uvod u algebarsku topologiju

Uvod u algebarsku topologiju

Ismar VoliWellesley College

Odsjek za matematikuPrirodno-matematikog fakulteta u Sarajevu

25. mart 2015.

Pregled predavanja

1 Topologija u kontekstu matematike

2 ime se bavi algebarska topologija?

3 Invarijante algebarske topologije

4 Neki nerijeeni problemi iz algebarske topologije

5 Neke nove primjene algebarske topologije

1. Topologija u kontekstu

Gruba (neoficijelna) podjela matematike je na:

Logika Kompleksna analiza

Algebra Funkcionalna analiza

Teorija brojeva Numerika analiza

Geometrija Diferencijalne jednaine

Topologija Vjerovatnoa i statistika

Realna analiza Primjenjena matematika


Po Amerikom Matematikom Drutvu, gruba podjela matematike je:


Topologija se moe nai u skoro pola polja:


Topologija je integralna u etiri od sedam milenijum problema za kojeClay Math Institute daje po milion dolara:

1 Yang-Mills equations

2 Riemann Hypothesis

3 P vs. NP problem

4 Navier-Stokes equation

5 Hodge conjecture

6 Poincar conjecture (rijeeno)

7 Birch and Swinnerton-Dyer conjecture

Skoro pola dobitnika Fields medalje su koristili topologiju na znaajannain u njihovom radu (Ahlfors, Douglas, Kodaira, Serre, Thom, Milnor,Atiyah, Grothendieck, Smale, Hironaka, Novikov, Mumford, Deligne,Quillen, Margulis, Connes, Thurston, Yau, Donaldson, Freedman, Jones,Mori, Witten, Kontsevich, Voevodsky, Perelman, Mirzakhani).


Razlog velike zastupljenosti topologije u matematici je da se onanalazi na presjeku algebre, analize, geometrije, kombinatorike,teorije kategorija, itd.

Radi toga se dalje i dijeli u razne podoblasti kao sto su

algebarska topologijageometrijska topologijadiferencijalna topologijanisko-dimenzionalna topologijatopologija mnogostrukostikombinatorijalna topologija

(Naravno ova podjela nije ista i svaki topolog radi u nekom presjekuovih kategorija.)

2. ime se bavi algebarska topologija?

Topoloki prostor je bilo koji skup sa sistemom otvorenihpodskupova. Topologija pokuava da prepozna i klasifikujetopoloke prostore. Klasifikacija se vri do na neku relacijuekvivalencije ili deformacije, kao na primjer

homeomorfizam: Prostori X and Y su homeomorfni akopostoji preslikavanje (neprekidna funkcija) f : X Y sainverznim preslikavanjem; ilihomotopnost: Prostori X and Y su homotopno ekvivalentniako postoje preslikavanja f : X Y and g : Y X takva dasu f g and g f homotopne identinom preslikavanju (X semoe deformisati u Y ); iliizotopija: Isto kao homotopnost, samo X and Y sumnogostrukosti i sva preslikavanja su ulaganja (X se moedeformisati u Y kroz injektivna preslikavanja); itd.

2. ime se bavi algebarska topologija?

Homotopnost Izotopnost

Eksplicitne konstrukcije deformacija nisu jednostavne, i cilj algebarsketopologije je da se pronau algebarske invarijante prostora koje nammogu rei da li se dva prostora mogu deformisati jedan u drugog. Dakleelimo dodijeliti svakom prostoru neki algebarski objekt tako da, ako sudva prostora ekvivalentna, njihovi algebarski objekti su izomorfni.

Mug_and_Torus_morph.movMedia File (video/quicktime)

monster-movie.mpgMedia File (video/mpeg)

3. Invarijante algebarske topologije

Najosnovnije invarijante su:Hk(X ), grupe (singularne) homologije;Hk(X ), grupe (singularne) kohomologije;k(X ), grupe homotopije.(k 0 u sva tri sluaja)Grupe homologije su bazirane na kombinatorici prostora usmislu da kodifikuju strukturu simpleksa (generalizacijetrougla). Njih je najtee definisati ali ih je najlake izraunati.Grupe kohomologije su dualne homologiji, ali imaju viestrukture pa su tee za izraunati.Grupe homotopije je lako definisati:k(X ) = {prelikavanja Sk X modulo relacija homotopije},ali su one najtee za izraunati.


Primjer (Euklidski prostor Rn)

Hk(Rn) = Hk(Rn) =

{Z, k = 00, k 6= 0

k(Rn) = 0 k

Primjer (Sfera Sn)

Hk(Sn) = Hk(Sn) =

{Z, k = 0, n0, k 6= 0, n

k(Sn) = generalno nepoznato

S obzirom da je homologija invarijanta homotopije, Rn i Sn nisuhomotopski ekvivalentni.


Primjer (Torus T 2 = S1 S1)

Hk(T 2) = Hk(T 2) =

{Z, k = 0, k = 2Z Z, k = 1

k(T2) =

{0, k = 0, k > 2Z Z, k = 1

Primjer (Realni projektivni prostor RPn)

Hk(RPn) =

Z, k = 0, k = n i n neparno;Z/2Z, 0 < k < n i k neparno;0, u ostalim sluajevima.

k(RPn) =

0, k = 0;Z, k = 1, n = 1;Z/2Z, k = 1, n > 1;k(S

n), k > 1, n > 0.

(kohomologija se moe dobiti kroz Poenkareovu dualnost)

4. Neki nerijeeni problemi

Ve smo spomenuli da su grupe k(Sn) generalno nepoznate. Ovoje jedan od najvecih nerijeenih problema algebarske topologije.

Spomenimo jos neke od veih otvorenih pitanja (po mom izboru vrlo je vjerovatno je da bi nekom drugom topologu neki druginerijeeni problemi bili interesantniji).

Prostori petlji

X je prostor petlji ako je ekvivalentan prostoru preslikavanja izkruga S1 u neki prostor Y . Dakle

X = Map(S1,Y ).

Prostori petlji se koriste u topolokoj kvantnoj teoriji polja, teorijistringova, a i osnovni su objekt u stabilnoj teoriji homotopije.

Problem klasifikacije (konanih) prostora petlji je otvoren (i vezanje za problem klasifikacije konanih p-kompaktnih grupa).


Generalna Poenkareova pretpostavka

Ako izgleda kao sfera, onda je sfera.

Pitanje je da li je svaka homotopska sfera, to znai prostor koji jehomotopski ekvivalentan sferi, homeomorfna sferi. Odgovor jepozitivan, osim u dimenziji 4, gdje se odgovor jo uvijek ne zna.

Difeomorfizmi mnogostrukosti

Glatka mnogostrukost M je topoloki prostor koji lokalno izgledakao Euklidski prostor i ima odreenu diferencijalnu strukturu.

Glatka preslikavanja, to jest difeomorfizmi, f : M M ine grupudifeomorfizama mnogostrukosti M, Diff(M). Ove grupe su takodjetopoloki prostori, sto ih ini Lijevim grupama.

Topoloka struktura Lijevih grupa Diff(M) je nepoznata u velikombroju sluajeva, recimo M = Sn.


Klasifikacija vorova

vor je ulaganje (glatko injektivno preslikavanje) S1 R3.

Teorija vorova ima primjene u biologiji, hemiji, i fizici (DNK,molekularno upetljavanje, teorija fluida, teorija stringova, itd.).

Dva vora su ekvivalentna ako su izotopna (jedan se moedeformisati u drugi kroz vorove; prije smo vidjeli film izotopije).


Iako znamo da postoji beskonano mnogo neizotopnih vorova,problem njihove organizacije, ili klasifikacije, je jo uvijek otvoren.

Tradicionalni pokuaj klasifikacije vorova je po broju presjeka.Nekih 6 milijardi vorova su tako klasifikovani ali ono to nemamoje formula za broj vorova sa datim brojem presjeka. Ovo je najveinerijeeni problem nisko-dimenzionalne topologije.


Invarijante vorova

Povezan sa prethodnim problemom je pitanje invarijanti vorova,dakle funkcija

f : K G ,

gdje je K skup vorova a G je obino neka abelova grupa, koja dajeistu vrijednost izotopnim vorovima.

Neke od poznatijih invarijanti vorova su minimalni broj presjeka,fundamentalna grupa komplementa, Alexander, HOMFLY, i Jonespolinomi, Vasiljevljeve invarijante, Khovanov homologija, itd.

Otvoreno pitanje je da li postoji invarijanta (ili skup invarijanti), sasvojstvom: Za svaka dva vora koji nisu izotopni, ta invarijanta (ilineka invarijanta u tom skupu) daje druge vrijednosti za ta dvavora. Takva invarijanta (ili skup) se zove kompletna invarijantakoja razdvaja vorove.

(Ovo i prethodno pitanje su ustvari: ta su H0(K) i H0(K)? Pitanje viihgrupa (ko)homologije prostora vorova je takoe otvoreno.)

5. Nove primjene algebarske topologije

Tokom veine svog postojanja (nekih 110 godina), algebarskatopologija je bila apstraktna i sluila je kao alat za druge oblastimatematike. Jo uvijek je uglavnom apstraktna, ali su u posljednjih10-12 godina razvijene neke njene nove direktne primjene kojeprivlae mnogo panje.

Topoloka analiza podataka

U sadanjoj eri velikih podataka, bitno je prepoznati strukture uoblacima podataka. Topoloka analiza podataka posmatra takveoblake kao topoloke prostore i izuava njene oblike.

Topologija, tanije homologija, moe prepoznati da li su podacipovezani, da li imaju rupe, da li sadre petlje, da li suorijentirani, koliko su podgrupe podataka daleko, itd.

Primjene ovakve analize podataka su raznovrsne: kompjuterskavizija, analiza oblika ili digitalna geometrija (prepoznavanje lica),medicina (podaci magnetnih rezonanci ili CT skanova), itd.


Razmak meu grupamaodreenih inhibitora pomaepri dizajnu lijekova za rak(Bak-Lerner)

Skup 3 3 piksela velikogkontrasta u fotografijama imaoblik Klajnove boce i to dajenovi algoritam za kompresiju(Carlsson-Ishkhanov-deSilva-Zomorodian)


Robotika

Algoritam planiranja kretanja je veliki problem u teoriji robotike. Naime,pitanje je kako programirati robota da iz take A ode u taku B a da pritome to uini

na kontinuiran nain, dakle ne udara u prepreke ili u druge robote; i

na najefikasniji nain, dakle izabere najkrai put od A do B.

Efikasno kretanje Neefikasno kretanje

Koordinirano kretanje je problem ne samo u robotici nego u svakoj prilicigdje je bitno izbjei sudaranje, kao na primjer u kontroli vazdunogsaobraaja.


Slino ovome je problem kretanja robotske ruke. Ovdje je cilj da seprogramira ruka, koja ima nekoliko stepena slobode,

tako da izbjegava prepreke na efikasan nain.


Iza oba problema stoji topologija.

Cilj je da se definie prostor koji je model za mogue pozicije gruperobota svaka taka u ovom prostoru odgovara jednoj poziciji grupe.Onda se oduzmu take koje odgovaraju sudarima. ta ostaje je prostorkoji je model za sve bezbjedne pozicije robota.

Putanje u ovom prostoru daju planiranje kretanja bez sudara.

Minimizacija svih moguih putanja daje efikasno planiranje kretanja.

Prostori koji modeluju kretanje grupe robota su konfiguracijskih prostori.Ako je X topoloki prostor, onda imamo konfiguracijski prostor n taakau X :

Conf(n,X ) = {(x1, x2, ..., xn) X n : xi 6= xj za i 6= j}

Ovo je X n bez velike dijagonale, znai sve dijagonale su uklonjene.

Konfiguracijski prostori su posvuda u matematici i fizici (teorijahomotopije, Lijeva theorija, vorovi i pletenice, klasina mehanika, fizikaestica, itd.)


Topologija prostora konfiguracija je veoma komplikovana. Naprimjer, grupe homotopije prostora Conf(n,Rm) su nepoznate, a dai ne govorimo o komplikovanijim prostorima od Rm.

PrimjerUzmimo prostore

, , .

Onda su ovo slike konfiguracijskih prostora dvije take na njima:

(Abrams-Ghrist)


Topologija se primjenjuju u mnogim drugim oblastima, kao to su

raunalna biologija (modeliranje i procesiranje podataka izbiolokih i drutvenih sistema),dinamika (promjene kroz vrijeme),mree komunikacija (tornjevi za mobilne telefone, internet,neuralne mree),procesiranje signala (transfer i procesiranje digitalnih,vizuelnih, audio, i drugih informacija), itd.

Evo jos nekih specifinih primjena (slike su iz serije MathMoments Amerikog Matematikog Drutva):

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Working Up a Lather

Bubbles, of little matter both in weight and presumed practical use, are the building blocks of foam. So that actually makes them crucial in many applica-tions ranging from the padding inside bicycle helmets to fire retardants. And as anyone who has observed foam knows, bubbles come in various sizes, they grow, form clusters (as below), and burstall of which has made describing foam quite difficult. Mathematicians recently successfully modeled clusters of hundreds of bubbles for the first time by treating different aspects of their interactions sepa-rately, such as the flow of fluid between connected bubbles. The key to their model was solving sets of linked partial differential equations, which allowed researchers to break up the problem into different components while making sure that the components could still be coupled together consistently.

A round soap bubble minimizes surface area: a sphere is the least-area way to enclose a given volume of air. One long-unresolved question, known as the double-bubble conjecture, asked if two bubbles that meet in the usual way provide a least-area way to enclose and separate two equal volumes of air. The proof that they do offers an illustration of patterns in some modern mathematics research: It involved computers, it was the work of many people, including under-graduates, and the research didnt end there. What about three or more bubbles? Shapes enclosing unequal volumes? Or those in higher dimensions? If only the answers would just pop up.

For More Information: Multiscale Modeling of Membrane Rearrangement, Drainage, and Rupture in Evolving Foams, Robert I. Saye and James A. Sethian, Science, May 10, 2013.

The Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture.

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Freeing Up Architecture

Many of todays most striking buildings are nontraditional freeform shapes. A new field of mathematics, discrete differential geometry, makes it possible to construct these complex shapes that begin as designers digital creations. Since its impos-sible to fashion a large structure out of a single piece of glass or metal, the design is realized using smaller pieces that best fit the original smooth surface. Triangles would appear to be a natural choice to represent a shape, but it turns out that using quadrilateralswhich would seem to be more difficultsaves material and money and makes the structure easier to build.

One of the primary goals of researchers is to create an efficient, streamlined process that integrates design and construction parameters so that early on architects can assess the feasibility of a given idea. Currently, implementing a plan involves extensive (and often expensive) interplay on computers between subdivisionbreaking up the entire structure into manageable manufacturable piecesand optimizationsolving nonlinear equations in high-dimensional spaces to get as close as possible to the desired shape. Designers and engineers are seeking new mathematics to improve that process. Thus, in what might be charac-terized as a spiral with each field enriching the other, their needs will lead to new

mathematics, which makes the shapes possible in the first place.

For More Information: Geometric computing for freeform architec-ture, J. Wallner and H. Pottmann. Journal of Mathematics in Industry, Vol. 1, No. 4, 2011.



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Bending It Like Bernoulli1

The colored strings you see represent air flow around the soccer ball, with the dark blue streams behind the ball signifying a low-pressure wake. Computational fluid dynamics and wind tunnel experiments have shown that there is a transition point between smooth and turbulent flow at around 30 mph, which can dramatically change the path of a kick approaching the net as its speed decreases through the transition point. Players taking free-kicks need not be mathematicians to score, but knowing the results obtained from mathematical facts can help players devise better strategies.

The behavior of a ball depends on its surface design as well as on how its kicked. Topology, algebra, and geometry are all important to determine suitable shapes, and modeling helps determine desirable ones. The researchers studying soccer ball trajectories incorporate into their mathematical models not only the pattern of a new ball, but also details right down to the seams. Recently there was a radical change from the long-used pentagon-hexagon pattern to the adidas +TeamgeistTM. Yet the overall framework for the design process remains the same: to approximate a sphere, within less than two percent, using two-dimensional panels.

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For More Information: Bending a Soccer Ball with CFD, Sarah Barber and Timothy P. Chartier. SIAM NEWS, July/August 2007.

1 Daniel Bernoulli (BurrNOOlee) was a Swiss mathematician who did pioneering work in fluid flow.



Putting Music on the Map

Mathematics and music have long been closely associated. Now a recent math-ematical breakthrough uses topology (a generalization of geometry) to represent musical chords as points in a space called an orbifold, which twists and folds back on itselfmuch like a Mbius strip does. This representation makes sense musi-cally in that sounds that are far apart in one sense yet similar in another, such as two notes that are an octave apart, are identified in the space.

This latest insight provides a way to analyze any type of music. In the case of Western music, pleasing chords lie near the center of the orbifolds and pleasing melodies are paths that link nearby chords. Yet despite the new connection between music and coordinate geometry, music is still more than a connect-the-dots exercise, just as mathematics is more than addition and multiplication.

For More Information: The Geometry of Musical Chords, Dmitri Tymoczko, Science, July 7, 2006.

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Making Connections

People in a society, neurons in the brain, and pages on the Web, along with theirconnections, are all examples of networks. Mathematicians study characteristics ofnetworks, such as the number and distribution of connections, to discover what suchattributes may reveal about the intrinsic nature of a network. For example, the colorsin the picture below indicate how disruptive deleting a node would be to thenetwork, in this case a living cell.The discovery and verification of network propertiessuch as this has significance for applications ranging from the microscopic to theworldwide, including the protection of both computers and humans against viruses.

The study of networks spawned the phrase six degrees of separation, the theme ofa game involving actors connections via common film appearances. In an experimentdone in the 1960s, over 100 randomly chosen people in the Midwest were found tobe connected to a Massachusetts stockbroker (by a friend of a friend of a friend, andso on) in an average of just six steps.That people halfway across the country couldbe so closely connected was quite a revelation and proved that even a large networkcould be a small world.Today, researchers use parameters from graph theory andprobability in analyzing networks to determine whether an elaborate network, be ita power grid or actors connecting to Kevin Bacon, is indeed a small world after all.

For more information: Scale-Free Networks, by Albert-Lszl Barabsi andEric Bonabeau, Scientific American, May 2003


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Deciphering DNA

Anyone who has used a garden hose knows that knots appear in strangeplaces. Scientists have found that a branch of mathematics called knottheory appears in many familiar places, including in our DNA. Mathematicsplays a key role in understanding how DNA functions and replicates itself.

Certain enzymes cut a strand of DNA at one point, pass another part of thestrand through the gap, and then seal the cut. Knot theory gives insight onhow frequently an enzyme has to act, from which one can infer how longthe enzyme might take to make a product. This kind of complex manipula-tion is significant in many cellular processesincluding DNA repair and generegulationand is the type of problem central to the theory of knots.

For More Information:Whats Happening in the Mathematical Sciences, Vol. 2, Barry Cipra.

Left: Photograph courtesy of Paul Thiessen.Right: Photograph courtesy of the University of Minnesota.

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uvod u algebarsku topologiju

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