Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 23, 2017.

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Thursday, February 23, 2017

11:00 am in 241 Altgeld Hall,Thursday, February 23, 2017

Partitions into $k$th powers of a fixed residue class

Amita Malik (Illinois Math)

Abstract: G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect $k$th powers, which was later proved by E. M. Wright. In this talk, we discuss partitions into parts from a specific set $A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb{N}, m\equiv a_0 \pmod{b_0} \right\}$, for fixed positive integers $k$, $a_0,$ and $b_0$. We give an asymptotic formula for the number of such partitions, thus generalizing the results of Wright and others. We also discuss the parity problem for such partitions. This is joint work with Bruce Berndt and Alexandru Zaharescu.

2:00 pm in Altgeld Hall,Thursday, February 23, 2017

Inequalities for products of power sums and the classical moment problem.

Bruce Reznick (UIUC)

Abstract: This is a partial repeat of a seminar I gave here in the early 1980s. For $x = (x_1,\dots x_n) \in \mathbb R^n$ and $r \in \mathbb N$, define the $r$-th power sum $M_r(x) = \sum_{i=1}^n x_i^r$. Upper bounds for many products of power sums come from the Hölder and Jensen inequalities. I will discuss some other cases: for example $M_1M_3/(nM_4)>-\frac 18$, where the lower bound is best possible, and the maximum and minimum values of $M_1M_3/M_2^2$ are $\pm \frac{3\sqrt 3}{16}n^{1/2} + \frac 58 + \mathcal O(n^{-1/2})$. In the first case, the classical Hamburger moment problem gives a particularly illuminating explanation. Most of this can be found in my paper: Some inequalities for products of power sums, Pacific J. Math., 104 (1983), 443-463 (MR 84g.26015), available at https://projecteuclid.org/euclid.pjm/1102723674

2:00 pm in 241 Altgeld Hall,Thursday, February 23, 2017

Some maximal curves obtained via a ray class field construction

Dane Skabelund   [email] (UIUC)

Abstract: This talk will be about curves over finite fields which are "maximal" in the sense that they meet the Hasse-Weil bound. I will describe some problems relating to such curves, and give a description of some new "maximal" curves which may be obtained as covers of the Suzuki and Ree curves.

3:00 pm in 347 Altgeld Hall,Thursday, February 23, 2017

A non-partitionable Cohen-Macaulay complex

Bennet Goeckner (The University of Kansas)

Abstract: Stanley conjectured in 1979 that all Cohen-Macaulay complexes were partitionable. We will construct an explicit counterexample to this conjecture, which also disproves a related conjecture about the Stanley depth of monomial ideals. This talk is based on joint work with Art Duval, Caroline Klivans, and Jeremy Martin. No prerequisite knowledge of simplicial complexes or commutative algebra will be assumed.

4:00 pm in 245 Altgeld Hall,Thursday, February 23, 2017

From Picard to Rickman: Mappings in spatial quasiconformal geometry

Pekka Pankka (University of Helsinki)

Abstract: One of the classical theorems in complex analysis is the Picard’s theorem stating that a non-constant entire holomorphic map from the complex plane to the Riemann sphere omits at most two points. From the conformal point of view, two dimensional geometry is special in this sense. Namely, by classical Liouville’s theorem from the same era, every conformal map from a domain of the n-sphere to the n-sphere is a restriction of a Möbius transformation for n>2. In particular, Picard’s theorem holds trivially in higher dimensions. An alternative for the overly rigid spatial conformal geometry is a so-called quasiconformal geometry; heuristically, instead of preserving the angles we allow them to distort by a bounded amount. In this talk, I will discuss the role of Picard’s theorem in quasiconformal geometry, which takes us from complex analysis to geometric topology.