Home page of the research group of
Katarzyna and Franz-Viktor Kuhlmann

Institute of Mathematics,
University of Szczecin,
ul. Wielkopolska 15,
70-451 Szczecin,
Poland
Email: katarzyna.kuhlmann [at] usz.edu.pl,
fvk [at] usz.edu.pl, fvk [at] math.usask.ca

• general valuation theory,
• applications of valuation theory to algebraic and arithmetic algebraic geometry: local uniformization and desingularization in arbitrary characteristic and arbitrary dimension,
• model theory of valued fields (generalizations of the Ax-Kochen-Ershov Theorem, quantifier elimination, decidability),
• valuations on function fields, Zariski spaces of places,
• spaces of real places,
• ordered sets, abelian groups and fields,
• hyperfields,
• ultrametric spaces,
• fixed point theory of contracting functions (ball spaces).

The Ph.D. students in our research group:

Hanna Ćmiel
Alessandro Linzi
Hanna Stojałowska (Hanna did her Masters degree on "Totally positive units of a real holomorphy ring" in 2019 with Katarzyna)
Piotr Szewczyk (Piotr did his Masters degree on the "Continuity of roots of polynomials over valued fields" in 2019 with FV)
Piotr Błaszkiewicz (Piotr did his Bachelor degree on "Algebraic Structures used for Quantifier Elimination in Valued Fields" in 2019, and his Masters degree on "Quantifier Elimination in valued fields with respect to the corresponding hyperfields" in 2021, both with FV)

Publications with participation of our Ph.D. students:

1. Hanna Ćmiel, Alessandro Linzi, Piotr Szewczyk and Piotr Błaszkiewicz: Caristi-Kirk and Oettli-Théra ball spaces, and applications, J. Fixed Point Theory Appl. 21 (2019), no. 4, Paper No. 98
   (pdf file)
2. Hanna Ćmiel, Franz-Viktor Kuhlmann and Katarzyna Kuhlmann: A generic approach to measuring the strength of completeness/compactness of various types of spaces and ordered structures, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 115, Article number: 156 (2021)
   (pdf file)
3. Alessandro Linzi and Hanna Stojałowska: Hypervaluations on hyperfields and ordered canonical hypergroups, to appear in the Iranian journal of mathematical sciences and informatics
   (pdf file)
4. Hanna Ćmiel, Franz-Viktor Kuhlmann and Piotr Szewczyk: On the continuity of roots of polynomials over valued fields, submitted
   (pdf file)
5. Katarzyna Kuhlmann, Alessandro Linzi and Hanna Stojałowska: Orderings and valuations in hyperfields, submitted
   (pdf file)

Currently suggested topics for Master's and Ph.D. studies in our group:

• Valuations on rational function fields in one variable: comparison of key polynomials, minimal pairs, generating sequences and other means of their classification and description, with applications to local uniformization.
• Valuations on rational function fields in several variables: their construction and description.
• Valuations on algebraic function fields: properties of their Zariski spaces, structure of valued function fields with defect.
• The valuation theory of deeply ramified fields, generalizations of the theory of tame fields.
• Algebra and model theory of extremal fields.
• Quantifier elimination in valued fields.
• Decidability of Laurent series fields over finite fields, and of their perfect hulls.
• Theorems on the continuity of roots.
• Which topological spaces can be realized as spaces of real places?
• General properties and classification of ball spaces, application to fixed point and coincidence theorems in various mathematical areas.
• Topological fixed point theorems.

Support for young mathematicians:

We are particularly active in the support of mathematicians early on in their career, giving advice on their research projects and articles in preparation. We invite them to visit us for these purposes at our Institute, and we organize conferences and workshops specially geared towards young mathematicians. Everyone who is interested in learning about our projects and areas of expertise is cordially invited to visit us or to contact us by email.

We support our students in building up connections with other researchers around the world, either by visiting them, or by inviting these researchers to visit us. Our students are actively participating in the conferences and workshops we organize, such as:

International connections and collaborations:

We have connections to individual researchers and mathematical centers at the following universities:
Brazil: ICMC - USP Sao Carlos,
Czech Republic: Institute of Mathematics of the Czech Academy of Sciences,
France: Institut Mathématique de Jussieu - Paris Rive Gauche, Toulouse,
Germany: Münster, Dresden, Hochschule RheinMain,
Poland: University of Wroclaw, University of Silesia at Katowice,
USA: University of Missouri - Columbia, University of Pennsylvania, University of Hawaii, Graduate Center New York.

The Valuation Theory Home Page:

We are maintaining a web site that gives information on what is going on in valuation theory, announces conferences andcontains a preprint repository:

Conferences:

We have organized and are organizing a number of conferences and workshops. Detailed information and links can be found on The Valuation Theory Home Page. Since the year 2000, FV has organized the Colloquiumfest, a small and mostly low budget conference, with various co-organizers. It also has taken place at various universities and venues: University of Saskatchewan, CUNY Graduate Center in New York, IHP Paris, Bogazici University in Istanbul, University of Valladolid, University of Central Lancashire, University of Szczecin, Federal University of São Carlos in Brazil, Institute of Mathematics of the Czech Academy of Sciences in Prague.
Would you like to co-organize a Colloquiumfest at your institution? Just contact us!

Furthermore, FV initiated and co-organized two big international Valuation Theory Conference, each of which was followed by the publication of proceedings volumes. The third one is to follow in the next years - perhaps in Poland?

Last update: February 20, 2022