## CIMPA Research School on Combinatorial and Computational Algebraic Geometry |
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## University of Ibadan, Oyo State, Nigeria, 11—24 June 2017. |
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## Preliminary Schedule## Information Sheet About School (.pdf) |
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We acknowlege additional support from
The University of Ibadan, Max Planck Institute for Mathematics in the Sciences, Leipzig, Department of Mathematics, Texas A&M University, Department of Mathematics and Computer Science, Emory University, and the French Embassy in Nigeria. |

We will be staying at Adebayo Akande Hall near the University of Ibadan Campus.

Registration:

Students who are not from Nigeria and who wish to attend are asked to register on the CIMPA website of the school. This website is now taking applications, and there is a deadline of February 5, 2017.

Those from Nigeria should contact Dr. H.P. Adeyemo (University of Ibadan, Ibadan, Nigeria),

We expect to fully support those students we accept up to the limits of our budget. We will also admit some students who have full or partial support from other sources. If you have any questions, please contact Dr. Adeyemo or Professor Sottile.

- Dr. H. Praise Adeyemo, (University of Ibadan, Ibadan, Nigeria), Co-chair.
- Erwan Brugallé, (École polytechnique), Co-chair.
- Victoria Powers, Emory University, Atlanta, Georgia, USA.
- Frank Sottile, Texas A&M University, College Station, Texas, USA.

- Prof. G.O.S. Ekhaguere, University of Ibadan .
- Prof. Ezekiel Ayoola, University of Ibadan .
- Prof. O.O. Ugbebor, University of Ibadan .
- Dr. V.F. Payne, University of Ibadan .
- Dr. U.N. Bassey, HOD Mathematics, University of Ibadan .
- Dr. Deborah Ajayi, University of Ibadan .
- Dr. H. Praise Adeyemo, University of Ibadan .

- Prof. A. Kuku (President, African Academy of Sciences).
- Prof. S.A. Ilori (AMMSI Regional Coordinator Zone 1 (Anglophone)).
- Prof. A.T. Solarin (President, African Mathematical Union).
- Prof. S.E. Onah (Director, National Mathematical Centre, Abuja).
- Prof. M.O. Ibrahim (President, Mathematical Association of Nigeria).
- Prof. N.I. Akinwande (President, Nigerian Mathematical Society).

This research school will introduce the participants to some basics of algebraic geometry with an emphasis on computational aspects, such as Groebner bases and combinatorial aspects, such as toric varieties and tropical geometry. We will also learn how to use the freely available software Macaulay2 for studying algebraic varieties. The lecturers for this school are all active in these areas and collectively have deep experience both as researchers and educators through the supervision of students; Ph.D. and postdoctoral, as well as the organization of and lecturing in short courses.

This CIMPA Research School will last two weeks, and will have seven short courses of
four or five lectures each, as described below.
These will range from foundational to provide background through more
advanced topics. We plan to have lectures in the morning and just after lunch
(approximately four hours each day) with afternoon exercise sessions, including computer labs to
gain experience using open-source software such as Macaulay2.
Each day of the research school will
conclude with a more advanced research talk given by participants and by some
distinguished Nigerian Mathematicians.
This model of lectures, exercise sessions, and research talks has been used successfully at
past summer schools.

This school is aimed at faculty and advanced graduate students from Nigeria and
neighboring countries. We are planning to have approximately 40 participants with 25 from
Nigeria and 15 from other countries from Africa, particularly from Nigeria's neighbours, both
anglophone and francophone.
If more funding is found, we will invite more students.

Quoting Sophie Germain, "Algebra is written geometry and geometry is drawing algebra". Algebraic geometry is the study of common zeroes associated to a set of polynomials. I will give a short course on algebraic sets and ideals. This will focus on Hilbert's theorems and their consequences which establish a dictionary between algebraic sets and ideals. This dictionary is the source of the strength of algebraic geometry, for it allows us to study algebraic sets through their defining polynomials and vice versa

Enumerative geometry is the area of mathematics which studies questions like: how many lines pass through two points (easy)? How many conics pass through five conics pass through five points (easy)? How many cubics with a crossing point pass through 8 points (less easy)?... The aim of this course is to give a basic introduction to this topic from different points of view: enumeration of algebraic curves, and enumeration of tropical curves. In particular, tropical geometry provides an efficient tool to solve enumerative problem via combinatorial methods.

Algebraic structures, polynomials, and ideals of polynomial rings. Gr&oum;obner basis basics, orderings on monomials, division algorithm, monomials ideals, Hilbert Basis Theorem and Gröobner Basis Buchberger's algorithm, ideal membership problem, implication problem and elimination theory.

If a real polynomial

Tropical geometry can be viewed as algebraic geometry over the max-plus semi-field. Its objects are polyhedral in nature and are of interest to classical algebraic geometers because tropical varieties appear via a degeneration procedures of classical varieties. Yet, in the world of tropical geometry there are many objects not arising in this way. Sometimes they arise purely from combinatorics.

In this way, tropical geometry provides a powerful bridge between the worlds of combinatorics and algebraic geometry and the benefits of this connection go both ways. Combinatorial methods can be used to solve problems in classical algebraic geometry. A major example of this is the tropical enumerative geometry of complex and real curves via Mikhalkin's correspondence theorem.

On the other hand, tools standard in algebraic geometry can be used to approach structures in discrete mathematics. This has been quite useful in the study of matroids, which, thanks to tropical geometry, can be treated with the tools and theorems of classical algebraic geometry. In particular, the development of the tropical intersection theory has led to the resolution of conjectures in matroid theory in the very recent work of Adiprasito, Huh and Katz.

Among the simplest ideals are toric ideals–prime ideals generated by binomials. These are the ideals of toric varieties, and they enjoy a strong relation to geometric combinatorics of point conficurations and fans. This course will introduce these objects and develop their basic properties with a focus on their relation to combinatorics and how this controls the geometry of the corresponding algebraic variey.

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi a decade ago, and it has found numerous applications. We discuss the use of orthogonal tensor decompositions in data analysis, and we present work with Abo and Seigal aimed at characterizing which configurations of vectors arise as the eigenvectors of some tensor. This short course also serves an invitation to applied algebraic geometry.

Last modified: Wed Apr 12 14:08:11 EDT 2017