Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.

Abstract. We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the Yang-Baxter equation. The images of Bn under these representations are finite groups, and we identify them precisely as extensions of extraspecial 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of Bn factoring over Temperley-Lieb algebras and the corresponding link invariants. 1.

We compute the generalized triangle inequalities explicitly for all rank 3 symmetric spaces of noncompact type. For SL(4, C) there are 50 inequalities none of them redundant by [KTW]. For both Sp(6, C) and Spin(7, C) there are 135 inequalities of which 24 are trivially redundant in the sense that they follow from the inequalities defining the Weyl chamber ∆. There are 9 more redundant inequalities for each of these two groups. One interesting feature is that these inequalities do not occur for the other system (and consequently must be redundant because the two polyhedral cones are the same by Theorem 1.8). The two equal polyhedral cones D3(B3) = D3(C3) have precisely 102 facets and 51 generators (edges). 1

Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when known. Background material and more specialized results are given in a series of appendices. We give a personal view of the field while remaining aware that there is much important and beautiful work that we have been unable to mention.

This paper is dedicated to Louis Billera on the occasion of his sixtieth birthday. Abstract: This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory gl nC and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov [1] about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each n ≥ 5 a counterexample, with arbitrarily increasing denominators as n grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when n is fixed. 1

Abstract. Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one–parameter family of flat connections ∇κ on h with values in any finite–dimensional g–module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group Bg of type g is a deformation of the action of (a finite extension of) W on V. The residues of ∇κ are the Casimirs κα of the subalgebras sl α 2 ⊂ g corresponding to the roots of g. The irreducibility of a subspace U ⊆ V under the κα implies that, for generic values of the parameter, the braid group Bg acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g–modules if g = sl3 but that this is not the case if g ≇ sl2, sl3. We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin’s braid group Bn on the zero weight spaces of all simple U�sln–modules for n ≥ 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self–dual g–modules and obtain complete classification results for g = sln and g2 and conjecturally complete results for g orthogonal or symplectic. Contents

Abstract. Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a Q-variation of Hodge structures of weight one on Y with Higgs bundle E 1,0 ⊕ E 0,1, coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the slopes satisfy µ(E 1,0) − µ(E 0,1) ≤ µ(Ω 1 Y). We prove a similar inequality in the higher dimensional case. If the latter is an equality, and if the discriminant of E 1,0 or the one of E 0,1 is zero, one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank> 1. Part of the results extend to variations of

2. The moduli space of polygons in Hm+1 6 2.1. Coadjoint orbits 6 2.2. The space of closed polygons 7

Abstract. J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits’s geometry and works through several examples. We also give a criterion for recognizing the automorphism of the geometry induced by an automorphism of the group. The E6 geometry is studied in depth. Contents 1. Tits’s geometry ΓP 3 2. A concrete geometry ΓV, part I 4

Let H be a subgroup of a finite group G. We use Markov chains to quantify how large r should be so that the decomposition of the r tensor power of the representation of G on cosets on H behaves (after renormalization) like the regular representation of G. For the case where G is a symmetric group and H a parabolic subgroup, we find that this question is precisely equivalent to the question of how large r should be so that r iterations of a shuffling method randomize the Robinson–Schensted–Knuth shape of a permutation. This equivalence is remarkable, if only because the representation theory problem is related to a reversible Markov chain on the set of representations of the symmetric group, whereas the card shuffling problem is related to a nonreversible Markov chain on the symmetric group. The equivalence is also useful, and results on card shuffling can be applied to yield sharp results about the decomposition of tensor powers. 1.