Students of active courses are reffered to ILIAS at the Universität zu Köln.
I am currently teaching and coordinating excersise classes for the lecture "Convex Optimization", held by Frank Vallentin.
I am co-supervising the block seminar on "Selected topics in convex optimization", with Frank Vallentin.
Arian Cyrus Joharian – Critical modular lattices of level 2 and 3 in the Gaussian core model, 2024
Lea Marie Hüpper – Lattices from Eisenstein matroids, 2024
Stephen Weißbach – The chromatic number of 4-dimensional lattices, 2023
Jana Pier – On the local optimality of the lattice A_n^* for lattice quantization, 2023
Toni Golian – SDD-Zertifikat für die Kusszahl tau_3, 2023
Antonia Thiemeyer – Lokale Optima der Gaußschen Masse, 6 month research project, with A. Heimendahl and F. Vallentin. This project was funded by HYPATIA.SCIENCE, building on this project we wrote the paper Critical even unimodular lattices in the Gaussian core model.
Lea Marie Hüpper – Computergestützte Analyse der Dichte von Gitterpackungen abgerundeter Tetraeder, 6 month research project, with F. Vallentin. This project was funded by HYPATIA.SCIENCE, in it we analyzed lattice packings of rounded tetrahedrons using computer aided methods.
I have taught the following courses. If you are interested in any materials associated to these courses that is no longer available through the course site please contact me.
This course was joint with Frank Vallentin. It consisted of 4SWS (4*45 minutes) of lectures and 2SWS of exercise classes each week.
Polynomial optimization represents a broad class of optimization problems. Here the objective function is a polynomial and the constraints are defined by polynomial inequalities. This versatile framework allows for the modelling of numerous optimization problems. Moreover, in practice, one can often solve these problems exactly or at least approximately. The goal of the lecture is to develop the theory of polynomial optimization problems, placing particular emphasis on algorithmic methodologies for identifying globally optimal solutions. The following topics have been covered:
Lecture 1 - 09.04. - Introduction
Lecture 2 - 12.04. - Languages, terms, and formulas
Lecture 3 - 16.04. - Theories, models, linear arithmetic
Lecture 4 - 19.04. - Quantifier elimination (in linear arithmetic)
Lecture 5 - 23.04. - Ideals, algebraic sets, Hilbert's Basissatz
Lecture 6 - 26.04. - Vanishing ideals, Hilbert's Nullstellensatz, ideal-algebraic set correspondence
Lecture 7 - 30.04. - Monomial orderings, multivariate polynomial division, monomial ideals
Lecture 8 - 03.05. - Gröbner bases, Buchberger's algorithm
Lecture 9 - 07.05. - Solving polynomial equations, model-completeness
Lecture 10 - 10.05. - Algebraically closed fields, a proof of Hilbert's Nullstellensatz
Lecture 11 - 14.05. - Sums of squares, the Gram matrix method
Lecture 12 - 17.05. - Hilbert's classification, the Motzkin polynomial, Sturm sequences
Lecture 13 - 28.05. - Sturm sequences and counting real roots
Lecture 14 - 31.05. - Real fields
Lecture 15 - 04.06. - Characterization of real closed fields
Lecture 16 - 07.06. - Model theory and real closed fields, Proof of Artin's theorem
Lecture 17 - 11.06. - Practical SOS
Lecture 18 - 14.06. - Real Nullstellensätze and Positivstellensätze
Lecture 19 - 18.06. - Proof of the Positivstellensatz of Krivine-Stengle
Lecture 20 - 21.06. - Schmüdgen's and Putinar's Positivstellensätze
Lecture 21 - 25.06. - Proof of Putinar's Positivstellensatz, SOS hierarchy for polynomial optimiztion
Lecture 22 - 28.06. - Moment sequences and matrices
Lecture 23 - 02.07. - The theorem of Curto, Fialkow, the quotient algebra
Lecture 24 - 05.07. - Finishing the proof of the theorem of Curto, Fialkow, the moment relaxation
Lecture 25 - 09.07. - Semidefinite programming hierarchies, Duality
Lecture 26 - 12.07. - Flat extensions
This course was joint with Frank Vallentin. It consisted of 4SWS (4*45 minutes) of lectures and 2SWS of exercise classes each week.
Polynomial optimization represents a broad class of optimization problems. Here the objective function is a polynomial and the constraints are defined by polynomial inequalities. This versatile framework allows for the modelling of numerous optimization problems. Moreover, in practice, one can often solve these problems exactly or at least approximately. The goal of the lecture is to develop the theory of polynomial optimization problems, placing particular emphasis on algorithmic methodologies for identifying globally optimal solutions. The following topics have been covered:
Lecture 1 - 09.04. - Introduction
Lecture 2 - 12.04. - Languages, terms, and formulas
Lecture 3 - 16.04. - Theories, models, linear arithmetic
Lecture 4 - 19.04. - Quantifier elimination (in linear arithmetic)
Lecture 5 - 23.04. - Ideals, algebraic sets, Hilbert's Basissatz
Lecture 6 - 26.04. - Vanishing ideals, Hilbert's Nullstellensatz, ideal-algebraic set correspondence
Lecture 7 - 30.04. - Monomial orderings, multivariate polynomial division, monomial ideals
Lecture 8 - 03.05. - Gröbner bases, Buchberger's algorithm
Lecture 9 - 07.05. - Solving polynomial equations, model-completeness
Lecture 10 - 10.05. - Algebraically closed fields, a proof of Hilbert's Nullstellensatz
Lecture 11 - 14.05. - Sums of squares, the Gram matrix method
Lecture 12 - 17.05. - Hilbert's classification, the Motzkin polynomial, Sturm sequences
Lecture 13 - 28.05. - Sturm sequences and counting real roots
Lecture 14 - 31.05. - Real fields
Lecture 15 - 04.06. - Characterization of real closed fields
Lecture 16 - 07.06. - Model theory and real closed fields, Proof of Artin's theorem
Lecture 17 - 11.06. - Practical SOS
Lecture 18 - 14.06. - Real Nullstellensätze and Positivstellensätze
Lecture 19 - 18.06. - Proof of the Positivstellensatz of Krivine-Stengle
Lecture 20 - 21.06. - Schmüdgen's and Putinar's Positivstellensätze
Lecture 21 - 25.06. - Proof of Putinar's Positivstellensatz, SOS hierarchy for polynomial optimiztion
Lecture 22 - 28.06. - Moment sequences and matrices
Lecture 23 - 02.07. - The theorem of Curto, Fialkow, the quotient algebra
Lecture 24 - 05.07. - Finishing the proof of the theorem of Curto, Fialkow, the moment relaxation
Lecture 25 - 09.07. - Semidefinite programming hierarchies, Duality
Lecture 26 - 12.07. - Flat extensions
This course was an introduction to discrete optimization. It consisted of 3SWS (3*45 minutes) of lectures and 1SWS of exercise classes each week. We covered the following topics:
Convexity and polyhedra: separation and representation theorems, the face lattice of convex polyhedra.
A quick recollection of linear programming.
Integer linear programming: totyaly unimodular matrices and applications of ILP to combinatoprial problems, the simplex method, cutting planes
An introduction to complexity theory: deterministic and nondeterministic Turing machines, the classes P and NP, NP-completeness.
This course was an introduction to discrete optimization. It consisted of 3SWS (3*45 minutes) of lectures and 1SWS of exercise classes each week. We covered the following topics:
Convexity and polyhedra: separation and representation theorems, the face lattice of convex polyhedra.
A quick recollection of linear programming.
Integer linear programming: totyaly unimodular matrices and applications of ILP to combinatoprial problems, the simplex method, cutting planes
An introduction to complexity theory: deterministic and nondeterministic Turing machines, the classes P and NP, NP-completeness.
This course was aimed at students in the master's program in their second year and PhD students. It consisted of 3SWS (3*45 minutes) of lectures and 1SWS of exercise classes each week.
We covered the following topics:
An overview of convex optimization with an emphasis on semidefinite programming.
The Lovasz theta number: it's application to graph coloring and the Shannon capacity of graphs, symmetry reduction.
Harmonic analysis on the sphere: Zonal spherical functions, symmetry reduction, basics of representation theory.
The linear programming bound on the sphere: bounding the size of spherical codes, energy minimization, Hermite interpolation on the sphere.
Least distortion embeddings of finite metric spaces.
This course was aimed at students in the master's program in their second year and PhD students. It consisted of 2SWS (2*45 minutes) of lectures and 2SWS of exercise classes each week. Lecture notes were made available (in german). An english version is in preparation. We covered the following topics:
1. lecture (06.04): Error correcting codes: introduction, equivalence, Hamming distance
2. lecture (13.04): duality and Hamming codes, the weight enumerator, the MacWilliams identity
3. lecture (20.04): extended Hamming codes, the binary Golay code
4. lecture (27.04): combinatorial t-designs
5. lecture (04.05): the LP bound (Delsarte bound), association schemes
6. lecture (11.05): association schemes and the Bose-Mesner algebra
7. lecture (15.05): the Hamming scheme, the LP bound in association schemes
8. lecture (25.05): energy minimization for codes
9. lecture (05.06): polynomial interpolation, optimality of the Golay code for energy minimization
10.lecture (19.06): spherical codes, Gegenbauer polynomials, the LP bound for spherical codes
11.lecture (22.06): Peter-Weyl theorem for the sphere, zonal spherical functions, O(R^n)-invariance and -irreducibility
12.lecture (26.06): proof of the Peter-Weyl theorem for the sphere, positive definite kernels
13.lecture (06.07): spherical t-designs, energy minimization for spherical codes
14.lecture (13.07): universal optimality on the sphere
This course was aimed at students in the master's program in their second year and PhD students. It consisted of 2SWS (2*45 minutes) of lectures and 2SWS of exercise classes each week. Lecture notes were made available (in german). An english version is in preparation. We covered the following topics:
An introduction to Euclidean lattices: duality, representations and spaces of lattices, on overview on classical lattice problems.
Voronoi reduction: the Voronoi cell and tessellation of a lattice, the Delaunay subdivision associated to a lattice, secondary cones and Voronoi reduction, Voronoi's principal domain of the first kind.
The lattice sphere covering problem: how to solve it using Voronoi reduction and determinant-maximization.
Graphs: spanning trees, the matrix-tree theorem.
Oriented matroids: axiomatics, duality, realizable oriented matroids, graphical and co-graphical oriented matroids, regular oriented matroids, deletion and contraction.
Zonotopal lattices: zonotopes, regular lattices, the Voronoi cell of zonotopal lattices, classification of zonotopal lattices, projections for graphical and co-graphical lattices.
Algorithms for problems on zonotopal lattices: the closest vector problem, lattice quantization.