Counterexamples to a conjecture on matching Kneser graphs by Moharram N. Iradmusa

“…In this note, we show that the conjecture is not true for snarks.…”
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Gopakumar-Vafa invariants and the Emergent String Conjecture by Tom Rudelius

“…Abstract The Emergent String Conjecture of Lee, Lerche, and Weigand holds that every infinite-distance limit in the moduli space of a quantum gravity represents either a decompactification limit or an emergent string limit in some duality frame. …”
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An extension to the planar Markus–Yamabe Jacobian conjecture by Marco Sabatini

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Nonspecial varieties and generalised Lang–Vojta conjectures by Erwan Rousseau, Amos Turchet, Julie Tzu-Yueh Wang

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The conjectures of Artin-Tate and Birch-Swinnerton-Dyer by S. Lichtenbaum, N. Ramachandran, T. Suzuki

“…We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of the generic fibre. …”
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An Improved Upper Bound for the Erdős–Szekeres Conjecture by Mojarrad, Hossein Nassajian, Vlachos, Georgios

“…In 1961, they obtained the lower bound 2n−2+1≤ES(n), which they conjectured to be optimal. In this paper, we prove that ES(n)≤(2n−5n−2)−(2n−8n−3+2)≈716(2n−4n−2).…”
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Finiteness of K3 surfaces and the Tate conjecture by Lieblich, Max, Maulik, Davesh, Snowden, Andrew WIlson

Subjects: “…Tate conjecture, twisted sheaves, K3 surfaces, Fourier-Mukai equivalence…”
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Some remarks concerning Voevodsky’s nilpotence conjecture by Bernardara, Marcello, Marcolli, Matilde, Trigo Neri Tabuada, Goncalo Jorge

“…In this article we extend Voevodsky’s nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. …”
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A categorical approach to the stable center conjecture by Kazhdan, David, Bezrukavnikov, Roman, Varshavsky, Yaakov

“…Let G be a connected reductive group over a local non-archimedean field F. The stable center conjecture provides an intrinsic decomposition of the set of equivalence classes of smooth irreducible representations of G(F), which is only slightly coarser than the conjectural decomposition into L-packets. …”
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Proof of a conjecture of Bergeron, Ceballos and Labbé by Postnikov, Alexander, Grinberg, Darij

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Arnold‐Thom Gradient Conjecture for the Arrival Time by Colding, Tobias, Minicozzi, William

“…If so, this would explain various conjectured phenomena.…”
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Goldfeld's conjecture and congruences between heegner points by Kriz, D, Li, C

“…We show that this conjecture holds whenever E has a rational 3-isogeny. …”
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LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN by Maulik, Davesh, Neguţ, Andrei

“…We prove a weak version of this conjecture when is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. …”
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The orthogonal vectors conjecture for branching programs and formulas by Kane, Daniel, Williams, Richard Ryan

“…In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among n Boolean vectors in d dimensions. The OV Conjecture (OVC) posits that OV requires n2−o(1) time to solve, for all d = ω(log n). …”
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An Improved Proof of the Rabin-Harmanis-Stearns Conjecture by Perry, Harold M.

“…We offer an improved presentation of Aanderaa's constructive proof of the Rabin-Hartmanis-Stearns conjecture: For all k≥2, there exists a language Lk such that Lk can be recoginzed by a k-worktape real time Turing machine but cannot be recognized by any (k-1)-worktape real time Turing machine.…”
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An introduction to the algebraic geometry of the Putman–Wieland conjecture by Landesman, Aaron, Litt, Daniel

“…We also explain how a hyperelliptic analogue of the Putman–Wieland conjecture fails, following work of Marković.…”
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The Sato-Tate conjecture on average for small angles by Baier, Stephan, Zhao, Liangyi

“…We obtain average results on the Sato-Tate conjecture for elliptic curves for small angles.…”
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A conjecture on product decompositions in simple groups by Liebeck, M, Nikolov, N, Shalev, A

“…We propose a conjecture concerning decompositions of finite simple groups as products of conjugate subgroups, and prove it for a large class of maximal subgroups. © European Mathematical Society.…”

The Surface Group Conjectures for groups with two generators by Gardam, G, Kielak, D, Logan, AD

“…The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. …”

Proof of the De Concini-Kac-Procesi conjecture by Kremnizer, K

“…In this paper we prove a conjecture by De Concini, Kac and Procesi \cite{CP} (Corollary \ref{conj}): The dimension of any $M\in U_q-\mood^\chi$ is divisible by $l^{codim_\mathcal{B}\mathcal{B}_\chi}$.…”