Topology
Quasi-polynomial algorithms for telling if a knot is trivial
Marc Lackenby announced a quasi-polynomial time algorithm to decide whether a given knot is the unknot! This is a big breakthrough. This question is known to be both in NP and in coNP. See this post, and updates there in the comment section.
Topology seminar, UC Davis, February 2021 and Oxford, March 2021 and today, May 20! 16:00 CET, in the Copenhagen-Jerusalem combinatorics seminar (see details at the end of the post).
Unknot recognition in quasi-polynomial time (The link is to the slides)
NP hardness for related questions regarding knots
This is a talk by Martin Tancer given at the Copenhagen-Jerusalem combinatorics seminar.
Title: The unbearable hardness of unknotting (link to the 2018 arXive paper)
Abstract:
During the talk, I will sketch a proof that deciding if a diagram of the unknot can be untangled using at most k Riedemeister moves (where k is part of the input) is NP-hard. (This is not the same as the unknot recognition but it reveals some difficulties.) Similar ideas can be also used for proving that several other similar invariants are NP-hard to recognize on links.
Joint work with A. de Mesmay, Y. Rieck and E. Sedgwick.
An approach toward exotic differentiable 4-dim spheres
Monumental work on Arnold’s conjecture
Arnold Conjecture and Morava K-theory by Mohammed Abouzaid, Andrew J. Blumberg
(I heard it from a retweet by Alex Kontorovich.)
Lower bound for the number of fixed points of a map is a central theme in topology (and other areas) with many great results. The Lefshetz fixed point theorem tells you that a certain signed sum of the fixed points is bounded from below . For symplectic manifold Arnold’s conjecture asserts that the number of fixed point is bounded below from the sun of Betti numbers. This have led and is connected to tremendous mathematics.
Abstract (part): We prove that the rank of the cohomology of a closed symplectic manifold with coefficients in a field of characteristic p is smaller than the number of periodic orbits of any non-degenerate Hamiltonian flow.
Graph theory
Large spanning graphs with odd degrees
Asaf Ferber and Michael Krivelevich solved an old conjecture in Graph theory. (In a 1994 paper by Yair Caro the problem is already referred to as part of the folklore of graph theory.)
Abstract: We prove that every graph G on n vertices with no isolated vertices contains an induced subgraph of size at least n/10000 with all degrees odd. This solves an old and well-known conjecture in graph theory.
Here is an article about it in Quanta Magazine. And here is a Quanta Magazine article on the solution of the Erdos-Hajnal conjecture for pentagon-free graphs (that I mentioned in this post).
The Betti numbers of the independence complex of a ternary graph
A ternary graph (aka Trinity graph) has no induced cycles of length divisible by three. Chudnovsky, Scott, Seymour and Spirkl proved that for any ternary graph G, the number of independent sets with even cardinality and the independent sets with odd cardinality differ by at most 1. Hehui Wu and Wentao Zhang proved the stronger conjecture that for ternary graphs, the sum of Betti numbers of the independent complex is at most one. (Both conjectures were proposed by Roy Meshulam and me.) Congratulations to Hehui and Wentao and all other people mentioned in this post.
For related advances see Alexander Engstrom, On the topological Kalai-Meshulam conjecture; Jinha Kim, The homotopy type of the independence complex of graphs with no induced cycle of length divisible by three
Here is a very nice survey by Paul Seymour and Alex Scott on chi-boundedness. Here is an earlier post on Trinity graphs and chi-boundedness.
A breakthrough in additive combinatorics
David Conlon, Jacob Fox and Huy Tuan Pham used a new unified approach to solve a variety of open problems around subset sums including the open problems of Burr and Erdős on Ramsey complete sequences (Erdős offered $350 for their solutions) and a problem of Alon and Erdős on estimating the minimum number 
The paper is: Subset sums, completeness and colorings,
P-adic local Langlands
Test Your Intuition: p-adic local Langlands edition!
This post over Persiflage will test your intuition about 2-dimensional crystalline deformation rings. Since I couldn’t understand even the first sentence, I decided to ask my FB and HUJI friend to explain it to me, so I may try to write about it later. Actually, a better idea is that Persiflage will try to give a vastly non-technical explanation from scratch on p-adic local Langlands conjecture, and where it stands and why it matters. (To be clear, I am sure it is a great staff! And perhaps it is not possible to explain it to a wide audience. But it worth a try.)
Meanwhile it reminded me of the following story to which I apparently was a witness.
At the time of my birth, the medical approach was that mother and child needed to stay at the hospital for at least four nights and even for an entire whole week. After two nights at the hospital, my mother decided that she was ready to go home and got into a lengthy discussion with her doctor about it. At some point the exhausted doctor told her “Madam, I studied seven years in medical school, and I know what is good for you.” My mother was not impressed and replied: “Had I studied seven years in medical school, I would have been a medical doctor as well. And now, let me go home.” At the end, my mother (and myself) left the hospital on the third day. Years later this became the medical standard. (My own approach is to follow medical doctor’s instructions.)
Mathematics over the media
Ramanujan machine: machine learning for producing mathematical conjectures
This is a cool and very interesting direction by very very young Israeli mathematicians that was reported all over the media. (See, here for a Nature article.)
(There are various groups all over the world attempting to use machine learning to produce mathematical conjectures.)
A new approach for solving quadratic equations
I heard it (I think it was a year ago) from a Facebook post written by the prime minister of Singapore! And then I saw it in other places all over the media (here on the NYT). I like it, and, in fact, I like the many things Po-Shen Loh (who is my grand academic nephew) does in the service of mathematics.
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Copenhagen-Jerusalem Combinatorics Seminar
It is my pleasure to invite you to the following talk:
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Thursday, May 20 Time: 16:00-18:00 CET, Jerusalem +1
Location: Zoom: https://ucph-ku.zoom.us/j/69204766431
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Marc Lackenby: Unknot recognition in quasi-polynomial time
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Abstract:
I will outline a new algorithm for unknot recognition that runs in quasi-polynomial time. The input is a diagram of a knot with n crossings, and the running time is 2^{O((log n)^3)}. The algorithm uses a wide variety of tools from 3-manifold theory, including normal surfaces, hierarchies and Heegaard splittings. In my talk, I will explain this background theory, as well as explain how it fits into the algorithm.