Department of

Mathematics


Seminar Calendar
for events the day of Thursday, April 2, 2015.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, April 2, 2015

11:00 am in 241 Altgeld Hall,Thursday, April 2, 2015

Around the Mobius function

Maksym Radziwill (Rutgers Univ. Math)

Abstract: The Mobius function plays a central role in number theory; both the prime number theorem and the Riemann Hypothesis are naturally formulated in terms of the amount of cancellations one gets when summing the Mobius function. In recent joint work with K. Matomaki we have shown that the sum of the Mobius function exhibits cancellations in "almost all intervals" of increasing length. This goes beyond what was previously known conditionally on the Riemann Hypothesis and allows us to settle a conjecture on correlations of consecutive values of the Liouville function (a close cousin of Mobius function). Our result holds in fact in greater generality. Exploiting this generality we show that between a fixed number of consecutive squares there is always an integer composed of only "small" prime factors. This settles a conjecture on "smooth numbers" and is related to the running time of Lenstra's factoring algorithm. Finally, in recent work with K. Matomaki and T. Tao we have been able to use the previously-mentioned general result to show that Chowla's conjecture (on correlations of the Mobius function) holds on average and we strengthened previous results (of Hildebrand) on patterns in the Liouville function.

1:00 pm in 347 Altgeld Hall,Thursday, April 2, 2015

Macroecology for microbes: modeling patterns of phylogenetic diversity

James O'Dwyer (UIUC Plant Biology)

Abstract: Macroecological patterns are aggregated over large numbers of individuals, and often display a kind of universal behavior across different systems, independent of differences in their underlying ecological processes. Physicists benefit from several principles which can underly universal behavior: for example, laws of large numbers. Can we identify similar principles in ecology, and understand which phenomena are universal and which are more contingent on mechanism? In this talk I will put these questions in context, and present new models and data which shed light on when and why ecological systems display emergent patterns.

1:00 pm in Altgeld Hall 243,Thursday, April 2, 2015

Complexity of finitely generated residually finite groups

Alexei Myasnikov (Stevens Institute of Technology)

Abstract: Finitely presented residually finite groups are usually thought of as nice from the algorithmic view-point, in particular, they have decidable word problem. In this talk I will address the following general questions for such groups G: how large could be the Dehn function of G? How large could be the gap between the complexity of the word problem and the Dehn functions of G? What is the time complexity of the classical McKinsey algorithm for the word problem in G (this is the only known uniform algorithm for the word problem in such groups)? How large could be the depth functions in G? The depth function measures how deep one has to go into finite index subgroups to separate a non-trivial element of a given length in G from the identity. These are joint results with O. Kharlampovich and M. Sapir. I will also discuss finitely generated recursively presented residually finite groups with really strange algorithmic properties, so called Dehn monsters. To build them we need Golod-Shafarevich construction and a forcing-type argument from logic. This is based on joint results with D. Osin and B. Khoussainov.

2:00 pm in 140 Henry Administration Bldg,Thursday, April 2, 2015

Some arithmetic properties of numbers of the form $\left\lfloor p^c \right\rfloor$

Zhenyu Guo (University of Missouri - Columbia)

Abstract: For fixed $c \in (1, 149/87)$, we derive an asymptotic formula for the number of primes $p \le x$ such that $\left\lfloor p^c \right\rfloor$ is squarefree, where $\left\lfloor . \right\rfloor$ is the floor function. We also prove that there are infinite many $p$ such that $\left\lfloor p^c \right\rfloor$ is an almost prime.

3:00 pm in 243 Altgeld Hall,Thursday, April 2, 2015

The Broken Circuit Complex and the Orlik-Terao Algebra of a Hyperplane Arrangement

Dinh Le Van (Universität Osnabrück)

Abstract: The Orlik-Terao algebra of an arrangement is a commutative analogue of the well-studied Orlik-Solomon algebra, which is the cohomology ring of the arrangement complement. Recently, it has attracted significant attention, mainly because it encodes subtle information missing in the Orlik-Solomon algebra: it was used by Schenck-Tohaneanu to characterize 2-fomality, a non-combinatorial property which is necessary for the arrangement to be free and for the complement space to be aspherical. Nevertheless, the Orlik-Terao algebra is a deformation of a combinatorial object, the broken circuit algebra. Exploiting this strong connection between the two algebras, I will give in this talk characterizations for the following properties of each of them: having a linear resolution, being a complete intersection, and being Gorenstein. If time permits, a generalization of Schenck-Tohaneanu's result will also be discussed.

4:00 pm in 245 Altgeld Hall,Thursday, April 2, 2015

Ancient solutions

Natasa Sesum (Rutgers)

Abstract: I will discuss ancient solutions in the context of the mean curvature flow, the Ricci flow and the Yamabe flow. I will discuss the classification result in the Ricci flow, construction result of infinitely many ancient solutions in the Yamabe flow. In the last part of the talk I will mention the most recent result about the unique asymptotics of non-collapsed ancient solutions to the mean curvature flow which is a joint work with Daskalopoulos and Angenent.