Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
The lecture will probably be held in English.
Requirements on examinations, assessments and coursework will be described in the supplements of the module handbooks to be published as part of the course cataloque by end of October.
Teacher: Amador Martín Pizarro
Assistant: Charlotte Bartnick
Language: in English
In this course the basics of geometric model theory will be discussed and concepts such as quantifier elimination and categoricity will be introduced. A theory has quantifier elimination if every formula is equivalent to a quantifier-free formula. For the theory of algebraically closed fields of fixed characteristic, this is equivalent to requiring that the projection of a Zariski-constructible set is again Zariski-constructible. A theory is called \(\aleph_1\)-categorical if all the models of cardinality \(\aleph_1\) are isomorphic. A typical example is the theory of non-trivial \(\mathbb Q\)-vector spaces. The goal of the course is to understand the theorems of Baldwin-Lachlan and of Morley to characterize \(\aleph_1\)-categorical theories.
necessary: Mathematical Logic \
useful: Algebra and Number Theory
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)
Machine Learning and Mathematical Logic
Lecture: Do, 14-16h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined
Requirements on examinations, assessments and coursework will be described in the supplements of the module handbooks to be published as part of the course cataloque by end of October.
Teacher: Maxwell Levine
Language: in English
Developments in artificial intelligence have boomed in recent years, holding the potential to reshape not just our daily routines but also society at large. Many bold claims have been made regarding the power and reach of AI. From a mathematical perspective, one is led to ask: What are its limitations? To what extent does our knowledge of reasoning systems in general apply to AI?
This course is intended to provide some applications of mathematical logic to the field of machine learning, a field within artificial intelligence. The goal of the course is to present a breadth of approachable examples.
The course will include a gentle introduction to machine learning in a somewhat abstract setting, including the notions of PAC learning and VC dimension. Connections to set theory and computability theory will be explored through statements in machine learning that are provably undecidable. We will also study some applications of model theory to machine learning.
The literature indicated in the announcement is representative but tentative. A continuously written PDF of course notes will be the main resource for students.
Background in basic mathematical logic is strongly recommended. Students should be familiar with the following notions: ordinals, cardinals, transfinite induction, the axioms of ZFC, the notion of a computable function, computable and computably enumerable sets (a.k.a. recursive and recursively enumerable sets), the notions of languages and theories and structures as understood in model theory, atomic diagrams, elementarity, and types. The concepts will be reviewed briefly in the lectures. Students are not expected to be familiar with the notion of forcing in set theory.
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective in Data (MScData24)
Seminar: Di, 16-18h, SR 127, Ernst-Zermelo-Str. 1
Preregistration: no preregistration
Preliminary Meeting 23.07., 12:15, Fakultätssitzungsraum 427, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
In HISinOne: no course registration, but exam registration until 8 October 2025.
Teacher: Heike Mildenberger
Assistant: Stefan Ludwig
Language: in German
The topics are: Finite and infinite graphs, Eulerian paths, connectivity properties, colourings, spanning trees, random graphs. If desired, more advanced subjects, such as the Rado graph and 0-1 laws or probabilistic methods, can also be presented.
Linear Algebra I and II, Analysis I and II
Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)
Lecture: Di, Do, 12-14h, HS II, Albertstr. 23b
Tutorial: 2 hours, date to be determined
Sit-in exam 28.07., 14:00-17:00, SR 318, Ernst-Zermelo-Str. 1
Sit-in exam (resit) 09.12., 14:30-17:30, SR 318, Ernst-Zermelo-Str. 1
Teacher: Amador Martín Pizarro
Assistant: Stefan Ludwig
Language: in German
This introductory course in mathematical logic consists of several parts. It the basics of predicate logic and a brief introduction to model theory and the axiom system as well as the axiom system of set theory. The aim of the lecture is to explain the recursion-theoretical content of the predicate calculus, in particular the so-called Peano-arithmetic and Gödel's incompleteness theorems.
Basic knowledge of mathematics from first semester lectures
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Mathematical Concentration (MEd18, MEH21)
Pure Mathematics (MSc14)
Elective (MSc14)
Elective (MScData24)
Seminar: Di, 16-18h, SR 403, Ernst-Zermelo-Str. 1
Preliminary Meeting 29.01., 13:15, Raum 313, Ernst-Zermelo-Str. 1
Preparation meetings for talks: Dates by arrangement
Teacher: Heike Mildenberger
Assistant: Maxwell Levine
Language: Talk/participation possible in German and English
Set Theory
Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Elective (MScData24)
Set Theory – Independence Proofs
Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined
Teacher: Maxwell Levine
Assistant: Hannes Jakob
Language: in English
How does one prove that something cannot be proved? More precisely, how does one prove that a particular statement does not follow from a particular collection of axioms?
These questions are often asked with respect to the axioms most commonly used by mathematicians: the axioms of Zermelo-Fraenkel set theory, or ZFC for short. In this course, we will develop the conceptual tools needed to understand independence proofs with respect to ZFC. On the way we will develop the theory of ordinal and cardinal numbers, the basics of inner model theory, and the method of forcing. In particular, we will show that Cantor's continuum hypothesis, the statement that \(2^{\aleph_0}=\aleph_1\), is independent of ZFC.
Required: Mathematical Logic
Elective (Option Area) (2HfB21)
Compulsory Elective in Mathematics (BSc21)
Mathematical Concentration (MEd18, MEH21)
Pure Mathematics (MSc14)
Mathematics (MSc14)
Concentration Module (MSc14)
Elective (MSc14)
Elective (MScData24)