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- Maths
- Trigonometry
- Geometry
- Statistics
- Algorithms
13 mathematical publications. 2 publications in "Bulletin of the London Mathematical Society". Book published at Cambridge University Press. 20 years of teaching experience.
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- Trigonometry
- Geometry
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About Grzegorz
I am a math expert. I have published 13 math papers. Among others, in the "Bulletin of the London Mathematical Society". I have written The Banach-Tarski Paradox, published in Cambridge University Press. I work with outstanding mathematicians. I have been teaching for 20 years. The first month on the internet.
Book:
The monograph of G. Tomkowicz and S.Wagon "The Banach-Tarski Paradox", Cambridge University Press, 2016.
A. Papers:
1. A free group of piecewise linear transformations, Colloq. Math. 125
(2011), 141-146.
In this paper I prove a conjecture of Jan Mycielski [My98] that there
exists a free group of piecewise linear transformations acting without fixed points on the punctured Euclidean plane. I was able to reduce some 48 cases to a single one. The technique of [My89] is used.
2. (with Jan Mycielski) The Banach-Tarski paradox for the hyperbolic
plane (II), Fund. Math. 222 (2013), 289-290.
In this paper we ll a gap in the proof of [My89]. We also add some
remarks on the geometry of the hyperbolic plane.
3. (with Stan Wagon) Visualizing paradoxical sets , Math. Intelligencer,
36:3 (2014), 36-43.
In this paper we give an alternative construction of the Hausdorff paradox and then apply it to get a simple geometric illustration of the paradox in the hyperbolic plane H2. We visualize also a set in H2 that has two points such that removing anyone of them we get a set isometric to the original. This answers a question of Jan Mycielski.
4. (with Jan Mycielski) On small subsets in Euclidean spaces , Bull. Polish
Acad. Sci. 64 (2016), 109-118.
In this paper we introduce a concept of small sets that is closely related
to a notion of smallness of sets introduced by A. Tarski in [Ta38a]. We refine also some results obtained by Tarski in [Ta38a] and [Ta38b], and prove a number of related theorems. I have used this work in my paper 7.
5. (with Krzysztof Nowak) Intersections of generic rotations in some
classical spaces, Bull. Polish Acad. Sci. 64 (2016), 105-107.
In this paper we show that for any two definable sets A and B in the
Euclidean space Rn or the hyperbolic space Hn or the sphere Sn (n > 1)
of dimension less than n we have dim (q(A) \ B) < min(dim A; dimB) for
a generic rotation q. This is analogous to the transversality theorem of R.
Thom and it is applied in my paper 4.
6. On decompositions of the hyperbolic plane satisfying many congru-
ences, Bull. of London Math. Soc. 49 (2017), 133-140.
In this paper I solve a problem in the theory of paradoxical decompositions and invariant measures stated by J. Mycielski and S. Wagon in [MW84]. I generalize a construction of J. F. Adams [Ad54]. As corollary of this result I derive that H2 contains a set E which is simultaneously a half and a third and a fourth and ... up to (concealed information) of H2.
7. Banach-Tarski paradox in some complete manifolds, Proc. Amer. Math. Soc. 145 (2017), 5359-5362.
In this paper, applying a theorem of M. Laczkovich [La92c], I obtain a
short uniform proof of Banach-Tarski like paradoxes in the classical spaces Rn, Hn, Sn and the spaces of connected non-solvable Lie groups.
8. (with Jan Mycielski) Shadows of the Axiom of Choice in the universe
L(R), Archive for Math. Logic 57 (2018), 607-616.
In this paper we show that a number of geometric consequences of the
uncountable Axiom of Choice can be proved using only the Axiom of Dependent Choices.
9. (with Robert Samuel Simon) A Bayesian Game without -equilibria,
Israel J. Math. 227 (2018), 215-231.
In this paper we show that there exists a Bayesian game for which there are no e-equilibria with Borel measurable strategies for small enough e > 0.
10. (with Jan Mycielski) Paradoxical sets and sets with removable points,
J. Geom. (2018) 109: 28. (concealed information)
11. A properly discontinuous free group of affine transformations, Geom. Dedicata 197 (2018), 91-95.
In this paper I sharpen a result of G. A. Margulis [Ma87] showing that there exists a free subgroup of SA3(Z) (special affine group with linear and translational parts having integer entries) acting on R3 without fixed points.
This solves a problem of Jan Mycielski [My98].
12. (with Jan Mycielski) On the equivalence of sets of equal measures by countable decomposition, Bull. of London Math. Soc. 51 (2019), 961-966.
In this paper, assuming that G is second countable locally compact topological group with at most countably many connected components or totally
disconnected, we show that: If X; Y G are Haar measurable sets with non-empty interiors then (X) = (Y ) if and only if there exist two countable
partitions of G into Borel sets A1;A2; ::: and B1;B2; ::: and elements g1; g2; ::: 2 G such that gi(X \ Ai) = Y \ Bi for i = 1; 2; :::. The conjecture that the above equivalence is true for all Haar o-finite locally compact topological groups was posed in [Chu76]. Our proof uses some of his ideas and the structural theorems on locally compact groups of van Dantzig and Yamabe.
13. (with Jan Mycielski) On graphs with two removable vertices and related concepts, Acta Math. Hungar. 163 (2021), 538-546.
In this paper we study graphs G that have two vertices u and v and there exist two endomorphisms f and g of G such that f(E) = E n fug and g(E) = E n fvg. We prove that Cayley subgraphs with two removable vertices have a very special presentation. Our result solves Question 7.22 of S. Wagon and G. Tomkowicz [TW] and it is related to the works of Mycielski and Straus on sets and groups with removable points.
14. (with Robert Samuel Simon) Paradoxical decompositions and finitary rules, Submitted,
In this paper we introduce and study a class of nitary rules defined on a Cantor space X such that any colouring of X that satisfies any rule from the class defines the sets that are not measurable with respect to any finitely additive, invariant measure defined on all subsets of X.
15. (with Tugkan Batu and Robert Samuel Simon), (concealed information)
In the paper show there that there exists a probabilistic colouring rule. This provides the first example of Bayesian game that fails to have "-Harsanyi
equilibria for sufficiently small E with respect to all proper finitely additive extensions; this game is constructed using only two players.
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About the lesson
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Hello
I am Grzegorz Tomkowicz
I am a math expert. Please write to me. I'll see if I can help you. I teach most of the mathematics taught at universities.
The first hour is free.
- Mathematical logic and foundations
- Combinatorics and graph theory
- Order, lattices, ordered algebraic structures
- General algebraic systems
- Number theory
- Field theory and polynomials
- Commutative rings and algebras
- Algebraic geometry
- Linear and multilinear algebra; matrix theory
- Associative rings and algebras
- Nonassociative rings and algebras
- Category theory, homological algebra
- K-theory
- Group theory and generalizations
- Topological groups, Lie groups
- Real functions and elementary calculus
- Measure and integration
- Functions of a complex variable
- Potential theory
- Several complex variables and analytic spaces
- Special functions including trigonometric functions
- Ordinary differential equations
- Partial differential equations
- Dynamical systems and ergodic theory
- Difference and functional equations
- Sequences, series, summability
- Approximations and expansions
- Fourier analysis
- Abstract harmonic analysis
- Integral transforms, operational calculus
- Integral equations
- Functional analysis
- Operator theory
- Calculus of variations and optimal control; optimization
- Geometry, including classic Euclidean geometry
- Convex and discrete geometry
- Differential geometry
- General topology
- Algebraic topology
- Manifolds and cell complexes
- Global analysis, analysis on manifolds
- Probability theory and stochastic processes
- Statistics
- Numerical analysis
- Computer science
- Mechanics of particles and systems
- Mechanics of deformable solids
- Fluid mechanics
- Optics, electromagnetic theory
- Classical thermodynamics, heat transfer
- Quantum Theory
- Statistical mechanics, structure of matter
- Relativity and gravitational theory
- Geophysics
- Operations research, mathematical programming
- Game theory, economics, social and behavioral sciences
- Biology and other natural sciences
- Systems theory; control
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