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1. On a problem of M. Talagrand

   https://doi.org/10.1002/rsa.21077  

   Frankston, Keith; Kahn, Jeff; Park, Jinyoung (December 2022, Random Structures & Algorithms)

   Abstract We address a special case of a conjecture of M. Talagrand relating two notions of “threshold” for an increasing family of subsets of a finite setV. The full conjecture implies equivalence of the “Fractional Expectation‐Threshold Conjecture,” due to Talagrand and recently proved by the authors and B. Narayanan, and the (stronger) “Expectation‐Threshold Conjecture” of the second author and G. Kalai. The conjecture under discussion here says there is a fixedLsuch that if, for a given , admits withand(a.k.a.  isweakly p‐small), then admits such a taking values in ( is‐small). Talagrand showed this when is supported on singletons and suggested, as a more challenging test case, proving it when is supported on pairs. The present work provides such a proof. 

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2. Big polynomial rings and Stillman's Conjecture

   https://doi.org/10.1007/s00222-019-00889-y  

   Erman, Daniel; Sam, Steven V; Snowden, Andrew (May 2019, Inventiones mathematicae)

   The purpose of this paper is to prove that certain limits of polynomial rings are themselves polynomial rings, and show how this observation can be used to deduce some interesting results in commutative algebra. In particular, we give two new proofs of Stillman's conjecture. The first is similar to that of Ananyan-Hochster, though more streamlined; in particular, it establishes the existence of small subalgebras. The second proof is completely different, and relies on a recent noetherianity result of Draisma. 

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3. Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes

   https://doi.org/10.1007/s00208-023-02597-8  

   Mehlhop, Nathan; Słomian, Wojciech (March 2024, Mathematische Annalen)

   Abstract We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subsets of primes. This is a contribution to the Rosenblatt–Wierdl conjecture (Lond Math Soc Lect Notes 205:3–151, 1995, Problem 4.12, p. 80) with averages taken over primes. These inequalities provide endpoints for ther-variational estimates obtained by Trojan (Math Ann 374:1597–1656, 2019). 

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4. Bounds on the dimension of lineal extensions

   https://doi.org/10.4171/JFG/161  

   Bushling, Ryan_E G; Fiedler, Jacob B (March 2025, Journal of Fractal Geometry, Mathematics of Fractals and Related Topics)

   LetE \subseteq \mathbb{R}^{n}be a union of line segments andF \subseteq \mathbb{R}^{n}the set obtained fromEby extending each line segment inEto a full line. Keleti’sline segment extension conjectureposits that the Hausdorff dimension ofFshould equal that ofE. Working in\mathbb{R}^{2}, we use effective methods to prove a strong packing dimension variant of this conjecture. Furthermore, a key inequality in this proof readily entails the planar case of the generalized Kakeya conjecture for packing dimension. This is followed by several doubling estimates in higher dimensions and connections to related problems. 

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5. Low-Depth Algebraic Circuit Lower Bounds over Any Field

   https://doi.org/10.4230/LIPIcs.CCC.2024.31  

   Forbes, Michael A (July 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   The recent breakthrough of Limaye, Srinivasan and Tavenas [Limaye et al., 2022] (LST) gave the first super-polynomial lower bounds against low-depth algebraic circuits, for any field of zero (or sufficiently large) characteristic. It was an open question to extend this result to small-characteristic ([Limaye et al., 2022; Govindasamy et al., 2022; Fournier et al., 2023]), which in particular is relevant for an approach to prove superpolynomial AC⁰[p]-Frege lower bounds ([Govindasamy et al., 2022]). In this work, we prove super-polynomial algebraic circuit lower bounds against low-depth algebraic circuits over any field, with the same parameters as LST (or even matching the improved parameters of Bhargav, Dutta, and Saxena [Bhargav et al., 2022]). We give two proofs. The first is logical, showing that even though the proof of LST naively fails in small characteristic, the proof is sufficiently algebraic that generic transfer results imply the result over characteristic zero implies the result over all fields. Motivated by this indirect proof, we then proceed to give a second constructive proof, replacing the field-dependent set-multilinearization result of LST with a set-multilinearization that works over any field, by using the Binet-Minc identity [Minc, 1979]. 

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