Automatic transcript from https://www.youtube.com/watch?v=MCL9R3CJCo0
http://www.math.harvard.edu/~knill/technology/2016/translated.txt   actual 

0:00welcome these slides deal with an integer valuation on finite simple
0:04grants it is the characteristic since I oliver cannot talk is faster computer
0:09voices do the comments to the slides are read by other and Ella
0:14characteristic is a quadratic valuation on graphs which sums up over pairs of
0:18complete intersecting subgraphs while not hamas will be invariant it is
0:22invariant under Perry's under subdivision and so defines a
0:25combinatorial invariant for compact manifolds on finite simple graphs it is
0:29a multiplicative quadratic valuation for discrete manifolds with boundary it is
0:34the younger characteristic of the Interior minus the characteristic of the
0:37boundary there is a guy responded formula for the characteristic the
0:40existence of quadratic in Somerville invariance confirms a conjecture of
0:44grunbaum
0:48the healer characteristic is by far the most important Valuation it is a homo
0:51tell the invariant behaved correctly with respect to films and products and
0:55invariant under their eccentric subdivisions the healer characteristic
0:58super accounts in places complete subgraph subgraph the even dimensional
1:0214 counted positively and the odd dimensional 14 counted negatively
1:09even so solid
1:11increase already the user polyhedral formula has not yet been known at that
1:15time the characteristic is named after Leonhard Euler who first prove the
1:19famous Formula D minus declassify equals two in the planar case more than 100
1:23years before you ever rented a car discovered some of the relations already
1:26wrote it down in a secret diary but did not prove it
1:30the formula was extended to higher dimensions starting with the Ludwig
1:33schlafly and Henri poincaré many important theorems deal with ruler
1:37characteristic examples are gas bonded poincaré hop you have one care freeman
1:42herwitz women rock or mckean singer
1:49here's an interesting things in place in the graph interactions between even
1:52dimensional dimensional sympathies are counted positively the interaction
1:56between even a dog dimensional cases discounted negatively let's call it the
1:59one characteristic afterwards done woohoo introduced these evaluations
2:02about sixty years ago the number thing not have been studied much
2:11who made his PhD under the guidance of charles dera man who had other famous
2:15students like Andre
2:19here is a simple example of
2:21valuation of the user characteristic Andrew characteristic of the graph the
2:24graph is the complete graph with two vertices the one simplex for the user
2:28characteristics we just counts in places since there are two even to mention of
2:32the places which are the vertices in 10 dimensional simplex the edge the EULAR
2:36characteristic is 2-1 equal one for the Wii characteristic first count the
2:40interactions between vertices they are to the interaction between the ages
2:44there is one and finally the interactions between vertices and edges
2:48there are four note that we look at the order of the pairs if we add everything
2:52up
2:56and here is another example the graphic
3:00triangle counting vertices edges and triangles gives one note that this is
3:04already Euler's formula V mind as he closed off for the one characteristic
3:08there are now already more possibilities we have the interactions between
3:12vertices we have the self interactions between ages we also have the
3:16interactions between different ages new is also the self interactions between
3:20triangles and the interactions between edges and triangles as well as
3:24interactions between vertices in triangles if we had everything up we get
3:28characteristic of the triangle is one
3:34we see already a pattern even dimensional simplifies the
3:37characteristic is 140 dimension of them places the characteristic 8-1 this will
3:42follow in general from a boundary formula and the fact that the wound
3:45characteristic is invariant under their eccentric subdivision
3:52was minus one for high-dimensional graphs and 14 even dimensional graphs we
3:56can write the user characteristics using Omega the characteristic is a quadratic
4:00valuation there are more general characteristics the cubic one for
4:04example sums up overall triples of symbols which simultaneously intersect
4:08for the cubicle characteristic all samples evaluate 21 again like for you
4:12are characteristic
4:17civil dimension d meaning that guarantees in place is present in the
4:20graph but no D plus one dimensional thin places if the UK counts the number of
4:24cases places in G one confirmed the F factor which contains all these numbers
4:28now take the dot product between us D plus one dimensional vector X with this
4:34defector this gives the value of the valuation
4:40valuations are described by
4:41matrix here denoted with capital G
4:45this matrix is the analog of the effector it counts the number of
4:48intersections of ice in places with Jason places to get the quadratic
4:52valuation take a symmetric matrix acts usually with integer injuries telling
4:56what value we give you an interaction of high-dimensional NJ dimensional
4:59intersecting some places then we formed a song
5:04the discrete had claimed and Rhoda sure that the space of linear evaluations on
5:09a graph is D plus one dimensional where D plus one is the Cliq number the
5:13maximum number of vertices with a complete graph into you can have their
5:16is a generalization of the discrete had wider remit give the dimension of the
5:20set of quadratic evaluations note that the space of valuation is still a linear
5:24space it is just that the action of evaluation on graft is now better
5:31you had
5:32mathematician had winners there is a celebrated their own internal geometry
5:36the theory in the continuum is well developed valuations have to be built
5:40carefully on like four measures valuations evaluate geometric objects
5:44with internal structure simply shell structure in the continuum the basic
5:49ingredients are convex sets
5:53the discrete version of head with your girl has been developed later that they
5:57can be found in the bucket and clean and Gian Carlo rota introduction to a metric
6:01probability from 1997 in the discrete also what does not measure the content
6:05of that but objects with internal structure simplicial complexes one can
6:09also formulated using graphs graphs have a natural simply complex structure the
6:14Whitney complex
6:18here is the standard base
6:19base of linear evaluations there is a similar standard basis for quadratic
6:23valuations there is another natural bases in the linear case and it is a
6:27very eccentric subdivision
6:3204 grant
6:34mistake is evaluated it is a cytosine and I mine they are the nitrogenous
6:39bases of the DNA the numbers on the vertices of the graphs are curvatures
6:43they add up to the characteristic
6:48here is going
6:49nations in particular applies to the characteristics there always exists a
6:54function k on vertices such at the same overall bertice is the value of the
6:58valuation this works for an evaluation and also for any multilinear evaluation
7:02of course the prototype is the UN characteristic that harem originally
7:07comes from the continuum where it had been proven to more and more generality
7:10culminating in the gaza
7:16ok the discrete version 4 u'r characteristic in a paper of Norman
7:19levitt but the statement had not been seen as a result in not in graph theory
7:26in the context
7:28param involves a healer curvature which is a complicated expression in terms of
7:31the Riemann curvature tensor it is a Fianna only to find for even dimensional
7:35manifolds in the discrete curvature is defined for all networks the fact that
7:40advantages for high dimensional geometric graphs is a special case of
7:43the 10 Somerville relations
7:48you are the main figures for the Gold Bond it in the continuum goes on and on
7:52it dealt only with the two-dimensional case after work of åland over and wheel
7:56chair and was the first to prove the result without investing in an ambient
7:58space
8:04multilinear evaluation here is an explicit expression of the curvature in
8:07the case of the characteristic the curvature depends on a disk of radius
8:11and has become a second-order difference operator like in the continuum the song
8:15is over all pairs of some places X&Y where it contains the vertex into
8:19simplex y intercept the cineplex know that the Simplex why does not
8:23necessarily have to contain the vertex you can try it out implemented in
8:26computer code contained in the tree print
8:31here is a simple example
8:33graph we compared to curb its shares of the EULAR characteristic in the loop
8:36characteristic you see the EULAR curvature is supported here on the
8:39boundary the characteristic curvature has moved slightly inside for a line
8:43graph in distance to a larger from the boundary the curvature of the line graph
8:47is 0 the graph is black bear
8:52this is an example
8:53discretization of the line is great variety it is the figure eight graph the
8:58curvature is only 90 near the intersection point which is the
9:01singularity the total figure 8 Rafa seven which is the characteristic you
9:07will see that the one characteristic is invariant under very eccentric
9:10subdivision this will mean that we can assign also characteristic 72 dilemmas
9:14lead
9:18wheel graphic the two-dimensional disc
9:20characteristic one this is true for all two-dimensional discs and more generally
9:23for all even dimensional balls the boundary of an even to mention Obama's
9:27another dimension of fear the characteristic that Siri 0 regular care
9:3423
9:37minus one the curvature is here located at only one central point the unit
9:41sphere the vertices connected to that vertex is an octahedron graph which is
9:46an even dimensional graph we see here already and illustrations Helen that the
9:49characteristic of a graph with boundaries that you are characteristic
9:52of the graph minus the characteristic of the boundary
9:58here is an exam
9:59it is a graph of maximal dimension one which has no closed loop the loop
10:03characteristic is here much larger than the EULAR characteristic you can observe
10:06that the curvature can become negative near an end point
10:12we look
10:12human characteristics and one characteristic in comparison the graph
10:16is a two-dimensional desk to the left you see the UN curvature the UN
10:20curvature in the interior is 1-2 lengths of the unit sphere at the boundary
10:23divided by six it is one-half minus the length of the unit sphere divided by six
10:28on the boundary eyes the UN curvatures in the two-dimensional case have been
10:31known since the 19th century especially through the work of Hinrich EEC and
10:35graph coloring
10:39and here is a circle
10:40which some of circles the healer characteristic 8-4 one can see this also
10:45from the fact that there are no tryin goals and five loops leading to a better
10:48number B one April 2-5 annular characteristic 1-5 equal 2-4 the
10:53characteristic is pretty large already which is due to the many interactions
10:56that the singularity with some is 76
11:02despite a
11:03fraph is usually considered a two-dimensional poly top we look at it
11:06as a one dimensional geometric object as it has no triangles you can think of it
11:11as a sphere with six holes the younger characteristic is 2-6 which is incall
11:152-4
11:16characteristic is 21 curvature is constant 5 half's
11:23here is another 10 will
11:24which
11:25finite simple graph and an evaluation we can attach to the vertices and index
11:29which adds up to the valuation an integer value function on vertices is
11:33also called advisor that the airline also applied for the Wii characteristic
11:41Cameron Highlands hop and requires the function to be regular enough so that
11:46critical points are isolated in the discrete the theorem holds for any
11:49finite simple graph and any linear or multilinear valuation it especially hold
11:54for the one characteristic
11:58evaluate
12:00like the
12:00simple user characteristic the index is given by this formula Caribbean X is the
12:05subgraph of the unit below the verdicts generated by the vertices on whichever
12:08is smaller or equal value taken on the central vertex the graphic sex is the
12:12subgraph of the unit sphere of the verdicts generated by the vertices on
12:16whichever is smaller than the value taken on the central vertex
12:22example where the valuation is the volume a particularly simple valuation .
12:26the volume is the number of facets the number of highest dimensional simple
12:30cyst which occur in a graph in this case the index is always not negative
12:37evaluation
12:39will be written as a bilinear form or intersection number in the case of the
12:42loop characteristic this is written as given
12:44overall ordered pairs of the places where the first simplex it in a in the
12:48second in B which intersect one scene like that but we characteristic is a
12:52self intersection number of a graph with itself
12:57here are two examples
12:59options if it all dimensional graph intersects nicely with a non-dimensional
13:02graph in the intersection number is equal to one to the right we see a curve
13:06intersecting at this intersection number
13:12and here is an example of two-dimensional discs
13:15transversely in a point in three space this looks like a cone but in
13:18four-dimensional space one to draw the embedding so that the intersection is
13:22perpendicular the intersection number is
13:27luminary step
13:29of the characteristic and a function at is given by a sum of intersection
13:33numbers this is now first written as a function on pairs of vertices the
13:37indexes 0 with the two vertices have distance larger than to the reason is
13:41that many balls of different vertices do not intersect so that intersection
13:44numbers are 0
13:48we can push the end
13:50ABC's become a scalar function on vertices it still is an integer the
13:54point hollyhock theorem states that the Sun is up to the Wii characteristic
14:00here is the judge
14:02why the characteristic is so important regular characteristic it is
14:06multiplicative the product of two graphs consists of all parents fax line of
14:11symbols is where X is a simplex in the first draft and why is a simplex in the
14:142nd Graf now to such shares are connected if one is contained in the
14:19other in the special case where one graph is the one point graph this is the
14:23very centre subdivision this product has all the properties we wish towards
14:27satisfies the Cuneta formula for coca-cola G
14:32here is an exam
14:34line graph a discrete interval and a circular graph a discrete circle the
14:38resulting graphic the discrete cylinder the healer characteristic of the line
14:42graph is one the EULAR characteristic of the second graph is 0 the cylinder has
14:46healer characteristic zero similarly the characteristic of the first graph is
14:49minus 12 with characteristic of the circle is zero in the cylinder again has
14:53with characteristic 0
14:59guys that are characteristic is invariant under very eccentric
15:01refinement refinement of a graph G has as vertices the set of some places in a
15:07few such a new vertices are connected if one is contained in the other how can
15:11one see that the younger characteristic is invariant the parody of the Simplex a
15:15number which is minus 14 out some places and 14 even dimensional some places is a
15:19scalar function on the vertices at the very center of refinement it turns out
15:23this is the poincaré hop index of a function health care hospitals that it
15:27sums up to the younger characteristic of the refined draft note that if you were
15:30characteristic is a much larger sum as it would sum up overall some places in
15:34the new graph
15:38my little blog called the quantity which
15:39variant under
15:40enteric refinement topological invariant but himself found a few interesting
15:44invariants like that in his early career topological invariants for graphs are
15:48interesting as they produced topological invariants for compact manifolds indeed
15:52an initial triangulation of a manifold becomes after successive bear eccentric
15:56refinement closer and closer to the manifold
16:02disagree
16:03affords the characteristic is related to the user characteristics and the
16:07characteristic of the boundary there is a general hair removal from 2010 telling
16:11that such a relation has to hold for a topological invariant the formula
16:15implies that for graphs without a boundary the UN characteristic of the
16:19characteristic agree it also implies that for a dimensional grass without
16:22very characteristic is minus the characteristic of the boundary
16:29three-dimensional both for example the characteristic is minus one the graph
16:33has a two-dimensional spheres boundary the characteristic is 1-2 which is minus
16:3811 could get a three-ball also through their eccentric refinement of a
16:41threesome flex
16:45yeah
16:46around the garden ideas of the set of complete subgraphs of a graph similarly
16:50the F matrix and chose the card in all the details on the set of pairs of
16:53complete subgraph of a graph intersecting in on Intergraph here is an
16:57example of a computation of the effector and matrix of a graph we can use the F
17:01factor to get the user characteristics and use the F matrix to get the wound
17:05characteristics
17:10Relations tel
17:11evaluations are 0 for grants which are geometric we call them D graphs Adi
17:16NEFAs affinis simple grapple with every unit sphericity here I D Series 80 graph
17:21which one becomes intractable the proof is very simple
17:25the curvature of it in Somerville valuation is it in Somerville valuation
17:28of the unit sphere
17:34x10 Duncan some of his victory clay
17:39masks in 1970 where the eff matrix satisfies them then Somerville equations
17:44he says then that it may be conjecture that the answer is affirmative we can
17:47confirm this context
17:54papers and books and combinatorial topology and discrete geometry
18:01here is how we get
18:02in Somerville relations take a victory which enters the EULAR characteristic
18:06ended in Somerville Victor B then produced the quadratic valuation has
18:09indicated it is their lundy graphs
18:15in
18:16here is how we write
18:17characteristic this is an example but only if he is not as the characteristic
18:21is then the UN characteristic and the UN characteristic or attend somerville
18:24invariant throughout dimensional graph
18:29another important
18:31withholding not only for linear evaluations but also for quadratic and
18:34higher multilinear evaluations is that if we averaged the indices overnight
18:38probability space of functions then we get curvature
18:44this
18:45averaging result proven by Thomas bank off in the 60 I is it is a general
18:49principle which connects points and dow spotted if we averaged a poincaré hop
18:53in disease we get your car
18:58here is an example which illustrate
19:01of a simple linear evaluation the volume the graphic the two-dimensional Disco
19:05vol 12 how do we get the index of a vertex v just count all the triangles
19:09which contain vie for which the value of effort to be as a maximum for every
19:12function if we get like this an index when averaging over all functions we get
19:16the curvature the verdict is the number of triangles attached to the verdicts
19:19divided by three
19:23the proof
19:25simple the valuation is given by definition as a sum over symbols is
19:28taking values and shot them equally to all the vertices of the Simplex in the
19:33case of a quadratic valuation we first have to get the interaction on this
19:37influx then distributed to the vertices
19:43to prove plank or just movie value of the Simplex to the verdicts on which
19:47jeff is maximal since we do not distribute the values they remain in
19:50tutors
19:53the multiplicative
19:55poverty can best be
19:56we get values on the vertices of the product RAF this turns out to be a
20:01poincaré hot index of the characteristic on the product RAF
20:07here is the code
20:08illustrates how one can
20:09characteristic in the Stanley wrice nearing the healer characteristic is
20:13just evaluating the rain element at a point where all vertices hope the value
20:16minus 12 get the wound characteristic dealer character is then subtract all
20:21the terms in the square of the things we need to remove all the Simplex pairs
20:24which do not intersect
20:30one has to
20:30right the sum of interactions according to and which simplex the interaction
20:33takes place now there are cases where both simple syntax why are different and
20:38not equal in cases where one of energy and in one case where both are see the
20:42upshot is that all the values together give either 1-1 depending on whether
20:45these even or odd dimensional in other words we get to you later characteristic
20:53to prove the day
20:54relations
20:55he'd like in the linear case and show that the curvature is equal to that in
20:58Somerville valuation on the unit sphere this allows to use induction
21:06the first is the 16 South a three-dimensional sphere and one of the
21:10six laconic solids in four dimensions it is the analog of the octahedron in four
21:14dimensions and its Respir you can check that any of its units your's is an
21:18octahedron its dual graph of this extends out is the tester at the
21:22four-dimensional cube note that this is a one-dimensional graph as it contains
21:26no triangles but if we triangulated we get to 30 again let's compute some
21:31quantities for these graphs
21:35we see here
21:3816 so the user characteristic is the dot product between the user victor with
21:42this vector it is like for any three-dimensional geometric graph this
21:46is already a classical in Somerville relation with the F matrix we can
21:49compute the week characteristic by applying it to the user Victor and again
21:52taking the dot product it is again 0 this illustrates already a quadratic in
21:57Somerville deletion
22:01the test
22:02or dementia
22:03but it is
22:04dimensional graph as it has no triangles the FBI has only two components the user
22:09characteristic is minus 16
22:11characteristic is $112 curvature is constant 7
22:18and here is the car
22:19mutation
22:20related to select which is now no more a plate Omix solid its allure
22:23characteristically characteristic are still there
22:30of classical attend somerville vectors we get the 10 Somerville matrix what
22:34always happens 40 grass is that the first row and first column of the matrix
22:37is zero these are the nonlinear 10 Somerville relations which grunbaum
22:41conjecture to hold
22:46finally here is an example
22:48graph the user characteristics of the factors 01 and 02 characteristics of the
22:53factors are five and eight the product has healer characteristic zero and we
22:57characteristic 40
23:01this is the end for more detailed look at the paper on the archive