August 1
(Monday) ～ August 5 (Friday), 2016
ABSTRACT
| August, 1 (Mon) | August, 2 (Tue) | August, 3 (Wed) | August, 4 (Thu) | August, 5 (Fri) |
| 9:30--10:30 | Fabrizio Zanello (Michigan Technological University) "Some recent interactions between commutative algebra and combinatorics" |
| Level and Gorenstein Hilbert functions, and their monomial and squarefree-monomial
counterparts, play an important role in both commutative algebra and combinatorics, thanks also to
their intriguing connections with a number of other mathematical topics.
This talk will attempt to review some of the old and new developments that have been shaping this field during the past 40 years. We will begin with R. Stanley's seminal contributions in the late Seventies, review some of the algebraic progress of the Nineties, and finally discuss the comeback center stage of the combinatorics of this topic during the last few years. The source for much of the recent material will be an AMS Memoir I co-wrote in 2012, On the shape of a pure O-sequence (arXiv:1003.3825), as well as the survey paper Pure O-sequences: known results, applications, and open problems (arXiv:1204.5247). Along the way, we will present several fascinating, and in some instances still unexplored, connections with other fields, such as finite geometries, design theory, plane partitions, and even matroid theory. The talk will include a selection of conjectures and open problems accessible to young researchers interested in algebra or in combinatorics. |
|
| 10:45--11:45 | Huy Tai Ha （Tulane University）
"Powers of sums of ideals" |
| Let $A = k[x_1, \dots, x_r]$ and $B = k[y_1, \dots, y_s]$ be standard graded polynomial rings over a field k, and let $I \subseteq A$ and $J \subseteq B$ be nonzero proper ideals. We shall discuss algebraic properties and invariants in powers, both symbolic and ordinary powers, of the sum $I+J \subseteq A \otimes_k B$. In particular, we shall give bounds and exact formulas for the depth and the regularity of powers of $I+J$. | |
| 13:15--14:15 | Akihiro Higashitani (Kyoto Sangyo University)
"On counterexamples of Cayley conjecture for lattice simplices" |
| It was proved by Nill that for any lattice simplex of dimension d with degree k which is not a lattice pyramid, the inequality $d \leq 4k － 2$ holds. In this talk, we present a complete characterization of lattice simplices satisfying the equality, i.e., the lattice simplices of dimension $(4k － 2)$ with degree k which are not lattice pyramids. As a by-product of this result, we show that such simplices are counterexamples for the conjecture known as ``Cayley conjecture'', which says that every lattice polytope of dimension d with degree less than $d/2$ can be decomposed into a Cayley polytope of at least $(d + 1 － 2k)$ lattice polytopes. | |
| 14:30--15:30 | Johannes Hofscheier （Otto-von-Guericke-University Magdeburg）
"Lattice simplices with a given degree" |
| We consider lattice simplices of a given degree r and relate them to certain subgroups of the (4r-1)-dimensional real euclidean space. It was shown by Lawrence that these subgroups form only finitely many maximal families. We present a classification of these maximal families for r=2 and apply our results to the description of the $h^*$-vectors of degree 2 simplices. This is joint work with Akihiro Higashitani. | |
| 15:45--16:45 | Benjamin Nill （Otto-von-Guericke-University Magdeburg） "Ehrhart theory for spanning lattice polytopes" |
| A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. In this talk we describe a new result in the Ehrhart theory of lattice polytopes that implies that the $h^*$-vector (also called delta-vector) of a spanning lattice polytope has no gaps. This generalizes a recent theorem by Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud-Goto conjecture. We also discuss how this relates to unimodality questions of lattice polytopes and previously achieved results on lattice polytopes of given degree. This is joint work with Johannes Hofscheier and Lukas Katthän. |