Seminars

The Departmental Scientific Seminar takes place on Thursdays from 8:15 AM to 10:00 AM in room PE1.

Its main goal is to collectively delve into selected areas of contemporary mathematics, and through this, intellectually engage those who participate. Based on the belief that—as with any advanced field of knowledge—using the subtle tools of modern analysis cannot be learned through passive participation, we strive to avoid dividing the lecturers and the audience. Each participant is encouraged to prepare and present, in collaboration with others, the next batch of material of interest—this is the best way to understand it in detail—and each significant result presented on the board should be thoroughly discussed by the participants, thus ensuring a thorough understanding. There are no stupid, unimportant, or irrelevant questions—if something is unclear, we do not proceed. Therefore, the goal is not to simply present the material, but to ensure that the methods, tools, and theorems discussed become the individual property of each of us.

We hope that, thus equipped, we will be more willing and effective in meeting the challenges facing us in our research and teaching. We are also pleased that staff and students from other departments, as well as from other universities, want to learn mathematics with us.

Over the past dozen or so years of the seminar's history, we have studied only a few (but slowly and thoroughly) monographs devoted to stochastic processes and related fields (a topic close to all department staff):

"Probability with Martingales" by D. Williams, Cambridge University Press, 1991,

"Functional Analysis" by P. D. Lax, 2002, Wiley (excerpts),

"Markov Chains" by J. Norris, Cambridge University Press, 1997 (first part),

"Poisson Processes" by J.F.C. Kingman, Oxford, 1992 (excerpts),

"Lectures on the Theory of Stochastic Processes" by A. D. Wentzel, PWN 1980,

"Generators of Markov Chains" by A. Bobrowski, Cambridge University Press, 2020.

We also participated several times (in a slightly smaller group) in intensive TULKA webinars. We explored the arcana of the theory of Hilbert spaces with reproducing kernels

"An Introduction to the Theory of Reproducing Kernel Hilbert Spaces," by V. Paulsen and M. Raghupathi, Cambridge University Press, 2016.

and the theory of Banach spaces

"A Short Course on Banach Space Theory," by N. L. Carothers, Cambridge University Press, 2005.

Those interested in participating in the seminar are asked to contact Dr. E. Ratajczyk (e.ratajczyk@pollub.pl).

The Department's teaching seminar takes place on Thursdays, from 10:15 AM to 12:00 PM in room PE1.

Its goal is to broaden the horizons of all Department staff in key areas of mathematics that are central to our teaching. For several months, we have been focusing on the most interesting aspects of mathematical analysis, discussing, among others:

Selected theorems for number sequences (including Archimedes' principle and Cantor's properties, Ascoli's theorem, theorems on monotonic sequences and subsequences, Cauchy's theorem, the connection between the completeness of the set of real numbers and the properties of monotonic sequences);

Convergence/divergence of selected sequences (including the irrationality and transcendentality of e, the divergence of the sequence (sin(n)) as n→∞, the set of its partial limits); Certain criteria for the convergence of sequences and their averages (including Stolz's theorem, Kronecker's and Toeplitz's theorem, and examples of applications to the laws of large numbers);

Boundaries of sets, upper and lower limits.

Number series (including the continuity principle, series with non-negative terms, Riemann's theorem on rearranging terms of conditionally convergent series);

Several theorems on real functions (including the Darboux property and the Bolzano-Cauchy theorem, monotonicity and continuity of one-to-one functions and functions with the Darboux property, Weierstrass's theorem on reaching the limit, one-sided limits of monotonic functions, and the set of points of discontinuity); Functional sequences (including uniform continuity of functions, Heine's theorem, pointwise and uniform convergence of functional sequences, the uniform norm, Cauchy's condition for uniform convergence of functional sequences, the theorem on the continuity of the limit of sequences of continuous functions, and the Stone-Weierstrass theorem).

Next, we will address selected topics in mathematical statistics.

Those interested in participating in the seminar are asked to contact Dr. E. Ratajczyk (e.ratajczyk@pollub.pl).