Recently I stumbled upon the following two conjectures about polynomials: The first one is the Jacobian conjecture: We have polynomials $$f_1, \ldots, f_n$$ in the variables $$x_1, \ldots, x_n$$ with coefficients in a field $$k$$ of non-zero characteristic. We define a function $$F\colon k^n \to k^n$$ by setting \[F(x_1, \ldots, x_n) := (f_1(x_1, \ldots, x_n), … Continue reading "Conjectures about polynomials"
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### Non-positive immersions

In a blog post about hyperbolicity of one-relator groups I mentioned the following property that a 2-dimensional complex $$X$$ might have: $$X$$ is said to have non-positive immersions if for every immersion of a finite, connected 2-complex $$Y$$ into $$X$$, we either have $$\chi(Y) \le 0$$ or $$Y$$ is contractible. For example, this property rules … Continue reading "Non-positive immersions"
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### Announcing a result on the arXiv

Today Rachel Greenfeld and Terence Tao announced via the arXiv that they can disprove the periodic tiling conjecture (arXiv:2209.08451). I do not want to discuss here in detail the contents of the conjecture or their approach to disprove it (if you are interested in this, you can read about it on Tao’s blog). What I … Continue reading "Announcing a result on the arXiv"
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### Predicting the future of arbitrary functions

Let $$S$$ be any non-empty set. If you have a function $$f\colon \mathbb{R} \to S$$ and you tell me its values on an interval $$(-\infty,t)$$, can I predict which value it will have at the time $$t$$? If the function is continuous, then of course I can; and in general not. Now interestingly, if you … Continue reading "Predicting the future of arbitrary functions"
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### Fields medalists 2022

Recently the Fields medalists 2022 (and all the other prizes given out by the IMU) were announced: official page. You can read about some of the achievements of these people on the blog of Gil Kalai (and of course also in the official laudationes).
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### Shaw Prize 2022

The Shaw Prize in Mathematical Sciences 2022 (link) is awarded to Noga Alon and to Ehud Hrushovski for their remarkable contributions to discrete mathematics and model theory with interaction notably with algebraic geometry, topology and computer sciences.
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### Finite subgroups of homeomorphism groups

If $$M$$ is a compact topological manifold, can one say something about its group of homeomorphisms $$\mathrm{Homeo}(M)$$? Of course one can, there is quite a lot to say about it, and there is also quite a lot which we still don’t know. Today I want to mention the following recent result by Csikós-Pyber-Szabó (arXiv:2204.13375): Let … Continue reading "Finite subgroups of homeomorphism groups"

### A quantitative coarse obstruction to psc-metrics

Recently, Guo and Yu pushed the following result to the arXiv (math.KT/2203.15003): For any $$R > 0$$ and positive integer $$m$$, there exists a constant $$k(R,m)$$ such that the following holds. If $$(M,g)$$ is a Riemannian spin manifold that admits a uniformly bounded, good open cover with Lebesgue number $$R$$ and $$R$$-multiplicity $$m$$, then \[\inf_{x \in M} \kappa_g(x) \le … Continue reading "A quantitative coarse obstruction to psc-metrics"
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