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1. PL-Genus of surfaces in homology balls

   https://doi.org/10.1017/fms.2023.126  

   Hom, Jennifer; Stoffregen, Matthew; Zhou, Hugo (January 2024, Forum of Mathematics, Sigma)

   Abstract We consider manifold-knot pairs$$(Y,K)$$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$$\Sigma $$in a homology ballX, such that$$\partial (X, \Sigma ) = (Y, K)$$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$$(Y, K)$$to any knot in$$S^3$$can be arbitrarily large. The proof relies on Heegaard Floer homology. 

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2. Cusps, Kleinian groups, and Eisenstein series

   https://doi.org/10.1017/fms.2023.73  

   Liu, Beibei; Wang, Shi (January 2023, Forum of Mathematics, Sigma)

   Abstract We study the Eisenstein series associated to the full rank cusps in a complete hyperbolic manifold. We show that given a Kleinian group$$\Gamma <{\operatorname{\mathrm{Isom}}}^+(\mathbb H^{n+1})$$, each full rank cusp corresponds to a cohomology class in$$H^{n}(\Gamma , V)$$, whereVis either the trivial coefficient or the adjoint representation. Moreover, by computing the intertwining operator, we show that different cusps give rise to linearly independent classes. 

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3. Higher Siegel–Weil formula for unitary groups: the non-singular terms

   https://doi.org/10.1007/s00222-023-01228-y  

   Feng, Tony; Yun, Zhiwei; Zhang, Wei (February 2024, Inventiones mathematicae)

   Abstract We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the$$r^{\mathrm{th}}$$ $r^{\mathrm{th}}$central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with$$r$$ $r$legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series. 

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4. Eisenstein cocycles in motivic cohomology

   https://doi.org/10.1112/S0010437X24007322  

   Sharifi, Romyar; Venkatesh, Akshay (October 2024, Compositio Mathematica)

   Several authors have studied homomorphisms from first homology groups of modular curves to$$K_2(X)$$, with$$X$$either a cyclotomic ring or a modular curve. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a$$1$$-cocycle from$$\mathrm {GL}_2(\mathbb {Z})$$to the second$$K$$-group of the function field of a suitable group scheme over$$X$$, from which the maps of interest arise by specialization. 

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5. A raising operator formula for Macdonald polynomials

   https://doi.org/10.1017/fms.2025.8  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G H (January 2025, Forum of Mathematics, Sigma)

   Abstract We give an explicit raising operator formula for the modified Macdonald polynomials$$\tilde {H}_{\mu }(X;q,t)$$, which follows from our recent formula for$$\nabla $$on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions$$\tilde {H}^{1,n}(X;q,t)$$that we call$$1,n$$-Macdonald polynomials, which reduce to a scalar multiple of$$\tilde {H}_{\mu }(X;q,t)$$when$$n=1$$. We conjecture that the coefficients of$$1,n$$-Macdonald polynomials in terms of Schur functions belong to$${\mathbb N}[q,t]$$, generalizing Macdonald positivity. 

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