- The PhD positions are fully funded for four years.
- The Starting date is October 1st, 2022.
- The
**Deadline for the applications **is **August** **1st 2022**.

The candidates for PhD positions should have a Master’s degree in Mathematics (obtained by September 11th 2022 at the latest), a very good knowledge of the English language, and should be of age up to 28 years (extension of this limit is possible in the case of using parental leave in the past).

Short description of the three open PhD positions:

**PhD position under the supervision of Prof. Dragan Marušič:**

The work of the selected PhD candidate will be primarily focused on studying symmetries of graphs.

There are many open problems that relate symmetry and combinatorial properties of graphs.

One of such problems is the question if every connected vertex-transitive graph admits a Hamilton path.

Although this is known to be true in some special cases, the general question remains open.

The PhD candidate will work on several different open problems in this area, the exact problems would be agreed with the candidate.

The candidate is expected to have a solid background in group theory.

Questions related to this PhD position should be sent to **dragan.marusic@upr.si**

**PhD position under the supervision of Prof. Bojan Kuzma:**

The work will be primarily focused on research in the fields of functional analysis, linear algebra, and graph theory.

The PhD student will be particularly involved in solving preserver problems that come from functional analysis and linear algebra with the help of the tools from graph theory. Typically, the preserver problems will be posed on subsets of operators acting on Banach spaces and on subsets of matrices over general fields. They will require characterizing all, possibly nonlinear, maps which preserve a given property. Since linearity of the preservers will not be assumed, an important approach which PhD student will investigate will be to build (an infinite) graph with an edge between two vertices if a given property holds for the two vertices. PhD candidate will try to infer algebraic relations between operators through the basic graph invariants (clique number, number of connected components …) of the corresponding vertices and, based on the obtained results, reduce the problem to already known preservers.

Questions related to this PhD position should be sent to **bojan.kuzma@upr.si**

**PhD position under the supervision of Prof. Enes Pasalic:**

The work will be primarily focused on cryptography and different topics of discrete mathematics.

APN (almost perfect nonlinear) functions are interesting discrete combinatorial objects that are defined over Galois fields (with the binary prime field) and have several different characterizations. These objects can be specified using certain combinatorial properties, coding theoretic concepts among others, but their most known characterization is given through their differential properties. Since F: GF(2)^n to GF(2)^n can be represented as a collection of n Bollean functions their structure may be also analyzed in terms of its coordinates. These Boolean functions are usually of some special form (bent or plateaued) which ensures that the APN property is satisfied. The proposed research topic aims to further investigate the structure of APN functions through its coordinate functions and to suggest a generic framework for their construction. Minimum requirements imposed on applicants include a good knowledge about finite fields, number theory and some elementary coding theory. It is of benefit if the applicant possesses a solid knowledge of group theory and algebraic geometry.

Questions related to this PhD position should be sent to **enes.pasalic@upr.si**

More information about the open call for PhD positions can be found under

https://www.upr.si/en/about-university/669-news/news-and-announcements/razpis-za-mlade-raziskovalce-v-letu-2022-