New timeslots: 
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Lectures:  Mo 16:00-17:30  Geom H3 and  Fr 10:15-11:45 Sed 19, 203
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Excercises: Mo 17:45-19:15  Geom H3
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We want to introduce to algebraic geometry so that we understand at least some basic facts necessary for the construction of the
Hilbert scheme of points on an algebraic surface.
A first course in algebra is a mandatory prerequisite for this course. Further knowledge of advanced algebra, function theory, differential geometry may occasionally be helpful. We start with a quick presentation of classical 
<a href="https://en.wikipedia.org/wiki/Algebraic_geometry">algebraic geometry</a> and then focus to the modern approach 
using the language of <a href="https://en.wikipedia.org/wiki/Scheme_(mathematics)">schemes</a>.
There are many good books on algebraic geometry and even more different flavours 
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<a href="https://mathoverflow.net/questions/2446/best-algebraic-geometry-textbook-other-than-hartshorne">Math overflow question: best algebraic geometry textbook?</a> 
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In this course we follow mostly these lecture notes:
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<a href="https://www.uio.no/studier/emner/matnat/math/MAT4215/data/masteragbook.pdf">Introduction to Schemes. G. Ellingsrud, J.C. Ottem</a> 
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Other accessible books (tbc):  
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<a href="https://link.springer.com/book/10.1007/978-3-642-19225-8">A Royal Road to Algebraic Geometry, A. Holme</a>
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Week 1: Informal introduction (cf. above links)
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Week 2: EO pp. 1-15  classical algebraic geometry
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Week 3: EO pp. 15-26 prime spectrum, Zariski topology,  residue fields, generic point 
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Week 4: EO pp. 26-41 affine spaces, irreducibility, connectedness, distinguished open sets, maps of prime spectra, fibres 
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Week 5: EO pp. 41-61 (pre-)sheaves, morphisms of (pre-)sheaves, stalks, locally ringed space
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Week 6: EO pp. 53-   affine schemes,...
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Week 7/8: EO pp. -- 100 glueing schemes, examples
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Week 9/10: EO pp. -- 128  finitness conditions, dimension
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Week 11/12:  Hilbert schemes of points 
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Additional literature:
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<a href="https://people.maths.ox.ac.uk/ritter/schemes.html">A. Ritter, Introduction to schemes</a>
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<a href="https://www.youtube.com/watch?v=JZKDmTIFR7A&list=PL8yHsr3EFj53j51FG6wCbQKjBgpjKa5PX">R. Borcherds, Lectures on algebraic geometry  </a> 
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<a href="https://www.youtube.com/watch?v=BX3jiLdehA4&list=PL8yHsr3EFj50Un2NpfPySgXctRQK7CLG-">R. Borcherds, Lectures on schemes</a> 
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<a href="https://link.springer.com/book/10.1007/978-1-4419-7400-6">L. Tu, An introduction to manifolds</a>
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03.05.2024: structure sheaf and locally ringed spaces
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<a href="https://link.springer.com/book/10.1007/978-3-540-37663-7">J. Neukirch: Algebraische Zahlentheorie</a> 
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<a href="https://link.springer.com/book/10.1007/978-1-84800-056-8">D. Perrin: Algebraic geometry</a>
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<a href="https://link.springer.com/book/10.1007/b97680">D.Eisenbud, J. Harris: The geometry of schemes<a>
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27.05.2024: Mathematics in Lean 
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<a href="https://www.imperial.ac.uk/media/imperial-college/faculty-of-engineering/computing/public/1819-ug-projects/Fernandez-I-MirR-Schemes-in-Lean.pdf">R.F. Mir: Schemes in Lean</a>
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<a href="https://arxiv.org/pdf/2405.10387" >K.Buzzard: GROTHENDIECK'S USE OF EQUALITY </a>, 
<a href="https://www.youtube.com/watch?v=Dp-mQ3HxgDE"> K. Buzzard: The future of mathematics</a>
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17.06.2024:  Hilbert Schemes of points
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<a href="https://math.umd.edu/~dbejleri/Hilbertschemes_gss.pdf">D. Bejleri: HILBERT SCHEMES: GEOMETRY, COMBINATORICS, AND REPRESENTATION THEORY</a>
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<a href="https://services.math.duke.edu/~ezra/MSRIworkshop/msriWorkshop.pdf">E. Miller:  Appendix: Hilbert schemes of points in the plane</a>, 
<a href="https://link.springer.com/book/10.1007/b138602">E. Miller, B. Sturmfeld: Combinatorial Commutative Algebra, Chapter 18</a>
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<a href="https://people.kth.se/~skjelnes/Pdffiler/forel_impa.pdf">M. Skjelnes: HILBERT SCHEMES OF POINTS ON SMOOTH SURFACES</a>