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Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are "ed without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Algebraic geometry --- Differential geometry. Global analysis --- 512.7 --- Algebraic geometry. Commutative rings and algebras --- Toric varieties. --- 512.7 Algebraic geometry. Commutative rings and algebras --- Toric varieties --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Addition. --- Affine plane. --- Affine space. --- Affine variety. --- Alexander Grothendieck. --- Alexander duality. --- Algebraic curve. --- Algebraic group. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Big O notation. --- Characteristic class. --- Chern class. --- Chow group. --- Codimension. --- Cohomology. --- Combinatorics. --- Commutative property. --- Complete intersection. --- Convex polytope. --- Convex set. --- Coprime integers. --- Cotangent space. --- Dedekind sum. --- Dimension (vector space). --- Dimension. --- Direct proof. --- Discrete valuation ring. --- Discrete valuation. --- Disjoint union. --- Divisor (algebraic geometry). --- Divisor. --- Dual basis. --- Dual space. --- Equation. --- Equivalence class. --- Equivariant K-theory. --- Euler characteristic. --- Exact sequence. --- Explicit formula. --- Facet (geometry). --- Fundamental group. --- Graded ring. --- Grassmannian. --- H-vector. --- Hirzebruch surface. --- Hodge theory. --- Homogeneous coordinates. --- Homomorphism. --- Hypersurface. --- Intersection theory. --- Invertible matrix. --- Invertible sheaf. --- Isoperimetric inequality. --- Lattice (group). --- Leray spectral sequence. --- Limit point. --- Line bundle. --- Line segment. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mixed volume. --- Moduli space. --- Moment map. --- Monotonic function. --- Natural number. --- Newton polygon. --- Open set. --- Picard group. --- Pick's theorem. --- Polytope. --- Projective space. --- Quadric. --- Quotient space (topology). --- Regular sequence. --- Relative interior. --- Resolution of singularities. --- Restriction (mathematics). --- Resultant. --- Riemann–Roch theorem. --- Serre duality. --- Sign (mathematics). --- Simplex. --- Simplicial complex. --- Simultaneous equations. --- Spectral sequence. --- Subgroup. --- Subset. --- Summation. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Toric variety. --- Unit disk. --- Vector space. --- Weil conjecture. --- Zariski topology.

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Robust optimization is a fairly new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. The authors are the principal developers of robust optimization.

Robust optimization. --- Linear programming. --- 519.8 --- 681.3*G16 --- 681.3*G16 Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- 519.8 Operational research --- Operational research --- Robust optimization --- Linear programming --- Optimisation robuste --- Programmation linéaire --- Optimization, Robust --- RO (Robust optimization) --- Mathematical optimization --- Production scheduling --- Programming (Mathematics) --- 0O. --- Accuracy and precision. --- Additive model. --- Almost surely. --- Approximation algorithm. --- Approximation. --- Best, worst and average case. --- Bifurcation theory. --- Big O notation. --- Candidate solution. --- Central limit theorem. --- Chaos theory. --- Coefficient. --- Computational complexity theory. --- Constrained optimization. --- Convex hull. --- Convex optimization. --- Convex set. --- Cumulative distribution function. --- Curse of dimensionality. --- Decision problem. --- Decision rule. --- Degeneracy (mathematics). --- Diagram (category theory). --- Duality (optimization). --- Dynamic programming. --- Exponential function. --- Feasible region. --- Floor and ceiling functions. --- For All Practical Purposes. --- Free product. --- Ideal solution. --- Identity matrix. --- Inequality (mathematics). --- Infimum and supremum. --- Integer programming. --- Law of large numbers. --- Likelihood-ratio test. --- Linear dynamical system. --- Linear inequality. --- Linear map. --- Linear matrix inequality. --- Linear regression. --- Loss function. --- Margin classifier. --- Markov chain. --- Markov decision process. --- Mathematical optimization. --- Max-plus algebra. --- Maxima and minima. --- Multivariate normal distribution. --- NP-hardness. --- Norm (mathematics). --- Normal distribution. --- Optimal control. --- Optimization problem. --- Orientability. --- P versus NP problem. --- Pairwise. --- Parameter. --- Parametric family. --- Probability distribution. --- Probability. --- Proportionality (mathematics). --- Quantity. --- Random variable. --- Relative interior. --- Robust control. --- Robust decision-making. --- Semi-infinite. --- Sensitivity analysis. --- Simple set. --- Singular value. --- Skew-symmetric matrix. --- Slack variable. --- Special case. --- Spherical model. --- Spline (mathematics). --- State variable. --- Stochastic calculus. --- Stochastic control. --- Stochastic optimization. --- Stochastic programming. --- Stochastic. --- Strong duality. --- Support vector machine. --- Theorem. --- Time complexity. --- Uncertainty. --- Uniform distribution (discrete). --- Unimodality. --- Upper and lower bounds. --- Variable (mathematics). --- Virtual displacement. --- Weak duality. --- Wiener filter. --- With high probability. --- Without loss of generality.