Mathematics Home / Algebra and Combinatorics 2017 Spring

**Spring 2017**

Time & Location: All talks are on Thursdays in Gibson 400A at 12:45 PM unless otherwise noted.

Organizer: **Tai Ha**

**January 26**

Introduction to the Hodge Conjecture

Al Vitter - TULANE UNIVERSITY

Abstract:

This will be the first of probably 2 talks. The Hodge Conjecture concerns smooth (non-singular) complex projective varieties. It relates the purely algebraic structure of the variety to its topological/complex-analytic structure via its cohomology groups. I will begin by discussing in a relatively untechnical way, some of the mathematics needed to state clearly the Hodge Conjecture. Then I will concentrate on some aspects that bring out (hopefully) the subtlety and importance of the conjecture.

**February 2**

Introduction to the Hodge Conjecture (part 2)

Al Vitter - Tulane University

Abstract:

This will be a continuation of my first talk. I will state the Hodge Conjecture and then talk about some aspects that show its subtlety and importance. The Hodge Conjecture concerns smooth (non-singular) complex projective varieties. It relates the purely algebraic structure of the variety to its topological/complex-analytic structure via its cohomology groups. I will begin by discussing in a relatively untechnical way, some of the mathematics needed to state clearly the Hodge Conjecture.

**February 9**

Classification of spherical diagonal actions of reductive groups

Mahir Bilen Can - Tulane University

Abstract:

In this talk we present our recent progress on the diagonal actions of a reductive groups on product varieties of the form X_1 x X_2, where X_1 is a symmetric space and X_2 is a partial flag variety. In particular, we classify all such actions.

**February 16**

Classification of spherical diagonal actions of reductive groups, Part II.

Mahir Bilen Can - Tulane University

Abstract:

In this talk we present our recent progress on the diagonal actions of a reductive groups on product varieties of the form X_1 x X_2, where X_1 is a symmetric space and X_2 is a partial flag variety. In particular, we classify all such actions.

**March 2**

The depth function of ideals in polynomial rings.

Tai Huy Ha - Tulane University

Abstract:

We shall discuss what could be the depth function of an ideal in a polynomial ring. That is, for which function f(n), there exists an ideal I in a polynomial ring R such that depth R/I^n = f(n) for all n > 0.

**March 9**

A DYNAMIC BUCHBERGER ALGORITHM

Prof. John Perry - University of Southern Mississippi

Abstract:

Gröbner bases are a major tool in commutative algebra, typically computed using the Buchberger algorithm. This algorithm is “static” in that it works with a fixed term ordering, required as input along with the ideal’s generators. In many cases, however, a “dynamic” Buchberger algorithm is more appropriate: it requires only the generators as input, and returns both a basis and an ordering that guarantees the Gröbner property. It computes the ordering by the guidance of a criterion inspired by an invariant of an ideal. This talk describes the algorithm, the traditional criterion to guide computation, and a new criterion.

**March 23**

Algebra & Combinatoric seminar

Points, symbolic powers and a conjecture by G. V. Chudnovsky.

Paolo Mantero - University of Arkansas

Abstract:

What is the smallest possible degree of an equation passing at least m times through t given points in the complex projective space P^N? The answer is not known (except in few special cases), however the complex analyst G. V. Chudnovsky in 1979 conjectured a lower bound, which until last year was only known to hold for points in P^2, general points in P^3 and certain extremal configurations in P^N.

In this talk we will survey the evolving framework of conjectures and results around the above question, and prove Chudnovsky's conjecture for any set of very general points in P^N.

**April 13**

The shortest path poset of Bruhat intervals

Prof. Saul Blanco - Indiana University (Host: Morris Kalka)

Abstract:

A Coxeter group W is a group generated by reflections; examples are the symmetric group and the hyperoctahedral group. These groups have many interesting combinatorial properties. For instance, one can define a partial order, called the Bruhat order, on the elements of W . If [u,v] is an interval in the Bruhat order, its Bruhat graph, B(u,v) includes the Hasse diagram of the poset [u,v] with edges directed upwards, as well as other edges that I will describe in the talk. While the longest u-v paths in B(u,v) are well-understood (they form a face poset of a regular cell decomposition of a sphere), not much is known about the other u-v paths in B(u,v). In this talk, I will describe what is known of the shortest u-v paths and point out connections to other areas.

**April 20**

Topic

Prof. Fabrizio Zanello - Michigan Tech University (Host: Tai H Ha)

Abstract: