The Measures with 𝐿²-Bounded Riesz Transform and the Painlevé Problem

Damian Dąbrowski X. Tolsa

Jul 2026 · Memoirs of the American Mathematical Society · 0 citations · 32 references

Abstract

In this work we provide a geometric characterization of the measures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with polynomial upper growth of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Riesz transform <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R mu left-parenthesis x right-parenthesis equals integral StartFraction x minus y Over StartAbsoluteValue x minus y EndAbsoluteValue Superscript n plus 1 Baseline EndFraction d mu left-parenthesis y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo> ∫ </mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mi>y</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mi>y</mml:mi> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}\mu (x) = \int \frac {x-y}{|x-y|^{n+1}}\,d\mu (y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belongs to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2(\mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . More precisely, it is shown that <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar script upper R mu double-vertical-bar Subscript upper L squared left-parenthesis mu right-parenthesis Superscript 2 Baseline plus double-vertical-bar mu double-vertical-bar almost-equals integral integral Subscript 0 Superscript normal infinity Baseline beta Subscript 2 comma mu Baseline left-parenthesis x comma r right-parenthesis squared StartFraction mu left-parenthesis upper B left-parenthesis x comma r right-parenthesis right-parenthesis Over r Superscript n Baseline EndFraction StartFraction d r Over r EndFraction d mu left-parenthesis x right-parenthesis plus double-vertical-bar mu double-vertical-bar comma"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mi> μ </mml:mi> <mml:msubsup> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi> μ </mml:mi> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mo> ≈

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F. Forstnerič, Á. Sigurðardóttir · Transactions of the American... · 1 citation

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Yanyan Han, Hongwei Huang, Jinghan Shao et al. · Proceedings of the American... · 0 citations

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Valentin Blomer, Jesse Thorner · Transactions of the American... · 0 citations

Open access Aug 2026

Effective bounds on characterising slopes for all knots

A slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p slash q"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p/q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is characterising for a knot <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K subset-of double-struck upper S cubed"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">K \subset \mathbb {S}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if the orientation-preserving homeomorphism type of the manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Subscript upper K Superscript 3 Baseline left-parenthesis p slash q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {S}^3_K(p/q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> obtained by performing Dehn surgery of slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p slash q"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p/q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> along <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> uniquely determines the knot <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We combine new applications of results from hyperbolic geometry with previous individual work of the authors to determine, for any given knot <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , an explicit bound <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue q EndAbsoluteValue greater-than script upper C left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">|q| > \mathcal {C}(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p slash q"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p/q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a characterising slope for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Furthermore, we find an optimal such <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper K right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi>

Patricia Sorya, L. Wakelin · Transactions of the American... · 0 citations

Open access Aug 2026

Integrality of GL₂×GL₂ Rankin-Selberg integrals for ramified representations

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 comma pi 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> π </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi _1,\pi _2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be irreducible admissible generic tempered representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper L Subscript 2 Baseline left-parenthesis upper F right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {GL}_2(F)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some finite extension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F slash bold upper Q Subscript p"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">F/\mathbf {Q}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of odd residue characteristic. Inspired by work of Loeffler and previous work of the author on unramified zeta-integrals, we introduce a natural general notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis pi 1 times pi 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> × </mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\pi _1\times \pi _2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - <italic>integral</italic> data at which the Rankin-Selberg zeta-integral can be evaluated. We then establish an integral refinement of Jacquet-Langland’s GCD-result for this zeta-integral, when evaluated at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis pi 1 times pi 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> × </mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\pi _1\times \pi _2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -integral data. This is compatible with the notion of integrality coming from the Fourier coefficients of newforms of even integral weights. Our approach relies on a reinterpretation of the Rankin-Selberg zeta-integral, and works of Assing and Saha on values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic Whittaker new vectors.

Unknown authors · Representation Theory · 0 citations

Open access Aug 2026

Conformally Invariant Fields Out of Brownian Loop Soups

Consider a Brownian loop soup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper D Superscript theta"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi> θ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathcal {L}_D^\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with subcritical intensity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta element-of left-parenthesis 0 comma 1 slash 2 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi> θ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\theta \in (0,1/2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in some 2D bounded simply connected domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We define and study the properties of a conformally invariant field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript theta"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi> θ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> naturally associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper D Superscript theta"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi> θ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathcal {L}_D^\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Informally, this field is a signed version of the local time of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript upper D Superscript theta"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi> θ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathcal {L}_D^\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the power <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 minus theta"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi> θ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1-\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta equals 1 slash 2"> <mml:semantics> <mml:mrow> <mml:mi> θ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\theta = 1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript theta"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi> θ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Gaussian free field (GFF) in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our construction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript theta"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mi> θ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">h_\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> relies on the multiplicative chaos <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M Subscript gamma"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mi>

A. Jego, Titus Lupu, Wei Qian · Memoirs of the American Math... · 0 citations