Time and place

Seminar: Di, 16-18h, SR 125, Ernst-Zermelo-Str. 1
Preliminary Meeting
Preparation meetings for talks: Dates by arrangement

Teacher: Heike Mildenberger
Assistant: Maxwell Levine
Language: Talk/participation possible in German and English

Elective (Option Area) (2HfB21)
Mathematical Seminar (BSc21)
Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)
Mathematical Seminar (MSc14)
Elective (MSc14)
Elective (MScData24)

Time and place

Lecture: Do, 14-16h, SR 226, Hermann-Herder-Str. 10
Tutorial: 2 hours, date to be determined and announced in class

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)

Time and place

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)

Time and place

Lecture: Di, Do, 12-14h, SR 404, Ernst-Zermelo-Str. 1
Tutorial: 2 hours, date to be determined and announced in class

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)

Time and place

Lecture: Mi, 12-14h, SR 127, Ernst-Zermelo-Str. 1

Teacher: Maxwell Levine

Time and place

Mo Di, 16-18h, SR 404, Ernst-Zermelo-Str. 1

Teacher: Heike Mildenberger
Assistant: Maxwell Levine

Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)

Time and place

Di, 12-14h, SR 404, Ernst-Zermelo-Str. 1

Teacher: Christian Ketterer
Assistant: Maxwell Levine

Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)

Time and place

Lecture: Do, 14-16h, SR 404, Ernst-Zermelo-Str. 1

Teacher: Maxwell Levine
Assistant: Maxwell Levine
general:

Time and place

Di, 16-18h, SR 125, Ernst-Zermelo-Str. 1

Teacher: Heike Mildenberger
Assistant: Maxwell Levine

Undergraduate Seminar (2HfB21, BSc21, MEH21, MEB21)

Time and place

Mo 16-18h, SR 404, Ernst-Zermelo-Str. 1, Fr, 16-18h, SR 403, Ernst-Zermelo-Str. 1

Teacher: Heike Mildenberger
Assistant: Maxwell Levine

Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)

Time and place

Lecture: Mi, 14-16h, SR 125, Ernst-Zermelo-Str. 1

Teacher: Maxwell Levine
general:

Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)

Time and place

Mo, 14-16h, SR 404, Ernst-Zermelo-Str. 1

Teacher: Heike Mildenberger
Assistant: Maxwell Levine

Supplementary Module in Mathematics (MEd18)
Compulsory Elective in Mathematics (BSc21)

Teacher: Heike Mildenberger
Assistant: Maxwell Levine

Compulsory Elective in Mathematics (BSc21)
Supplementary Module in Mathematics (MEd18)

Time and place

Exercise session: Do, 16-18h, -, -
Q&A session: Do, 11-12h, -, -
Sit-in exam 22.07., 10:00-13:00

Teacher: Heike Mildenberger
Assistant: Maxwell Levine
general:

Pure Mathematics (MSc14)
Mathematical Concentration (MEd18, MEH21)
Compulsory Elective in Mathematics (BSc21)

Time and place

Lecture: Do, 14-16h, , online
Exercise session: Mo, 12-14h, , online

Teacher: Maxwell Levine
Assistant: Christian Bräuninger
general:

Supplementary Module in Mathematics (MEd18)
Compulsory Elective in Mathematics (BSc21)

Teacher: Heike Mildenberger
Assistant: Maxwell Levine

Supplementary Module in Mathematics (MEd18)
Compulsory Elective in Mathematics (BSc21)