# Algebraic Geometry

### Varieties, Polyhedra, Computation

##### Winter Semester 2019/2020

# Preparatory Courses Summer 2019

These prepatory courses and seminars are part of the Thematic Einstein Semester on Algebraic Geometry of the newly installed MATH+ excellence cluster on application-oriented mathematics within the Berlin research landscape. There will be five courses and two seminars aiming to provide the theoretical basis to participate actively in a project group, the fall school, as well as the opening conference. Students of the three Berlin universities can earn ECTS credits (according to the respective course or seminar homepage).

## Condition: The Geometry of Numerical Algorithms

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**Lecturer: **Peter Bürgisser

**Time:** Tuesday, 14-16 and 16-18

**Place:** TU Berlin, H 2038 (14-16) and MA 548 (16-18)

**Description:** Design and analysis of numerical algorithms in nonlinear algebra, emphasizing the role of condition. Focus on polynomial equations. Complexity analysis by methods from probability and geometry.

## Variations on Bezout's Theorem

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**Lecturer: **Peter Bürgisser

**Time:** Thursday, 16-18

**Place:** TU Berlin, EW 184

**Description:** Bezout's Theorem is a fundamental result of algebraic geometry. There are important extensions to polynomial system with few terms (fewnomials) that build a bridge to convex geometry (toric varieties, Newton polytopes, mixed volume). The goal of the lecture is to prove and discuss these results from different angles. We shall mainly work over the complex numbers and rely also on tools from differential geometry or integral geometry, that will be devoloped in the lecture.

## Real Algebra

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**Lecturer: **Mario Kummer

**Time:** Thursday, 14-16

**Place:** TU Berlin, MA 644

**Description:** Real aspects of algebra, such as counting
real zeros or the study of systems of inequalities, are
dealt with. A major goal will be the solution of Hilbert's
17th problem: Each globally non-negative polynomial is a
sum of squares of rational functions. Other keywords:
ordered fields, real closed fields, semialgebraic sets,
Tarski principle, real spectrum.

## Algebraic Curves: Enumerative, Combinatorial and Computational Aspects

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**Lecturer: **Gavril Farkas

**Time:** Monday and Thursday, 9-11

**Place:** HU Berlin, Johann von Neumann-Haus, seminar room 20

**Description:** The course will focus on presenting an up to date survey of the many facets of the
modern theory of algebraic curves (Riemann surfaces) and their moduli spaces, with an emphasis on concrete and computational issues. Topics to be covered include the geometry of the moduli space of curves,
an introduction to the Hurwitz theory of counting covers of algebraic curves, Brill-Noether theory (both the
classical and the tropical approach) and the connection between curves and graph theory.

## Polyhedral Methods in Algebraic Geometry

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**Lecturer: **Christian Haase

**Time:** Tuesday, 10-12 and 14-16

**Place:** FU Berlin, Arnimallee 3, seminar room 024

**Exercise class:** Wednesday, 14-16, Arnimallee 3, seminar room 119

**Description:** This course has (at least) two target audiences: on the one hand, this course is for students of algebraic geometry who are curious to see their field interact with polyhedral geometry, and to even see it applied to statistics, economics and optimization. On the other hand, this course is for students of discrete geometry, optimization or combinatorics, from numerical analysis, stochastics or other fields who want to get a glimpse of what algebraic geometry is about, and see examples of applications to their favorite objects.

# Seminars Summer 2019

## Real Geometry and Optimization

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**Lecturer: **Rainer Sinn

**Time:** Wednesday, 16-18

**Place:** FU Berlin, Arnimallee 3, seminar room 130

**Description:** The goal of the seminar is to cover foundations in real algebraic geometry in the area of sums of squares (the keyword is Positivstellensatz) and applications to optimization (here the keyword is moment hierarchies) as well as combinatorial applications (like MAX CUT).

## Numerical Nonlinear Algebra

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**Lecturer: **Paul Breiding and
Sascha Timme

**Time:** Friday, 10-12

**Place:** TU Berlin, MA 316

**Description:** The Seminar is about applications and problems
related to the numerical solution of polynomial systems.
Participants should pick one of several projects and present
their own solution using the
Julia
package HomotopyContinuation.jl.
The seminar can be seen as the practical counterpart
of the course "Condition: The geometry of numerical algorithms".

For any question about these courses and seminars, you may contact the responsible lecturer.