# Clemens Brüser

PhD student

NameClemens Brüser

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Welcome to my personal homepage. I have started my PhD at TU Dresden with Mario Kummer in November 2022. My field of research is real algebraic geometry in a wider sense, more specifically I am working on determinantal representations of positive polynomials. Personally I am furthermore interested in historic aspects and developements of mathematics - though this is not my own field of research.

I did my Master's degree on Quantifier Elimination in Matrix Algebras under the supervision of Tim Netzer at the University of Innsbruck.

Previously, I have completed a teacher training program in the subjects Latin and Mathematics at the University of Innsbruck. In my thesis I studied the connection between mathematics and panegyric in the neo-Latin text Problema Austriacum by Albert Curtz. My supervisor was Martin Korenjak.

I am one of four elected representatives of scientific staff of the Insititute's Council of the intitute of geometry.

**Upcoming Events and Talks:**

Short stay, Institute of Algebra at University of Innsbruck, 07.-08.11.2024.

**Important Links:**

Geometrie-Seminar/Graduate Lectures: Seminar of the institute.

AGK-Seminar: Shared seminar on Algebra, Geometry and Combinatorics.

## Conferences

**Mathematics**

Real Algebraic Geometry and Hodge Theory, Summer School, TU Dresden, 09.-13.09.2024.

MEGA - Effective Methods in Algebraic Geometry, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, 29.07.-02.08.2024.

MoPAT - Moments and Polynomials: Applications and Theory, University of Konstanz, 11.-14.03.2024.

Let's get **R**eal, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, 07.-08.06.2023.

New Directions in Real Algebraic Geometry, Mathematisches Forschungsinstitut Oberwolfach, 19.-24.03.2023.

- Reporter of the Oberwolfach Report 15/2023.

Macaulay2 Bootcamp, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, 13.-14.12.2022.

**Latin**

Sapiens Ubique Civis VIII, Szeged, 01.-03.09.2021.

## Talks

**Mathematics**

Short stay, Institute of Algebra at University of Innsbruck, 07.-08.11.2024.

*Quadratic Determinantal Representations of Positive Polynomials*. MoPAT - Moments and Polynomials: Applications and Theory, University of Konstanz, 11.-14.03.2024.

*What is ... a semi-algebraic set?* Planck Institute of Molecular Cell Biology and Genetics, 08.03.2024.

*Real root counting algorithms*. Let's get **R**eal, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, 07.-08.06.2023.

*Positive Semidefinite Quadratic Determinantal Representations*. Geometry Seminar/Graduate Lectures TU Dresden, 16.05.2023.

*Positive Semidefinite Determinantal Representations*. Dresden-Leipzig-Seminar Algebra und Geometrie, Leipzig, 28.04.2023.

**Latin**

*The choir in Agamemnon suimet victor by Joseph Resch*. Sapiens Ubique Civis VIII, Szeged, 01.-03.09.2021.

**Mathematics**

*Quantifier Elimination in Matrix Algebras*. Master thesis, University of Innsbruck, 2022.

B. C., Kummer Mario: (Positive) Quadratic Determinantal Representations of Quartic Curves and the Robinson Polynomial [in preparation].

**Latin**

*Zum Chor in Joseph Reschs Agamemnon suimet victor*. Humanistica Lovaniensia [in preparation].

*Mathematics and Panegyric*. *The Problema Austriacum by Albert Curtz*, LIAS 2021/1, 35-61.

*Albert Curtz: „Problema Austriacum”*.* Text mit Einleitung, Übersetzung, Anmerkungen und didaktischen Überlegungen*, Diploma thesis, University of Innsbruck, 2019.

Semester |
Course |
Room |
Time |

Summer term 2024 | Conscructive Geometry | WIL/C 107 | Tuesday 07:30-09:00 (even week) |

Conscructive Geometry | WIL/A 221 | Wednesday 09:20-10:50 (odd week) | |

Summer term 2023 | Conscructive Geometry | Z21/217 | Wednesday 16:40-18:10 (even week) |

Conscructive Geometry | Z21/217 | Wednesday 16:40-18:10 (odd week) |

In the past, I supervised exercise classes in the following subjects at the University of Innsbruck and the University of Freiburg:

- Linear Algebra 1-2
- Analysis 1
- Algebra und Number Theory
- Commutative Algebra
- Elementary Geometry

For me, the connection between Latin studies and Mathematics is primarily of historic interest, which is rooted in the following reasons.

- For the most part of modern time (~ 1450-1750), Latin was
*the*language of science. Many groundbreaking results were first published in Latin. Jakob Bernoulli published his ideas on probability theory in 1713 in the treatise*Ars Coniectandi*. Other famous examples for the significance of the Latin language are Gauß'*Disquisitiones Arithmeticae*(1801) or the diverse works of Leonhard Euler. In 1889 still, Giuseppe Peano published his*Arithmetices Principia*in Latin. These works and ideas cannot be read in their original form without knowledge of the Latin language. - More general one should ask about the history of Mathematics. From this perspective, many recent and elaborate results can be motivated. As an example, axiomatic geometry can be developed purely from incidence realtions and related concepts. But its origins lie in Euclid's
*Elements*- along with the extensive discussion (often in Latin) of the famous parallel postulate in the subsequent centuries. Based on this approach, today's axioms are just a natural abstraction of the intuition in ancient times. This natural process is traced very well in the book*Geometry: Euclid and Beyond*(Robin Hartshorne, 2000). - Neo-Latin mathemtaical treatises are interesting in and of themselves, precisely because they do not conform to today's norms on mathematical literature. They often convey a completely different intent, but do so in the language of Mathematics. As an example, in a New Year's letter to a friend, Johannes Kepler writes about sexangular snowflakes (
*De nive sexangula*, 1611), but also states Kepler's Conjecture, which has been proven only recently.

*Nihil equidem magis opto, quam ut iis, quibus scientiarum incrementa cori sunt, placeant, quae vel hactenus desiderate explent, val aditum ad nove aperiunt.* (C.F. Gauß, *Disquisitiones Arithmeticae*, p. XII)

And I desire nothing more than that all those, to whom the promotion of scientific knowledge is important, will find joy in these ideas, which either solve problems that had been unsolved so far, or which show the way to new insight.