Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 20, 2014.

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Questions regarding events or the calendar should be directed to Tori Corkery. 
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Thursday, February 20, 2014

11:00 am in 241 Altgeld Hall,Thursday, February 20, 2014
Number Theory Seminar

Stern poynomials, continued fractions, and two theorems of Gauss

Karl Dilcher (Dalhousie University)

Abstract: In this talk I will present a polynomial analog of Stern's "diatomic" sequence, along with many of its properties. I will then define two subsequences of these polynomials. Various properties of these two interrelated subsequences are obtained; in particular, they have 0-1 coefficients and can be seen as extensions or analogues of the Fibonacci numbers. These sequences are then used to define two analytic functions as their respective limits. As an application we obtain evaluations of certain finite and infinite continued fractions whose partial quotients are doubly exponential. In a case of particular interest, the set of convergents has exactly two limit points. (This is joint work with K.B. Stolarsky). If time allows, I will also present joint work with John Cosgrave on mod $p^3$ analogs of binomial coefficient congurences due to Gauss and Jacobi, along with the related concept of a Gauss factorial.

Submitted by ford
1:00 pm in Altgeld Hall 347,Thursday, February 20, 2014
Group Theory Seminar

Intersecting free subgroups in virtually free groups

Alexander Zakharov (Moscow State University)

Abstract: We prove an estimate for the rank of intersection of free subgroups in fundamental groups of finite graphs of groups with finite edge groups. It is analogous to the Hanna Neumann inequality for free groups and the Ivanov-Dicks estimate for free products. An estimate for virtually free groups is obtained as corollary.

Submitted by kapovich
2:00 pm in 140 Henry Administration Building,Thursday, February 20, 2014
Graduate Student Number Theory Seminar

An introduction to infinite log-concavity

Armin Straub   [email] (UIUC Math)

Abstract: The purpose of this talk is to introduce and advertise the concept of log-concavity and its recent generalizations. A sequence $a_n$ is log-concave if $L(a_n) = a_n^2 - a_{n-1} a_{n+1} \ge 0$, and many important sequences from various fields, such as combinatorics, geometry or number theory, are known or believed to be log-concave. Following Boros and Moll, a sequence $a_n$ is $m$-log-concave if $L^j(a_n) \ge 0$ for all $j = 0, 1, \ldots, m$. A motivating example is the case of binomial coefficients which have been conjectured to be infinitely log-concave. While a recent result of Brändén shows that this is indeed true for rows of Pascal's triangle, the case of columns remains open. Time permitting, we will report on joint work with Luis Medina in which we investigate sequences which are fixed by a power of the log-concavity operator $L$. Surprisingly, we find that sequences fixed by the non-linear operators $L$ and $L^2$ are, in fact, characterized by a linear 4-term recurrence. Plenty of open problems will be mentioned along the way.

Submitted by astraub
4:00 pm in 245 Altgeld Hall,Thursday, February 20, 2014
Mathematics Colloquium

Periodicity and Complexity

Bryna Kra (Northwestern University)

Abstract: A beautiful example of a global property being determined by a local one is the Morse-Hedlund Theorem; for an infinite word in a finite alphabet, it provides the relation between the global property of periodicity and local information on the complexity. I will discuss higher dimensional versions of this problem, the use of local conditions on complexity to determine global properties of the configuration, and the relation to dynamics. This is joint work with Van Cyr.

Submitted by seminar