[Settfa] [Set Theory] Special, Thursday 28 May, Jan Lang

[Adam Bartoš]

Besides the standard seminar on Tuesday, we have also a special seminar 
on Thursday afternoon this week.

Thursday, 28 May 2026 - 15:00 to 16:30
Place: IM, konírna

Speaker: Jan Lang, Ohio State University, USA
Title: Notes about modular-based topologies

Abstract

This talk concerns the topology generated by modular convergence in 
vector spaces equipped with a convex modular $\rho$, with particular 
emphasis on the case where $\rho$ does not satisfy the 
$\Delta_2$-condition. We show that the modular topology is a topological 
vector space topology precisely when $\rho$ satisfies $\Delta_2$. In the 
absence of this condition, several familiar features of normed spaces 
break down: modular balls need not be open, they may have empty 
interior, and modular convergence may be strictly weaker than Luxemburg 
norm convergence.

The general theory is illustrated in the variable exponent spaces 
$\ell^{(p_n)}$ and $L^{p(.)}(\Omega)$, where unbounded exponents lead to 
genuinely non-normable modular phenomena. We also discuss applications 
to Dirichlet energy minimization and weak solutions of boundary value 
problems for the p(x)-Laplacian with unbounded exponent.

For more information see the seminar web page at
https://www.math.cas.cz/index.php/events/seminar/6 
<https://www.math.cas.cz/index.php/events/seminar/6>

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