### 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 .

### Location: Department of Mathematics, Altgeld Hall, University of Illinois at Urbana-Champaign

The talks will held at 245 Altgeld Hall. Coffee-breaks  will be held down the hall in front of 273 Altgeld Hall.

List of participants

 Purdue UIUC Andrei Gabrielov *Uli Walther *Saugata Basu *Giulio Caviglia * (s)Manoj Kummini * P (s)Kyungyong Lee * PPeter Scheiblechner * PKuei-Nuan Lin SMelissa Lindsey SLloyd West SChristine Berkesch SAmanda Kinzel SVita Kala SYoungsu Kim SPaolo Mantero SLan Nguyen SJonathan Montano S Alex Yong *Hal Schenck * (s)Bruce Reznick * Sheldon Katz *Tom Nevins *Li Li * P (s)Andrew Schultz * PJimmy Shan SDominic Searles SJacob Hall SAlex Seceleanu SSer-Wei Fu SBen Reinger SRadoslav Kirov SMatthew Yancey SImmanuel McLaughlin SJinwon Choi SStephen Maguire S * facultyP postdocS Graduate student(s) Speaker

TBA

### Title: 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.

### 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.

### Abstract:

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 Information

Dinner will be at 18h in  Mandarin Wok (approximate cost \$10 per head).

### Local Accomodation Information

Homewood Suites  (holding rooms for AlGeCom)1417 S. Neil, Champaign, IL   217-352-9960Other hotels in the area:Hampton Inn1200 W. University, Urbana, IL  217-337-1100  They have a shuttle to and from campus.Eastland Suites1907 N. Cunningham, Urbana, IL  217-367-8331    They have a shuttle to and from campus.Hawthorne Suites101 Trade Center, Champaign, IL  217-398-3400I-Hotel1900 S. First, Champaign, IL 217-819-5002   They have a shuttle to and from campus.Hilton Garden Inn1501 S. Neil, Champaign, IL  217-352-9970Holiday Inn1001 Killarney Rd, Urbana, IL  217-328-7900Holiday Inn Express1002 Killarney Rd., Urbana, IL 217-328-0328