Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, February 22, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2018          February 2018            March 2018
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6                1  2  3                1  2  3
7  8  9 10 11 12 13    4  5  6  7  8  9 10    4  5  6  7  8  9 10
14 15 16 17 18 19 20   11 12 13 14 15 16 17   11 12 13 14 15 16 17
21 22 23 24 25 26 27   18 19 20 21 22 23 24   18 19 20 21 22 23 24
28 29 30 31            25 26 27 28            25 26 27 28 29 30 31



Thursday, February 22, 2018

11:00 am in 241 Altgeld Hall,Thursday, February 22, 2018

#### Euler Systems and Special Values of L-functions

###### Corey Stone (University of Illinois)

Abstract: In the 1990s, Kolyvagin and Rubin introduced the Euler system of Gauss sums to derive upper bounds on the sizes of the p-primary parts of the ideal class groups of certain cyclotomic ﬁelds. Since then, this and other Euler systems have been studied in order to analyze other number-theoretic structures. Recent work has shown that Kolyvagin’s Euler system appears naturally in the context of various conjectures by Gross, Rubin, and Stark involving special values of L-functions. We will discuss these Euler systems from this new point of view as well as a related result about the module structure of various ideal class groups over Iwasawa algebras.

2:00 pm in 241 Altgeld Hall,Thursday, February 22, 2018

#### Multiples of long period small element continued fractions to short period large elements continued fractions

###### Michael Oyengo (UIUC)

Abstract: We construct a class of rationals and quadratic irrationals having continued fractions whose period has length $n\geq2$, and with "small'' partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large'' partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of these continued fractions and study properties of polynomials arising from these generalizations.

2:00 pm in 243 Altgeld Hall,Thursday, February 22, 2018

#### Basic, order basic, and bibasic sequences in Banach lattices

###### Vladimir Troitsky (University of Alberta)

Abstract: Recall that sequence in a Banach is a (Schauder) basis if every vector admits a unique series expansion which converges to the vector in norm. In a Banach lattice, one may replace norm convergence with order or uniform convergence. This leads to several types of order bases and order basic sequences. We will discuss connections between these types of sequences. This is a joint project with M.Taylor.

4:00 pm in 245 Altgeld Hall,Thursday, February 22, 2018

Abstract: Hadwiger's conjecture from 1943 states that every simple graph with no $K_t$ minor can be properly colored using t-1 colors. This is a far-reaching strengthening of the four-color theorem and appears to be currently out of reach in its full generality. In the last three years, however, several relaxations have been proven. In these relaxations one considers colorings such that every color class induces a subgraph with bounded maximum degree or with bounded component size. We will survey recent results on such improper colorings of minor-closed classes of graphs. Based on joint work with Zdenek Dvorak and with Alex Scott, Paul Seymour and David Wood.