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The fields of values of characters of degree not divisible by p Forum Math. Pi (IF 3.857) Pub Date : 2021-02-15 Gabriel Navarro; Pham Huu Tiep
We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where $p=2$, we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character
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Brill-Noether theory for curves of a fixed gonality Forum Math. Pi (IF 3.857) Pub Date : 2021-01-08 David Jensen; Dhruv Ranganathan
We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties
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Bounds for sets with no polynomial progressions Forum Math. Pi (IF 3.857) Pub Date : 2021-01-05 Sarah Peluse
Let $P_1,\dots ,P_m\in \mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of $\{1,\dots ,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots ,x+P_m(y)$ has size $|A|\ll N/(\log \log {N})^{c_{P_1,\dots ,P_m}}$. Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.
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Proof of a conjecture of Galvin Forum Math. Pi (IF 3.857) Pub Date : 2020-12-21 Dilip Raghavan; Stevo Todorcevic
We prove that if the set of unordered pairs of real numbers is coloured by finitely many colours, there is a set of reals homeomorphic to the rationals whose pairs have at most two colours. Our proof uses large cardinals and verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.
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Modules over algebraic cobordism Forum Math. Pi (IF 3.857) Pub Date : 2020-12-17 Elden Elmanto; Marc Hoyois; Adeel A. Khan; Vladimir Sosnilo; Maria Yakerson
We prove that the $\infty $-category of $\mathrm{MGL} $-modules over any scheme is equivalent to the $\infty $-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbf{P} ^1$-loop spaces, we deduce that very effective $\mathrm{MGL} $-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along
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GLOBAL NEARLY-PLANE-SYMMETRIC SOLUTIONS TO THE MEMBRANE EQUATION Forum Math. Pi (IF 3.857) Pub Date : 2020-08-05 LEONARDO ABBRESCIA; WILLIE WAI YEUNG WONG
We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the
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ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS Forum Math. Pi (IF 3.857) Pub Date : 2020-05-28 GEORGE LUSZTIG; ZHIWEI YUN
For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with
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BRANCH GROUPS, ORBIT GROWTH, AND SUBGROUP GROWTH TYPES FOR PRO- $p$ GROUPS Forum Math. Pi (IF 3.857) Pub Date : 2020-05-26 YIFTACH BARNEA; JAN-CHRISTOPH SCHLAGE-PUCHTA
In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro- $p$ group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between $n^{(\log n)^{2}}$ and $e^{n}$ . Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition
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HALF-SPACE MACDONALD PROCESSES Forum Math. Pi (IF 3.857) Pub Date : 2020-05-26 GUILLAUME BARRAQUAND; ALEXEI BORODIN; IVAN CORWIN
Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space
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-ADIC -FUNCTIONS FOR UNITARY GROUPS Forum Math. Pi (IF 3.857) Pub Date : 2020-05-06 ELLEN EISCHEN; MICHAEL HARRIS; JIANSHU LI; CHRISTOPHER SKINNER
This paper completes the construction of $p$ -adic $L$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $p$ -adic $L$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$ -adic $L$ -functions (Part I). Building on more recent
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THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE Forum Math. Pi (IF 3.857) Pub Date : 2020-04-20 HONG LIU; OLEG PIKHURKO; KATHERINE STADEN
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this
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ON THE COHOMOLOGY OF TORELLI GROUPS Forum Math. Pi (IF 3.857) Pub Date : 2020-04-13 ALEXANDER KUPERS; OSCAR RANDAL-WILLIAMS
We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $\#^{g}S^{n}\times S^{n}$ relative to a disc in a stable range, for $2n\geqslant 6$ . Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.
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THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE Forum Math. Pi (IF 3.857) Pub Date : 2020-04-06 BRAD RODGERS; TERENCE TAO
For each $t\in \mathbb{R}$ , we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $\unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp
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SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE Forum Math. Pi (IF 3.857) Pub Date : 2020-03-25 DANIEL LE; BAO V. LE HUNG; BRANDON LEVIN; STEFANO MORRA
We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$ . This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline
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ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES Forum Math. Pi (IF 3.857) Pub Date : 2020-03-23 VINCENT DELECROIX; ÉLISE GOUJARD; PETER ZOGRAF; ANTON ZORICH
A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an
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PRIMES REPRESENTED BY INCOMPLETE NORM FORMS Forum Math. Pi (IF 3.857) Pub Date : 2020-02-06 JAMES MAYNARD
Let $K=\mathbb{Q}(\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D714}$ the root of a degree $n$ monic irreducible polynomial $f\in \mathbb{Z}[X]$ . We show that the degree $n$ polynomial $N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$ in $n-k$ variables takes the expected asymptotic number of prime values if $n\geqslant 4k$ . In the special case $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$
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CHARACTER LEVELS AND CHARACTER BOUNDS Forum Math. Pi (IF 3.857) Pub Date : 2020-01-24 ROBERT M. GURALNICK; MICHAEL LARSEN; PHAM HUU TIEP
We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and
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MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS Forum Math. Pi (IF 3.857) Pub Date : 2020-01-20 ADAM J. HARPER
We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ , where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$ . In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$ . In particular, we find that $\mathbb{E}|\sum