Fri, November 6, 2020
Public Access


Category:
Category: All

06
November 2020
Mon Tue Wed Thu Fri Sat Sun
            1
2 3 4 5 6 7 8
9 10 11 12 13 14 15
16 17 18 19 20 21 22
23 24 25 26 27 28 29
30            
8:00am  
9:00am  
10:00am  
11:00am  
12:00pm  
1:00pm  
2:00pm  
3:00pm  
4:00pm  
5:00pm  
6:00pm [6:30pm] Claudia Polini, University of Notre Dame, IN, USA
Description:
Date and Time: 6 November 2020, 6:30pm IST/ 1:00pm GMT/ 08:00am EDT (joining time: 6:15 pm IST - 6:30 pm IST) Speaker: Claudia Polini, University of Notre Dame, IN, USA Google meet link: meet.google.com/urk-vxwh-nri Title: The core of monomial ideals Abstract: Let $I$ be a monomial ideal. Even though there may not exist any proper reduction of $I$ which is monomial (or even homogeneous), the intersection of all reductions, the core, is again a monomial ideal. The integral closure and the adjoint of a monomial ideal are again monomial ideals and can be described in terms of the Newton polyhedron of $I$. Such a description cannot exist for the core, since the Newton polyhedron only recovers the integral closure of the ideal, whereas the core may change when passing from $I$ to its integral closure. When attempting to derive any kind of combinatorial description for the core of a monomial ideal from the known colon formulas, one faces the problem that the colon formula involves non-monomial ideals, unless $I$ has a reduction $J$ generated by a monomial regular sequence. Instead, in joint work with Ulrich and Vitulli, we exploit the existence of such non-monomial reductions to devise an interpretation of the core in terms of monomial operations. This algorithm provides a new interpretation of the core as the largest monomial ideal contained in a general locally minimal reduction of $I$. In recent joint work with Fouli, Montano, and Ulrich, we extend this formula to a large class of monomial ideals and we study the core of lex-segment monomial ideals generated in one-degree.