# Selected Publications

### On Kazhdan's property (T) for $\operatorname{Aut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$

We prove that $\operatorname{Aut}(F_n)$ has Kazhdan’s property (T) for every $n \geqslant 6$. Our proof relies on relating the Laplace operators of $\operatorname{SAut}(F_n)$ for various $n$ via symmetrisation by torsion. The method works also for $\operatorname{SL}_n(\mathbb{Z})$ and for $n \geqslant 3$ yields a new proof of the fact that $\operatorname{SL}_n(\mathbb{Z})$ has property (T). We also provide new, explicit lower bounds for the Kazhdan constants of $\operatorname{SAut}(F_n)$ (with $n \geqslant 6$) and of $\operatorname{SL}_n(\mathbb{Z})$ (with $n \geqslant 6$) with respect to natural generating sets.
Preprint, 2018.

### $\operatorname{Aut}(\mathbb{F}_5)$ has property (T)

We give a constructive, computer-assisted proof that $\operatorname{Aut}(\mathbb{F}_5)$, the automorphism group of the free group on $5$ generators, has Kazhdan’s property (T).
Preprint, 2017.

### Certifying Numerical Estimates of Spectral Gaps

We establish a lower bound on the spectral gap of the Laplace operator on special linear groups using conic optimisation. In particular, this provides a constructive (but computer assisted) proof that these groups have Kazhdan property (T).
Accepted in Groups Complexity Cryptology, 2017.

### Constructions of exotic actions on product manifolds with an asymmetric factor

Given $M\times S^n$, where $M$ is an asymmetric manifold, i.e. a manifold which does not admit any non-trivial action of a finite group, we try to answer how many (non-product) actions are there on such oddities?
In review, 2016.

# Publications

#### Effective topological complexity of spaces with symmetries

In Publicacions Matemàtiques Volume 62, Number 1 (2018), p. 55-74

#### On equivariant and invariant topological complexity of smooth $\mathbb{Z}/p$-spheres

In Proceedings of American Mathematical Society Volume 145, Number 9 (2017), p. 4075-4086

#### Group actions on complex projective spaces

In The Topology and the Algebraic Structures of Transformation Groups, Proceedings of RIMS – Kokyuroku, Volume 1922, (2014), p. 147-153

#### On Representation of the Reeb Graph as a Sub Complex of Manifold

In Topological Methods in Nonlinear Analysis, Volume 45, Issue 1 (March 2015), p. 287-307

#### Non Symplectic Actions on Complex Projective Spaces

In Journal of Symplectic Geometry, Volume 10, Issue 1 (2012), p. 17-26

# Teaching

In the winter semester of 2017 I am teaching@AMU the following courses:

• Introduction to Mathematics for CS (Mon 11:45 A2-10; Thu 11:45 A2-8)
• Calculus 2 for CS (Thu 15:30 A2-22)
• Elementary Mathematics (Mon 15:30 A2-24; Thu 13:45 A2-22)
• Elements of Statistics (Tue 14:15; Tue 17:00; Fri 8:00)

In the summer semester of 2018 I will be teaching@AMU the following courses:

• Abstract Algebra 2 (TBA)
• Elements of Applied Mathematics (TBA)

# Contact

• kalmar@amu.edu.pl
• (0048 61 829) 5497
• Room B1-30, Faculty of Mathematics and Computer Science, ul. Umultowska 87, 61-614 Poznań, Poland
• Tuesday 10:30 – 12:00, or appointment by email