## Announcements and Notes

#### Urgent stuff

The 3rd (and last) HW has been posted here below.

NB! (i) A crash-course in duality and the proof of the Dehn-Sommerville Equations, and (ii) solutions to the first HW assignment have been posted here below.

Please Note! There will weekly talks at our Combinatorics, Algebra and Geometry Seminar (CAGS) on Fridays (Exploratory Hall, room 4106, at 12:30 pm). Many of the talks there are relevant to what we are covering in class. -- For more info check the following CAGS website.

#### General Info

Here below is a pdf version of the promotional poster for the course, it contains a rough outline of our plan of attack:

NB! All handouts for this class will be listed with an itemized icon as with the Syllabus.

For general more detailed information regarding this course, the Syllbus will be posted here below shortly. Please read it carefully as it outlines the overall plan for this course. It should answer all practical questions.

In the above figure we see a regular soccer ball, which is actually a convex polyhedron or a 3-dimensional convex polytope if each pentagonal or hexagonal face is not curved but rather a flat two-dimensional surface. This polytope is called the Truncated icosahedron, and is one of the 13 Archimedean solids, which are semi-regular convex polyhedrons composed of regular polygons. A convex polytope is basically a multidimensional convex polyhedron with vertices, edges and faces of various dimensions. A 2-dimensional polytope is a convex polygon and trivially each polygon has equally as many edges as vertices, which can be of any order, and just one 2-dimensional face, namely the internal of the polygon itself. In 3-dimensions it gets a tad more complicated but still there are some very nicely stated conditions on the number of vertices, edges, and 2-dimensional faces due to Stenitz from 1922. The interesting (and annoying!) thing is that in dimensions 4 and higher, it is unknown how to describe the conditions the number of vertices, edges and i-dimensional faces must satisfy in order for them to count (or represent) the number of vertices, edges and faces of a polytope. Finding such a description has been, and still is, an active area of research in algebraic combinatorics. The reason this is a branch in algebraic combinatorics is that it commutative algebra has turned out to be indispensable tool in obtaining important and descriptive results about polytopes and related combinatorial topics.

This topics course will serve as an introduction to the application of abstract algebra in combinatorics, and the hope is that, after this course, it should be clear that a firm knowledge in commutative algebra is necessary for the working combinatorist. -- I will assume basic knowledge in algebra (Math 621, Algebra, first year graduate course, or equivalent), and some combinatorics (Math 325, Discrete Mathematics II, or equivalent) is preferred. However, the most important thing is to have mathematical maturity and an open mind.

There is no required text for the course. The lectures will be self contained and be based in part of some of the following books, which also will serve as a list of references.

1. Takayuki Hibi, Algebraic Combinatorics; on Convex Polytopes, Carslaw Publications, (1992). (Out of print).
2. Richard P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Progress in Mathematics (PM-41), Birkhauser, Boston, (1996).
3. Ezra Miller; Bernd Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics (GTM-227), Springer Verlag, New York, (2005).
4. Arne Brøndsted, An Introduction to Convex Polytopes, Graduate Texts in Mathematics (GTM-90), Springer Verlag, New York, (1983).
In addition, and if time permits, we might discuss some results from selected research articles.

#### Weekly recap of Lectures and Problems

Here below are some notes to recap the lecture each week. Also, occasional problems and/or their solutions might be posted here as well.

1. First week, January 23.
• Part I, Convex polytopes and simplicial complexes
In this first lecture we introduced a convex polytope, one of the fundamental object of algebraic combinatorics. We defined a convex polytope as the convex hull of finitely many points in the N-dimensional Euclidean space, with its usual topology. Each polytope has faces which are precisely the non-empty intersections of the polytope and its supporting hyperplanes. Each face is itself a polytope and an i-face is a face of dimension i. By convention for a polytope of dimension d we call its 0-faces vertices, its 1-faces edges and its (d-1)-faces facets. We showed that each polytope is precisely the convex hull of its vertices, and that each face is the precisely the convex hull of those vertices contained in the face. This shows that the polytope can be presented in a finite way, so to speak, as a set-system or hypergraph on a finite set of vertices, i.e. a collection of subsets of a given finite set we call a vertex set. -- Next week we will continue to discuss the properties of polytopes and introduce some parameters attached to them.
2. Second week, January 30.
• Part I, cont.
In this second lecture we used two fundamental properties of a polytope to define a polyhedral complex, as a finite collection of convex polytopes such that (i) every face of a polytope in the complex is also contained in the complex, and (ii) the intersection of two polytopes in the complex is also in the complex. We have seen that if we know the vertices of a polytope, then every face is uniquely determined by a subset of vertices of the polytope. However, not every subset will yield a face. -- A simplex is precisely a polytope where every subset of its vertices will yield a face. Simplices are the fundamental polytopes partily for this very reason. A polyhedral complex is simplicial if every polytope in it is a simplex of some dimension. Likewise, a polytope is simplicial if every face is a simplex. Two polyhedral complexes are of the same combinatorial type if their face posets are order isomorphic.
3. Third week, February 6.
• Part I, cont.
The f-vector of a complex of dimension d-1 is the d-tuple where the i-th coordinate denotes the number of faces of dimension i for each i from 0 to d-1. We also defined the corresponding h-vector which is a (d+1)-tuple. Two complexes of the same combinatorial type have the same f- and h-vectors, but the converse is far from true, as easy examples show us. Characterizing the vectors that are f-vectors or h-vectors is a daunting task and only small parts have been solved. The f-vector and the h-vector of a polyhedral complex are equivalent, in that one can be obtained from the other. Since the f-vector directly describes the number of faces of each dimension, one might ask why bother with the h-vector? One reason for the h-vector is that each simplex of dimension d has (1,1,1,...,1) as its h-vector. Another reason is that there are certain equations or condictions easier to state using the h-vector (we will see later that the Dehn-Sommervill Equations have an elegant representation using the h-vector.)
• Part II, The Katona-Kruskal-Shutzenberger theorem
The information of the f-vector or the h-vector for a polyhedral complex is by no means enough to characterize the complex, as we have seen with simple examples, but it provides the first tool to distinguish complexes from one another. Some integer vectors are f-vectors for polyhedral complexes others are not. Hence, it is of great interest to characterize which vectors are indeed f-vectors for polyhedral complexes. In general this is unknown, but the characterization of f-vectors for simplicial complexes is known and has a nice solution. The theorem describing this characterization is due to Katona, Kruskal and Schutzenberger (KKS Thm), all who obtained the same characterization independently.
4. Forth week, February 13.
• Part II, cont.
To describe the characterization in the KKS theorem, one must use the i-binomial representation of a positive integer, and the reverse lexicographical ordering of subsets of [n] of cardinality i. The proof of the KKS Thm is unusual from most mathematical theorems, in that when proving a characterization, the direction to prove that the given conditions in (iii) of the f-vector imply the existence of a concrete simplicial complex with the given vector as its f-vector as in (i) is realitively easy. The hard part is actually to show that the f-vector of every simplicial complex in (i) actually does satisfies the given conditions in (iii). To do that one must rely on a fundamental theorem of the combinatorics of finite sets, that the boundary of the compression of a given collection of sets is contained in the compression of the boundary of the same collection of sets. The proof of this theorem, which we just used and didn't prove, can be found in I. Anderson, Combinatorics of finite sets, Oxford University Press, 1987.
• Part III, Simplicial polytopes and spheres
Here we will discuss (i) The Dehn-Sommerville Equations (DSE), (ii) The Upper Bound Theorem (UBT), and (iii) The Lower Bound Theorem (LBT) for simplicial spheres, which are simplicial complexes whose underlying sets are homeomorphic to topological shperes. Clearly, the boundary complex of a polytope is homeomorphic to a sphere, so a simplicial sphere does generalize the boundary complex of a simplicial polytope. In dimension 3 simplicial spheres are exactly boundary complexes of simplicial polytopes, but in dimension 4 there are simplicial spheres that are not boundary complexes of any polytope. The DSE actually also hold for any simplicial Eulerian poset. (i.e. a ranked poset such that any interval has the same number of elements of even rank as elements of odd rank, and it is simplicial if every down-set (the set of elements less than or equal to a given element) is actually a boolean algebra (as a hyper-cube)). -- The neat think about the DSE is that among the d entries of the f-vector, the latter half are a linear combination of the first half (for odd d the first "half" really means the lower integer part [d/2] of d/2). Hence, if we know the enties of the f-vector labelled with 0,1,...,[d/2]-1 then we can compute the rest of the entries as linear (in fact, integer) combination of the first ones. This greatly restricts the structure of an f-vector for a general (d-1)-sphere, and hence the boundary complex of a polytope.
5. Fifth week, February 20.
• Part III, cont.
The cyclic polytope C(n,d) is a simplicial polytope generated by n distinct points on the moment curve in the d-dimensional space. The cyclic polytope has the obvious maximum number (n choose i+1) of i-faces for i among 0,1,...,[d/2]-1. The Upper Bound Theorem (UBT) states that among every simplicial (d-1)-sphere, the cyclic polytope C(n,d) has the maximum number of i-faces for every i, that is for i among 0,1,...,d-1. The UBT for simplicial polytopes was proved by McMullen in 1070. -- NB! The cyclic polytope C(n,d) has n choose i+1 faces of dimension i only for the first half of the i-faces.! -- The proof of the UBT used the fact that each polytope has the combinatorial property of being shellable (something we will discuss soon, intuitively it means that we can build the boundary complex up and close it as if we are building closed brick-tower or an igloo). Since (d-1)-spheres are in general not shellable, this proof of McMullen does not carry over to (d-1)-spheres. To prove the UBT (which was done be Richard Stanley at MIT) one must use that a (d-1)-sphere is Cohen-Macaulay, which we will discuss in the coming weeks.
6. Sixth week, February 27.
• Part IV, M-vectors and Cohen-Macaulay rings
This part is at the heart of our coverage since it shows how commutative algebra is important in combinatorics. Here R is a graded, commutative algebra over a given field. We further assume that R is finitely generated (by part/summand nr. 1, as part/summand nr. 0 is the field itself). The version of the Hilbert Basis Theorem (HBT) that we will be using states that "If I is a graded ideal and R is finitely generated, then I is generated by finitely many homogeneous elements of R". The "finitely generated"-part here is indeed a special version of the HBT which states that "any ideal of the polynomial algebra over a field in finitely many variables/indeterminates, is finitely generated." Again, this is a special case of the purest form of the HBT that states that "if R is Noetherian, then so is R[x], the polynomial ring over R." -- The Hilbert function of R is simply the dimension of the n-th summand of R over the given field, and the Hilbert series is then the generating function of this function. Although it can many times be hard to express the dimension of each summand in a graded algebra R, the Hilbert series might actually sum to a nicely expressable function, something which is informative when investigating these dimensions. We will demonstrate the usufulness of the Hilbert functions and series in lectures to come.
7. Seventh week, March 6.
• Part IV, cont.
We say that a graded R is standard if all the generating variables have the same degree (of one). For a standard R we have the Noether Normalization Lemma (NNL), which basically states that R (being finitely generated) always contains a copy of the free polynomial ring over the base field k in some number of variables. The maximum number d of these variables turns out to be unique and is called the Krull dimension of R. In that case the elements of R representing these free variables are then called homogeneous system of parameters (h.s.o.p., or just "hsop") for R. The hsop for R are called regular if every element of R can be written uniquely as a linear combination of certain given elements of R, where the coefficients are polynomials in the hsop's. As we saw, to show that hsop is regular can take quite a bit of work, when done directly.
8. Eighth week, March 20.
If a standard graded algebra R has a regular hsop, then it turns out that every hsop for R is regular. In this case we call R a Cohen-Macaulay ring. To figure out whether or not a given set of hsop is regular can be a daunting task if we prefer to do that directly. Luckily, we can instead compute the Hilbert series for R and decide from its form whether the given set of hsop is regular or not. Note that computing the Hilbert series can itself be involved, but it is much more direct, and can (for that matter) be programmed for a computer to compute. In short computing the Hilbert series is mechanical.
9. Nineth week, March 27.
• Part IV, cont.
An M-vector (or O-sequence) is simply the vector corresponding to the Hilbert series of a finitely dimensional standard algebra S. The entries of an M-vector must all be positive and less than or equal to a binomial coefficient describing the number or monimial of a certain total degree. This gives us the first criterion to check if a given vector is indeed an M-vector. A nice algebraic condition describing exactly when a given vector is an M-vector, is given by Macaulay's Theorem (hence the "M" in M-vector!). This theorem is very similar to the previous KKS theorem we discussed.
• Part V, Stanley's idea of the proof of the UBC
Stanley's proof of the Upper Bound Conjecture (UBC), a purely combinatorial result, involves both commutative algebra and algebraic topology. The main tool is the face ring aka Stanley-Reisner ring of a given simplicial complex. This face ring is a graded standard algebra over a field, which is a quotient algebra of the polynomial ring in n variables, where n is the number of vertices of the simplicial complex. One nice property of the face ring is that its Hilbert series is given by a rational function in lambda, where the numerator is precisely the polynomial given by the h-vector of the complex. This result alone serves as a vital justification for studying the h-vector of a simplicial complex.
10. Tenth week, April 3.
• Part V, cont.
What Stanley needed to show, essentially, was that the face ring of a (d-1)-sphere is Cohen-Macaulay. That result was provided by a seminal paper by Gerald Allen Reisner, which gives topological criteria on when exactly the face ring is Cohen-Macaulay. From Reisner's result if follows that any (d-1)-sphere is Cohen-Macaulay, and hence Stanley was able to prove the UBC.
• Part VI, Reisner's topological criteria
In a seminal paper, Gerald Allen Reisner gave necessary and sufficient conditions for the face ring of a simplicial complex to be Cohen Macaulay (CM). These conditions turn out to be topological, in the sense that if two simplicial complexes are homeomorphic, then they are either both CM or neither is CM. -- In order to fully understand and appreciate Reisner's theorem, some algebraic topology is needed, in particular the reduced simplicial homology spaces of a simplicial complex. The word "group" is used if the coefficients are integers. The word "reduced" is here due to the fact that for i = -1 then in the reduced vector space case Ci is generated by one element, namely the empty set (so its dimension is f-1 = 1 consistent with the h-vector), but in the usual topological (abelian group) case, Ci = (0) corresponding to dimension zero.) Having the boundary operator and the fact that the composition of any two consecutive bounfary operators is the zero map, justifies the definition of the i-th reduced homology quotient space over k.
11. Eleventh week, April 10.
• Part VI, cont.
As we noticed in class the only reduced homology groups we can spot right away are for i = -1,0, where for i = -1 the i-th reduced homology group is zero if the complex is non-empty, but it is homomorphic to k if it is empty, and for i = 0 the dimension of the 0-th reduced homology group over k equals the number of connected components of the complex minus one. Hence we see that the 0-th reduced homology group only depends on the topological property of its connectivity. This holds in general, in that if two complexes are homeomorphic, then their reduced homology groups are homomorphic as vector spaces over the field k. We also discussed two examples that illustrated the reduced homology groups for the (d-1) dimensional ball and (d-1) dimensional sphere. -- To state and understand Reisner's theorem, we define the link of a face of a given simplicial complex. The link is itself a simplicial complex of dimension given by the dimension of the simplicial complex minus the number of vertices of the face. Reisner's theorem states that the face ring of a simplicial complex is Cohen Macaulay (CM) if and only if for every face F (including the empty face!) the i-th homology of the link of F is trivial, or equal to (0), for all i not equal to the dimension of the link of F.
12. Twelfth week, April 17th.
• Part VI, cont.
As an important corollary of Reisner's theorem, one obtains that if a simplicial complex is homeomorphic to either the (d-1) dimensional ball or (d-1) dimensional sphere, then the face ring of the simplicial complex is CM over any field. One important point, that was pointed out by the topologist James R. Munkres in 1984, is that Reisner's criterion is topological in the sense that if two simplicial complexes have homeomorphic geometric realizations then the face rings of them are either both CM or neither is CM. As an example, if a simplicial complex has a geometric realization homeomorphic to the torus, then its face ring is not CM. Another example shows that if a simplicial complex has a geometric realization homeomorphic to the real projective plane, then the face ring is CM if and only if the base field k is not of characteristic 2. -- We know that if the face ring is CM then the h-vector of the complex is an M-vector, but we don't, in general, have the converse. However, by Stanley's theorem from 1977, we do have that for a given h-vector/tuple it is an h-vector of a simplicial complex whose face ring is CM if and only if it is an M-vector.
• Part VII, Cohen Macaulay complexes
A simplicial complex is said to be Cohen Macaulay (CM) over the field k if its face ring over k is CM. For each simplicial complex this definition clearly does depend on the field k (notably on the characteristic!), but for most our examples it does not. Hence, we can just talk about a cimplicial complex being CM. It turns out that being CM is quite geometrical in many ways: in particular if our complex is CM, then (i) each link of the complex (itself a simplicial complex) is also CM, (ii) if the dimension of the complex is one or more, then its geometric realization is connected, and (iii) the complex is pure, which means that the complex is composed of simplices, all of the same dimension.

EOF/EOS