Frank Sottile Report on Activities: 1 January -- 15 June 2004 One of my main activities in this period was chairing the organizing committee of the Winter 2004 program at thet Mathematical Sciences Research Institute on the "Topological Aspects of Real Algebraic Geometry". Some of my more specific duties for the program were to help organize the Introductory Workshop (12-16 January 2004), a weekend workshop on Real Algebraic Geometry and Geometric Modeling (3-4 April), and the Workshop on Algorithmic, Combinatorial, and Applicable Real Algebraic Geometry (12-16 April). I was a mentor for the MSRI postdoc Seongchun Kwon and also for Frederic Bihan, a postdoc whom I supported on a grant. Other activities included co-organizing a special session on algebraic geometry at the Joint AMS-SMM meeting in Houston (13-15 May), as well as planning for two other meetings that I also co-organized---a Banff International Research Station meeting on Combinatorial Hopf Algebras (28 August-2 September 2004) and an AMS special session on the Modern Schubert Calculus at the Evanston AMS meeting (23-24 October 2004). I also served on the program committee for the 16th Internatinal Conference on Formal Power Series and Algebraic Combinatorics in Vancouver (28 June-2 July 2004). Finally, I was also a member of the editorial board for the SIAM Journal on Discrete Mathematics. My research program included many collaborations; I discuss those for which substantial work was done during this period. The bibliography at the end lists all papers and projects that were worked on (including revisions for publication and initial research) while I was a Clay Mathematical Institute Research Scholar. With co-organizer Vicki Powers, MSRI member Claus Scheiderer and Bruce Reznick, we wrote a paper "A New Approach to Hilbert's Theorem on Ternary Quartics", which will appear in the Comptes Rendus Academy of Sciences (Paris). Here, we strengthen Hilbert's theorem, showing that a postive ternary quartic is a sum of three squares in exactly eight inequivalent ways. A major focus of my work concerned real solutions to polynomial equations and geometric problems. With Evgenia Soprunova, we largely wrote a paper "Lower Bounds for Real Solutions to Sparse Polynomial Systems", which we completed in the Summer of 2004. This gives lower bounds on the number of real solutions to some sparse polynomial systems. With the Postdoc Benoit Bertrand and my visitor and MSRI member Frederic Bihan, we started a project to give upper bounds to some sparse polynomial systems which are significantly lower than the Khovanski or Kouchnirenko bounds. This will soon result in a paper "Polynomial systems with few real zeroes". With Soprunova, graduate student (and MSRI visitor) Jim Ruffo, we used MSRI computers to complete most of the experimentation (12 gigaHertz years) on a project to investigate the Schubert calculus on a flag manifold, numerically. The paper describing this project is currently being written, and will be entitled "Experimentation and conjectures in the real Schubert calculus". Another paper recently completed was "Cremona Convexity, Frame Convexity, and a Theorem of Santal\'o". This applied classical constructions in algebraic geometry to unbderstand notions of convexity for lines in n-space. This was a direct result of the environment at MSRI, involving Jacob Goodman, who co-organized the Autumn program on Discrete and Computational geometry, Autumn MSRI member Andreas Holmsen, year-long MSRI member Kristian Ranestad, and Ricky Pollack, who visited MSRI for parts of both the Autumn and Spring programs. Marcelo Aguiar and I completed two papers on combinatorial Hopf algebras that we had begun in 2003, "Cocommutative Hopf algebras of permutations and trees", and "Structure of the Loday-Ronco Hopf algebra of trees". I also completed two papers in the combinatorics of the Schubert calculus with Cristian Lenart, "Grothendieck Polynomials via Permutation Patterns and chains in the Bruhat Order" and "A Pieri-type formula for the K-theory of a flag manifold", the first paper was also with Shawn Robinson. I started and worked extensively on two other projects also while at MSRI, but these are further from completion. One, with (then) Berkeley graduate student Kevin Purbhoo, establishes Horn-type inequalities for so-called minuscule flag varieties; these extend the Horn inequalities for the Littlewood-Richardson numbers to this other setting. The other, with Tom Braden and Linda Chen, has resulted in a presentation of the equivariant cohomology ring of the Quot scheme for the Grassmannian. Bibliography Accepted Papers * Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, with Marcelo Aguiar. Advances in Mathematics, to appear. * Real k-flats tangent to quadrics in R^n, with Thorsten Theobald. Proc. Amer. Math. Soc., to appear. * A New Approach to Hilbert's Theorem on Ternary Quartics, with Victoria Powers, Bruce Reznick, and Claus Scheiderer, Comptes Rendus (Paris), to appear. Papers in Review * Cocommutative Hopf algebras of permutations and trees, with Marcelo Aguiar. * Grothendieck Polynomials via Permutation Patterns and chains in the Bruhat Order, with Cristian Lenart and Shawn Robinson. * A Pieri-type formula for the K-theory of a flag manifold, with Cristian Lenart. * Structure of the Loday-Ronco Hopf algebra of trees, with Marcelo Aguiar. * Cremona Convexity, Frame Convexity, and a Theorem of Santal\'o, with Jacob E. Goodman, Andreas Holmsen, Richard Pollack, and Kristian Ranestad. * Lower Bounds for Real Solutions to Sparse Polynomial Systems, with Evgenia Soprunova. Manuscripts in Progress * Polynomial systems with few real zeroes, with Benoit Bertrand and Frederic Bihan. * Lines Tangent to Four Triangles, with Herv\'e Br\"onnimann, Olivier Devillers, and Sylvain Lazard. * Experimentation and conjectures in the real Schubert calculus for flag manifolds, with Jim Ruffo, Yuval Sivan, and Evgenia Soprunova. * The recursive nature of minuscule Schubert calculus, with Kevin Purbhoo. * Equivariant cohomology of the quot scheme, with Tom Braden and Linda Chen. * Conic decompositions of polytopes, with Matthias Beck, Sinai Robbins, and Jonathan Weitsman.