung-Lin-Yau eigenvalue-diameter-curvature estimate: A mixture of Li-Yau and Lichnerowicz.

Lecture 12 Bonnet-Myers type diameter bounds; Rigidity: Characterizing the equality in gradient estimates.

Lecture 13 Distance functions as shifted eigenfunctions; Global combinatorial structure of graphs in rigidity cases.

Lecture 14 Distance-regular graph rigidity; Combinatorial properties of Bakry-Emery curvature.

Lecture 15 A combinatorial characterization of the hypercube; Ollivier/Lin-Lu-Yau curvature: An introduction.

Lecture 16 Wasserstein distance: diameter estimate; linear programming; Kantorovich duality.

Lecture 17 Bubley-Dyer; Gradient estimate of discrete-time heat equations; Eigenvalue estimates; Lin-Lu-Yau curvature and updated diameter and eigenvalue estimates. notes

Lecture 18 Lin-Lu-Yau curvature of Cartesian product graphs. notes

Lecture 19 Limit-free expression of LLY curvature: piecewise linearity of idleness function (I)

Lecture 20 Limit-free expression of LLY curvature: piecewise linearity of idleness function (II); Amply regular graph and Konig Theorem

Lecture 21 Direct transport and duality formula for LLY curvature

Lecture 22 Combitorial formula for LLY curvature on regular graphs; Cocktail party graphs.

Lecture 23 More examples: Johnson graphs, halved cube graphs, Gosset graphs; Cartesian products; Characterization of Bonnet-Myers sharp graphs via strongly spherical property; Laplacian of the distance function to a vertex.

Lecture 24 Laplacian of the distance function to an edge.

Lecture 25 Laplacian of the distance function to the antipodal edge; Finishing the proof: all regular self-centered Ollivier-Bonnet-Myers sharp graphs are strongly spherical; Removing the condition of self-centeredness in the case of diameter 2; Diameter of d-regular Ollivier-Bonnet-Myers sharp graphs is at most d.

Lecture 26 Removing the condition of self-centeredness in the case of diameter d.

Lecture 27 Comparison of Bakry-Emery and Ollivier/LLY Bonnet-Myers sharpness; Cutoff semigroup.

Lecture 28 Time-derivatives of cutoff semigroups; Relation to the Dirichlet heat kernel.

Lecture 29 Gradient estimate for cutoff semigroups and heat semigroups; Characterizations of Lin-Lu-Yau curvature lower bound.

Lecture 30 Equivalence between continuous and discrete time curature.

Homework 1