Seminar for Geometry, education and visualization with applications
PROGRAM
MATEMATIČKI INSTITUT SANU
Seminar geometriju, obrazovanje i vizualizaciju sa primenama
PLAN RADA ZA DECEMBAR 2022.
ČETVRTAK, 01.12.2022. u 17:15, sala 301f Matematickog instituta SANU i On-line
Srđan Vukmirović, Matematički fakultet, Beograd
LEVO INVARIJANTNE METRIKE NA LIJEVIM GRUPAMA
(deo 2/3 - geodezijske na $H^n$ i Hajzenbergova grupa)
Videćemo kako izračunati geodezijske linije levo invarijantne metrike na Lijevoj grupi na primeru Hipeboličkog prostora. Posle toga uvodimo Hajzenbergovu grupu i nalazimo njenu vezu sa simplektičkom geometrijom.
ČETVRTAK, 08.12.2022. u 17:15, sala 301f Matematickog instituta SANU i On-line
Tijana Šukilović, Matematički fakultet, Beograd
UVOD U SUB-RIMANOVU GEOMETRIJU (mini kurs)
Sub-Rimanova geometrija predstavlja uopštenje Rimanove geometrije u sledećem smislu: sub-Rimanovu mnogostrukost možemo posmatrati kao Rimanovu mnogostrukost sa skupom ograničenja za dopustive pravce kretanja. Osnovni primer dolazi iz mehanike: stanje objekta u pokretu jedinstveno je određeno njegovim položajem u prostoru i impulsom. Prirodno, neke trajektorije nisu dopustive. Na primer, ne možete promeniti brzinu kretanja bez promene položaja.
Osim u klasičnoj mehanici, sub-Rimanova geometrija ima brojne primene u teoriji upravljanja, simplektičkoj i kontaktnoj geometriji, teoriji hipoeliptičkih operatora, geometrijskoj teoriji grupa itd. U fizici, sub-Rimanov prostor je model za razne interesantne strukture od problema parkiranja do mačke u slobodnom padu! Neke od ozbiljnijih oblasti primena su robotika, finansije, kvantna mehanika, neurobiologija. Na predavanju će biti reči o Hajzenbergovoj grupi $H_3$ i njenoj vezi sa izoperimetrijskim problemom i problemom Didone: Koja je najkraća kriva koja obuhvata zadatu površ? Biće pokazano da je to osnovni netrivijalni primer sub-Rimanove mnogostrukosti.
ČETVRTAK, 15.12.2022. u 17:15, sala 301f Matematickog instituta SANU i On-line
Tijana Šukilović, Matematički fakultet, Beograd
UVOD U SUB-RIMANOVU GEOMETRIJU (deo 2)
Glavna tema ovog predavanja će biti geodezijske jednačine. U Rimanovoj geometriji sve geodezijske linije su normalne (regularne), odnosno dobijaju se kao rešenja geodezijske jednačine. Specifičnost sub-Rimanove geometrije je postojanje abnormalnih (singularnih) geodezijskih krivih. Primeri za oba tipa geodezijskih krivih će biti dati u dimenziji 3. U drugom delu predavanja će biti reči o egzistenciji geodezijskih krivih, tj. opštije, o egzistenciji horizontalne krive koja povezuje dve tačke na sub-Rimanovoj mnogostrukosti (teoerema Chow-Rashevskii).
ČETVRTAK, 22.12.2022. u 17:15, sala 301f Matematickog instituta SANU i On-line
Ivan Limonchenko, HSE University, Russia
THE AANDERAA-KARP-ROSENBERG CONJECTURE AND TORIC TOPOLOGY (part 1/2)
By property of a graph we mean a boolean function on the set of all graphs; it is called invariant if relabelling of vertices of a graph does not change the value of the property on it. In order to check a certain property of a graph, one needs to ask a number of questions about edges of a graph. If a (simple) graph has $n$ vertices, then $m=n(n-1)/2$ is the maximal possible number of its edges. The Aanderaa-Rosenberg conjecture (now proved) states that there exists a positive constant $C$ such that at least $Cm$ questions are needed to check any (non-trivial) monotonic invariant property. A stronger Aanderaa-Karp-Rosenberg conjecture (still open) asserts that one can always assume $C=1$ above. The topological approach developed to attack the last conjecture relates it to the study of fixed point sets of finite group actions on cellular spaces.
In this talk, we'll be interested in a version of the Aanderaa-Karp-Rosenberg conjecture, in which one considers all non-trivial monotonic properties. We're going to interpret a monotonic boolean function of $m$ variables as a simplicial complex with $m$ vertices, and then apply the results of Bjorner-Lovasz on algorithmic complexity of polyhedra to the corresponding polyhedral products. Finally, we'll deduce a version of the original Aanderaa-Rosenberg conjecture for non-invariant monotonic properties from the version of the Toral Rank Conjecture proved for moment-angle complexes by Ustinovskii. This new perspective provides new connections between toric topology, theoretical informatics, and probably even artificial intelligence.
The talk is based on the ongoing research project j.w. Anton Ayzenberg and Fedor Vylegzhanin.
ČETVRTAK, 22.12.2022. u 17:15, sala 301f Matematickog instituta SANU i On-line
Ivan Limonchenko, HSE University, Russia
THE AANDERAA-KARP-ROSENBERG CONJECTURE AND TORIC TOPOLOGY (part 2/2)
By property of a graph we mean a boolean function on the set of all graphs; it is called invariant if relabelling of vertices of a graph does not change the value of the property on it. In order to check a certain property of a graph, one needs to ask a number of questions about edges of a graph. If a (simple) graph has $n$ vertices, then $m=n(n-1)/2$ is the maximal possible number of its edges. The Aanderaa-Rosenberg conjecture (now proved) states that there exists a positive constant $C$ such that at least $Cm$ questions are needed to check any (non-trivial) monotonic invariant property. A stronger Aanderaa-Karp-Rosenberg conjecture (still open) asserts that one can always assume $C=1$ above. The topological approach developed to attack the last conjecture relates it to the study of fixed point sets of finite group actions on cellular spaces.
In this talk, we'll be interested in a version of the Aanderaa-Karp-Rosenberg conjecture, in which one considers all non-trivial monotonic properties. We're going to interpret a monotonic boolean function of $m$ variables as a simplicial complex with $m$ vertices, and then apply the results of Bjorner-Lovasz on algorithmic complexity of polyhedra to the corresponding polyhedral products. Finally, we'll deduce a version of the original Aanderaa-Rosenberg conjecture for non-invariant monotonic properties from the version of the Toral Rank Conjecture proved for moment-angle complexes by Ustinovskii. This new perspective provides new connections between toric topology, theoretical informatics, and probably even artificial intelligence.
The talk is based on the ongoing research project j.w. Anton Ayzenberg and Fedor Vylegzhanin.
Sednice seminara odrzavaju se u zgradi Matematickog instituta SANU, Knez Mihailova 36, na trecem spratu u sali 301f.
Rukovodilac Seminara dr Srđan Vukmirović
