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Trying to calculate , I’ve proved (not yet quite thoroughly checked for errors) the following partial result:

**Proposition** for some proper filters , , .

Currently the proof is located in this file.

I have proved that join of two connected (regarding a funcoid) filters, whose meet is proper, is connected. (I remind that in my texts filters are ordered *reverse* set-theoretic inclusion.)

The not so complex proof is available in the file addons.pdf. (I am going to move it to the book in the future.)

I added more on connectedness of filters to the file addons.pdf (to be integrated into the book later).

It is *a rough incomplete draft*. Particularly the proof, that the join of two connected filters with proper meet is connected, is not complete. (Remember that I order filters reversely to set-theoretic inclusion.)

This is now an important open problem to solve.

I have corrected some errors in my book about connectedness of funcoids and reloids.

In some theorems I replace like with and arbitrary paths with nonzero-length paths.

I also discovered (not yet available online) some new results about connected funcoids.

Please read my math research and decide if it is worth the prize. If you consider my research worth the prize, please nominate me.

Nominations for the Breakthrough Prize and New Horizons Prize in mathematics are now open.

The Breakthrough Prize in Mathematics is a $3,000,000 prize for transformative breakthrough(s) in mathematics, with special attention to recent developments. In selecting the prize winner we will pay particular attention to results from the last 10 years, although earlier contributions may also be taken into account. This is not intended as a lifetime achievement award, but rather it is intended to recognize someone currently making outstanding contributions. There are no age limits nor nationality restrictions. In exceptional circumstances, when a prize is awarded for joint work, the committee may split the prize. The recent winners were:

2015 Ian Agol

2016 Jean Bourgain

2017 Christopher Hacon and James McKernan jointly

There are up to 3 New Horizons Prizes in Mathematics of $100,000 each for promising young researchers, who have already produced very important work. Candidates should not have been awarded a PhD before January 1st, 2008. (This requirement can be waived in exceptional circumstances, such as an interrupted career path.) There are no nationality restrictions. In exceptional circumstances, when a prize is awarded for joint work, the committee may split the prize. Previously these prizes were offered to:

2015 Larry Guth, André Neves and Peter Scholze

2016 Mohammed Abouzaid, Hugo Duminil-Copin, and jointly Ben Elias and Geordie Williamson

2018 Aaron Naber, Maryna Viazovska and jointly Zhiwei Yun and Wei Zhang.

Nominations may be submitted online at

https://breakthroughprize.org/Nominations

A short statement is required from the nominator, along with between 1 and 3 letters of reference from other experts. The closing date for nominations is April 30th. If you have questions about the nomination process, queries can be addressed to Rob Meyer (meyer@breakthroughprize.org).

I have defined sides of a surface (represented by such things as a set in a topological space) purely topologically. I also gave two (possible non-equivalent) definitions of special points of a surface (such “singularities” as points of the border of a closed disk).

Currently these definitions and questions are presented in the file addons.pdf.

It is very interesting and intriguing. The theory is considered not in the framework of topological spaces, but in more general theory of funcoids.

“Micronization” was a thoroughly wrong idea with several errors in the proofs. This section is removed from the book.

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