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1. Klt varieties with conjecturally minimal volume

   https://doi.org/10.1093/imrn/rnad047  

   Totaro, Burt (March 2023, International Mathematics Research Notices)

   We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anti-canonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or alpha-invariant) exactly; it is extremely large, roughly 2^{2^n} in dimension n. These examples give improved lower bounds in Birkar’s theorem on boundedness of complements for Fano varieties. 

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2. The Kodaira dimension of complex hyperbolic manifolds with cusps

   https://doi.org/10.1112/S0010437X1700762X  

   Bakker, Benjamin; Tsimerman, Jacob (March 2018, Compositio Mathematica)

   We prove a bound relating the volume of a curve near a cusp in a complex ball quotient $$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$$ to its multiplicity at the cusp. There are a number of consequences: we show that for an $$n$$ -dimensional toroidal compactification $$\overline{X}$$ with boundary $$D$$ , $$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$$ is ample for $$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$$ , and in particular that $$K_{\overline{X}}$$ is ample for $$n\geqslant 6$$ . By an independent algebraic argument, we prove that every ball quotient of dimension $$n\geqslant 4$$ is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture. 

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3. On boundedness of singularities and minimal log discrepancies of Kollár components

   https://doi.org/10.1090/jag/822  

   Zhuang, Ziquan (January 2024, Journal of Algebraic Geometry)

   Recent study in K-stability suggests that Kawamata log terminal (klt) singularities whose local volumes are bounded away from zero should be bounded up to special degeneration. We show that this is true in dimension three, or when the minimal log discrepancies of Kollár components are bounded from above. We conjecture that the minimal log discrepancies of Kollár components are always bounded from above, and verify it in dimension three when the local volumes are bounded away from zero. We also answer a question from Han, Liu, and Qi on the relation between log canonical thresholds and local volumes. 

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4. A deterministic gradient-based approach to avoid saddle points

   https://doi.org/10.1017/S0956792522000316  

   Kreusser, L. M.; Osher, S. J.; Wang, B. (December 2022, European Journal of Applied Mathematics)

   Abstract Loss functions with a large number of saddle points are one of the major obstacles for training modern machine learning (ML) models efficiently. First-order methods such as gradient descent (GD) are usually the methods of choice for training ML models. However, these methods converge to saddle points for certain choices of initial guesses. In this paper, we propose a modification of the recently proposed Laplacian smoothing gradient descent (LSGD) [Osher et al., arXiv:1806.06317 ], called modified LSGD (mLSGD), and demonstrate its potential to avoid saddle points without sacrificing the convergence rate. Our analysis is based on the attraction region, formed by all starting points for which the considered numerical scheme converges to a saddle point. We investigate the attraction region’s dimension both analytically and numerically. For a canonical class of quadratic functions, we show that the dimension of the attraction region for mLSGD is $$\lfloor (n-1)/2\rfloor$$ , and hence it is significantly smaller than that of GD whose dimension is $n-1$ . 

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5. Improved Lower Bounds for 3-Query Matching Vector Codes

   https://doi.org/10.4230/LIPIcs.ITCS.2025.2  

   Aggarwal, Divesh; Dutta, Pranjal; Li, Zeyong; Obremski, Maciej; Saraogi, Sidhant (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   A Matching Vector (MV) family modulo a positive integer m ≥ 2 is a pair of ordered lists U = (u_1, ⋯, u_K) and V = (v_1, ⋯, v_K) where u_i, v_j ∈ ℤ_m^n with the following property: for any i ∈ [K], the inner product ⟨u_i, v_i⟩ = 0 mod m, and for any i ≠ j, ⟨u_i, v_j⟩ ≠ 0 mod m. An MV family is called r-restricted if inner products ⟨u_i, v_j⟩, for all i,j, take at most r different values. The r-restricted MV families are extremely important since the only known construction of constant-query subexponential locally decodable codes (LDCs) are based on them. Such LDCs constructed via matching vector families are called matching vector codes. Let MV(m,n) (respectively MV(m, n, r)) denote the largest K such that there exists an MV family (respectively r-restricted MV family) of size K in ℤ_m^n. Such a MV family can be transformed in a black-box manner to a good r-query locally decodable code taking messages of length K to codewords of length N = m^n. For small prime m, an almost tight bound MV(m,n) ≤ O(m^{n/2}) was first shown by Dvir, Gopalan, Yekhanin (FOCS'10, SICOMP'11), while for general m, the same paper established an upper bound of O(m^{n-1+o_m(1)}), with o_m(1) denoting a function that goes to zero when m grows. For any arbitrary constant r ≥ 3 and composite m, the best upper bound till date on MV(m,n,r) is O(m^{n/2}), is due to Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14).In a breakthrough work, Alrabiah, Guruswami, Kothari and Manohar (STOC'23) implicitly improve this bound for 3-restricted families to MV(m, n, 3) ≤ O(m^{n/3}). In this work, we present an upper bound for r = 3 where MV(m,n,3) ≤ m^{n/6 +O(log n)}, and as a result, any 3-query matching vector code must have codeword length of N ≥ K^{6-o(1)}. 

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