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Modern and applications in algebraic and

This course is both an introduction to modern combinatorics and to . Combinatorial methods are used nowadays in many areas of , and discrete com- binatorial objects, despite their simplicity of appearance, reveals indeed diverse and deep facets and structures. In this course, we aim to emphasize on the analogies and interactions between combinatorial objects such as graphs, matro¨ıdsand , and algebraic geometric objects such as algebraic , surfaces, or higher dimensional varieties. This course and the working group which follows will cover recent topics of research interest and applications of combinatorial methods in algebraic and , or recent applications of ideas from algebraic geometry to resolve prob- lems in combinatorics, very different in nature on the first sight, which leaves us thinking that deeper links between these objects should exist. Starting from basic definitions, we will arrive at the end of the course to the frontiers of current research in the topics, covering on the way several beautiful areas of combinatorics. For those who like to have a first research experience, some interesting research problems are presented during the lectures and the working group, that we could discuss in group or individually, and which could be also the topic of a parallel research internship.

The prerequisite are basic knowledge in at the level of the courses Algebra 1 and 2. The following are some topics which will be covered in the lectures and in the working group.

1. Introduction to

2. Algebraic geometry of graphs

3. Introduction to algebraic geometry, theory of algebraic curves and surfaces

4. Polytopes and toric varieties

5. Brill-Noether theory for graphs and algebraic curves

6. Uniform bounds on the of rational bounds on curves over number fields

7. Convex bodies and applications in algebraic geometry

8. Matroids and universality

9. Some interesting classes of associated to graphs

10. Zeta functions of graphs

1 Some references

[1] O. Amini, Geometry of graphs and applications in arithmetic and algebraic geometry, available at www.math.ens.fr/~amini. [2] M. Baker and D. Jensen, Degeneration of linear series from the tropical of view and applications, available at http://arxiv.org/abs/1504.05544.

[3] H. Chen, In´egalit´ed’indice de Hodge en g´eom´etrieet arithm´etique: une approche probabiliste, available at https://www-fourier.ujf-grenoble.fr/~huayi/Recherche/hodge.pdf. [4] W. Fulton, Introduction to toric varieties. No. 131. Princeton University Press, 1993.

[5] J. Huh, Milnor of projective hypersurfaces and the chromatic of graphs, Journal of the American Mathematical Society 25.3 (2012): 907–927.

[6] K. Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of points, graded and intersection theory, Annals of Mathematics 176 (2012), 925–978.

[7] E. Katz, Matroid theory for algebraic geometers, available at http://arxiv.org/abs/1409. 3503.

[8] E. Katz, J. Rabinoff, D. Zureick-Brown, Uniform bounds for the number of rational points on curves of small Mordell–Weil rank, available at http://arxiv.org/abs/1504.00694.

[9] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings I. Lecture notes in Mathematics 339 (1973).

[10] R. Lazarsfeld and M. Mustata, Convex bodies associated to linear series. Ann. Sci. Ec.´ Norm. Sup. 42 (2009), 783–835.

[11] Q. Liu, Algebraic geometry and arithmetic curves, Oxford graduate texts in mathematics, 2006.

[12] G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and theta functions, Contemp. Math. 465 (2008), Amer. Math. Soc., Providence, RI, 203–230.

[13] H. M. Stark and A. A. Terras. Zeta functions of finite graphs and coverings. Advances in Mathematics 121.1 (1996): 124-165.

[14] T. Tao, Algebraic combinatorial geometry: the polynomial method in arithmetic combina- torics, incidence combinatorics, and , disponible au http://arxiv.org/abs/ 1310.6482.

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