Mathematics

More shards of proofiness about the Stern sequence

Bruce Reznick (UIUC)

Abstract: The Stern sequence is defined recursively by s(0) = 0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1). We will talk about the behavior of s(n) mod d, for various d, intending to present material of interest to those who are not yet fans of Stern. Most of the work is taken from the speaker's ongoing course Math 595 SNT.

Celular resolutions via the hull complex

Alexandra Seceleanu (UIUC Math)

Abstract: In this talk I will present the way a geometric realization of a monomial ideal can give rise, through its associated cellular complex, to a free resolution of the ideal. The talk will focus on obtaining the hull complex of a monomial ideal. We will study its associated hull resolution and, time permitting, minimality issues.

Axioms for classes of Nakano spaces

Pedro Poitevin (UIUC Math)

Abstract: Nakano spaces are generalizations of classical L_p-spaces in which the p exponent is allowed to vary randomly with the underlying measure space. The class of Nakano spaces L_p for which p has its essential range included in a given compact subset K of the interval [1,¥) has been proven to be closed under Banach-lattice ultraproducts and ultraroots, and therefore it is axiomatizable in the continuous signature of Banach lattices. I will indicate how to find axioms for it in that signature. If time allows, I will also indicate how to find axioms for the slightly more interesting class of Nakano spaces L_p for which p has K as its essential range.

Sperner implies Brouwer

Prof. Mary-Elizabeth Hamstrom (UIUC Department of Mathematics)

Abstract: I will give two proofs of the implication of the title, that Sperner's Lemma implies the Brouwer Fixed Point Theorem. For details, see the Geometric Potpourri sign in 250 Altgeld Hall.

Edge-Colorings of Graphs

Michael Stiebitz (Technische Universität Ilmenau)

Abstract: The chromatic index chi'(G) of a graph G (multiple edges allowed) is the minimum number of colors needed to color the edges of G such that adjacent edges receive distinct colors. The edge coloring problem is NP-hard. There are two elementary lower bounds. Always chi'(G) >= D(G), where D(G) is the maximum degree of G. Also, chi'(G) >= W(G), where W(G) is the maximum, over all subgraphs H of G, of e(H)/\floor[n(H)/2].

Graphs with chi'(G)=W(G) are called elementary. Goldberg and Seymour independently conjectured in the 1970s that all graphs G with chi'(G) >= D(G)+2 are elementary. For an integer m, let J_m denote the class of all graphs G such that chi'(G) > (mD(G)+m-3)/(m-1). Shannon's Theorem implies that J₃ is empty. Always J_m\subseteq J_m+1, and the union of all J_m with m >= 3 consists of all graphs G such that chi'(G) >= D(G)+2.

The conjecture has previously been proved for all graphs in J_m for odd m up to 11. We use an extension of the method Tashkinov used for m=11 to prove that every graph in J₁₃ is elementary. The proof yields a polynomial-time algorithm to properly color the edges of every graph G with at most floor[(13chi'(G)+10)/12] colors.

A General Fatou Lemma for the Gelfand Integral.

Peter Loeb (UIUC Math)

Abstract: In joint work with Yeneng Sun, a general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space, or for a stronger conclusion, Banach lattice. A corollary sharpens previous results in the finite dimensional setting even for the case of scalar measures. Examples show that a vector measure space formed from Lebesgue spaces will not suffice as the underlying space for the result.

Spring Department Faculty Meeting