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Seminarium WM

• The maximal dimensions of path and graph algebras
  prof. dr hab. Piotr M. Hajac, IMPAN, Warszawa
  2026-10-28 godz. 13:15 - 14:15
  Graph theory is one of the most accessible parts of combinatorics, and one often uses graphs to visualize and study abstract objects in a plethora of mathematical contexts ranging from algebra and functional analysis to K-theory. Better still, the ubiquity of graphs goes beyond mathematics, e.g., Dijkstra’s graph-search algorithm is used in Google Maps. This talk will commence with an abridged hitchhiker's guide to the galaxy of applications of directed graphs. Then we will focus on solving a concrete optimization problem concerning the maximal dimension of both the path algebra and the graph algebra of a connected acyclic directed graph with N edges. The former is a broadly studied universal object, like the tensor algebra, and the latter is simply a matrix algebra. Finally, we will pose some reverse optimization problems: Given a dimension, what is the minimal sufficient number of edges? In particular, we will discuss the optimization-gap problem for matrix algebras, and speculate on its relation with the Cunningham chains of the first kind of prime numbers. (Based on joint work with Ł. Kaczmarczyk, M. Lowiel, and E. A. Pacheco.)

Seminaria w jednostkach

• TBA
  Zakład Fizyki Matematycznej, 2026-10-16 godz. 9:45 - 11:15
• Invariant von Neumann subalgebras of crossed products and tensor products
  Tattwamasi Amrutam,
  Zakład Analizy Funkcjonalnej, 2026-10-16 godz. 11:30
  Let $\Gamma$ be a countable discrete group acting on a tracial von Neumann algebra $(Q,\tau_Q)$ in a trace preserving way. We study invariant von Neumann subalgebras of $Q\rtimes\Gamma$ and of $L(\Gamma)\bar\otimes Q$, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation is scalar is necessarily contained in $L(\Gamma)$. Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant crossed-product sub algebras with non-trivial intersection with $Q$. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. If time permits, we will construct essentially free, prime, pointwise weakly mixing p.m.p. actions of $\F_n$ for every $2\leq n\leq\infty$, as well as non-prime examples with explicitly described factors. This is a joint work with Artem Dudko, Yongle Jiang, and Adam Skalski.

Ostatnio wydane publikacje

1. M. Iwanowska, A. Rybak, B. Wąsowska-Narojczyk
   Wpływ nauczania problemowego na wybrane aspekty efektywności kształcenia matematycznego
   Diagnozowanie uczenia się ucznia, studenta, nauczyciela, Między wartościami, doświadczeniem i wiedzą, (B. Niemierko, M.K. Szmigiel Ed(s).), (publ. by) TOMAMI sp. z o.o., Kraków, Poland, 2026, pp. 207-222.
2. J.-P. Gazeau
   Electromagnetic Field From Affine Symmetry of Punctured Plane
   Geometric Methods in Physics XLII, Workshop, Białystok, Poland, 2025, (A. Dobrogowska, D. Fernández, T. Goliński, P. Kielanowski Ed(s).), Trends in Mathematics , (publ. by) Birkhauser Verlag, 2026, pp. 69-76.
   DOI: 10.1007/978-3-032-21578-9_8
3. T. Goliński
   A Note on Precotangent Bundles: The Example of Grassmannians
   Geometric Methods in Physics XLII, Workshop, Białystok, Poland, 2025, (A. Dobrogowska, D. Fernández, T. Goliński, P. Kielanowski Ed(s).), Trends in Mathematics , (publ. by) Birkhauser Verlag, 2026, pp. 193-201.
   DOI: 10.1007/978-3-032-21578-9_17
4. A. Dobrogowska, G. Jakimowicz
   Eigenvalue Problem and the Description of Lie Algebras
   Geometric Methods in Physics XLII, Workshop, Białystok, Poland, 2025, (A. Dobrogowska, D. Fernández, T. Goliński, P. Kielanowski Ed(s).), Trends in Mathematics , (publ. by) Birkhauser Verlag, 2026, pp. 219-228.
   DOI: 10.1007/978-3-032-21578-9_19
5. K. Bardadyn, B. Kwaśniewski
   Banach Algebra Crossed Products by Inverse Semigroup Actions
   Geometric Methods in Physics XLII, Workshop, Białystok, Poland, 2025, (A. Dobrogowska, D. Fernández, T. Goliński, P. Kielanowski Ed(s).), Trends in Mathematics , (publ. by) Birkhauser Verlag, 2026, pp. 317-333.
   DOI: 10.1007/978-3-032-21578-9_27