Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day, informal meeting of mathematicians from the University of Illinois, Purdue University and nearby universities, with interests in algebra, geometry and combinatorics (widely interpreted).Further details will be posted here as they become available. Or you may contact the University of Illinois organizers Hal Schenck and Alexander Yong, or the Purdue organizers Uli Walther and Saugata Basu .First event : Fall, 2009Date: September 26, 2009Location: Department of Mathematics, Altgeld Hall, University of Illinois at Urbana-ChampaignThe talks will held at 245 Altgeld Hall. Coffee-breaks will be held down the hall in front of 273 Altgeld Hall.List of participants
Speakers and schedule:Coffee and pastries 8h30
Giulio Caviglia (Purdue) 9h30 - 10h30 TBA Manoj Kummini (Purdue) 11h - 12hTitle: Arithmetic rank of monomial ideals.Abstract:The arithmetic rank of an ideal $I$ (denoted $\mathrm{ara} I$) in a polynomial ring $R = \Bbbk[x_1, \ldots, x_n]$ is the least number $r$ such that there exists polynomials $f_1, \ldots, f_r$ such that the ideals $(f_1, \ldots, f_r)$ and $I$ have the same radical. It is the least number of equations to required to define the variety of $I$. In this survey talk, we will discuss combinatorial constructions that have been used to determine arithmetic rank of various classes of monomial ideals. Coffee and snacks 13h30Li Li (UIUC) 14h - 15h Title: On the algebra and combinatorics of q,t-Catalan numbers.Abstract: Haiman proved that the q,t-Catalan number is the Hilbert series of the graded vector space M=\oplus M_{d_1,d_2} spanned by a minimal set of generators for the ideal of the diagonal locus of (C^2)^2. It is natural to ask for a combinatorial construction of such generators. In this talk we give upper bounds for the dimension of M_{d_1, d_2} in terms of partition numbers, and find all bi-degrees (d_1,d_2) that acheive equality. For these bi-degrees, we answer the aforementioned question. This is joint work with Kyungyong Lee. Hal Schenck (UIUC) 16h - 17hTitle: Syzygies of plane curve singularitiesAbstract:Let C=V(Q) be a projective plane curve, reduced but not irreducible. Question: when is the Jacobian ideal of Q arithemtically Cohen-Macaulay? When Q is a product of linear forms, it is conjectured that the combinatorics of the line arrangement determines the Cohen-Macaulay property. In the case of lines, there is an inductive operation, which corresponds to the algebraic geometry operation of elementary modification of rank 2 bundles on a surface. When Q is not a product of linear forms, subtle behavior of the singularities enters the picture (quasihomogenity). I will report on recent work on this problem, partly with S. Tohaneanu (CMH, 2009) and partly with H. Terao and M. Yoshinaga. Dinner InformationDinner will be at 18h in Mandarin Wok (approximate cost $10 per head). Local Accomodation InformationHomewood Suites (holding rooms for AlGeCom) |