Let me summarize the result quickly for you if you don’t want to read the other blog post. Almost a hundred years ago Jensen and Pólya proved that the Riemannian hypothesis is equivalent to the statement that a certain infinite family of polynomials \(J_{n,d}\) has only real roots.

In February of this year a quite short paper was posted on the arXiv:1902.07321 ( meanwhile the paper is published in PNAS ) proving that for each \(d\) the polynomials \(J_{n,d}\) have only real roots for almost all \(n\).

]]>There is an intimate relation between the Laplace operator on X and the fundamental group of M. One example of this is the result of Brooks from 1981: the fundamental group of M is amenable if and only if the Laplace operator on X has a spectral gap around 0.

There are further relations between harmonic functions on X and the fundamental group G of M. The first one we want to discuss is the Liouville property:

- Lyons and Sullivan showed 1984 that if G is non-amenable, then there exist non-constant bounded harmonic functions on X.
- Kaimanovich on the other hand proved in 1986 that if G has sub-exponential growth or is polycyclic, then any bounded harmonic function on X is constant.
- In the remaining case, i.e., that G is an amenable group of exponential growth (and is not polycyclic), it is only known by a 2004 result of Erschler that there exists a closed manifold M with solvable fundamental group G and such that the universal cover of it admits a non-constant bounded harmonic function.

The strong Liouville property asks about the existence of non-constant positive harmonic functions on the universal cover:

- Lyons and Sullivan showed (in the same paper from 1984) that if G has polynomial growth, then any positive harmonic function on M is constant.
- In an arXiv preprint from yesterday Polymerakis showed that if G has exponential growth, then there are non-constant positive harmonic functions on the universal cover X.
- The remaining unsolved case is now the one of G having intermediate growth. But note that though there are some examples of groups with intermediate growth, none of these examples is finitely presentable. Hence it is currently unknown whether the remaining case here actually occurs (since fundamental groups of compact manifolds are finitely presentable).

Note that this week there was actually a school on harmonic maps: link.

]]>As I expected, the corresponding arXiv article ( arXiv:1708.06607 ) has already undergone several iterations (even claiming in the second one a proof of the Lindelöf hypothesis) and does by now not claim major progress on the Lindelöf problem, but only a novel approach which might become useful. See also the short discussion at MathOverflow: link.

Why am I writing about this here? There are many “proofs” of, e.g., the Riemannian hypothesis posted regularly and there is no need in general to talk about them. In this case, I enjoyed a bit reading the above cited news article and I did not know about the Lindelöf hypothesis – hence I wanted to share this with you.

]]>There was a meta-study published in Nature Communications (doi:10.1038/s41467-018-06292-0) about the performance of girls and boys in STEM fields. I learned about it from here.

To summarize a bit the results of the meta-study, let me quote a few sentences from its abstract:

*According to the ‘variability hypothesis’, this over-representation of males [persuing careers in STEM] is driven by gender differences in variance; greater male variability leads to greater numbers of men who exceed the performance threshold.*- Their meta study confirms the greater variability (but less average performance) of men:
*In line with previous studies we find strong evidence for lower variation among girls than boys, and of higher average grades for girls.* - But the variability hypothesis is disproven:
*However, the gender differences in both mean and variance of grades are smaller in STEM than non-STEM subjects, suggesting that greater variability is insufficient to explain male over-representation in STEM.*

I was a bit surprised by this, since I thought that such examples should be already known. Apparently not …

One reason why constructing such examples is hard, is the following result of Deligne-Sullivan from the late 70s: every compact hyperbolic manifold is virtually stably parallelizable, i.e., every compact hyperbolic manifold admits a finite-sheeted covering whose tangent bundle becomes trivial after taking the direct sum with a trivial bundle.

Note that being stably parallelizable implies that all Stiefel-Whitney classes vanish, which is much stronger than being spin (which just needs the first two Stiefel-Whitney classes to vanish). So especially, every compact hyperbolic manifold is virtually spinnable.

]]>Let me start with a picture of the speaker of the SPP, Bernhard Hanke, explaining at the beginning of the conference the next steps leading to the second funding period of the SPP. (I unfortunately forgot to take a picture of Carsten Balleier from the DFG saying a few words at the beginning.)

Next a few pictures from some of the plenary talks.

A picture from a coffee break at the Schloßgarten Cafe.

The next is a picture of Grigori Avramidi quoting how Thurston explained to Sullivan what a horocycle is (you can read this story in the December 2015 issue of the Notices of the AMS – the Thurston memorial issue – on page 1329, link-to-pdf).

And last, two pictures from the Conference Dinner.

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Though probably for many too long to read in detail, reading just the introductions is already interesting.

]]>Recently, there was a preprint posted on HAL (link) in which the authors provide an algorithm which runs in \(O(n\log(n))\)-time.

A nice article about this discovery may be found at the QuantaMagazine: link.

Further, it was also recently proven in another preprint (link) that \(O(n\log(n))\)-time is conditionally (i.e., if a certain conjecture in network coding is true) the best possible.

Update (May 7th): the results in the last linked preprint (link) are extremely strong and hence one should, at least for the time being, take it with a grain of salt (further information: link).

]]>In March 14th, 2019, the **L’Oréal-UNESCO For Women in Science International Awards** (homepage, UNESCO webpage, wikipedia) were presented to five women. One of the five laureates is

The other one is the **Ars legendi-Fakultätenpreis Mathematik** (webpage), which will be awarded to **Robert Rockenfeller** (homepage). Actually, since I’m quite interested in the art of teaching mathematics, I invited him next summer to Regensburg to our department colloquium to talk about what he did which earned him this prize; and I would actually recommend others to do the same – at least last year I was positively astonished by the talk (and what he did teaching-wise) by last-years laureate.