For more information please visit: https://scgp.stonybrook.edu/archives/43116
Public holiday
Title: Murmurations of arithmetic L-functions
Speaker: Andrew Sutherland
Abstract: I will report on ongoing efforts to understand murmurations in families of arithmetic L-functions, including L-functions of modular forms, elliptic curves and their symmetric powers, and of abelian surfaces
Title: Murmurations of Maass forms
Speaker: Min Lee
Abstract: In this talk, I will present joint work with Andrew R. Booker, David Lowry-Duda, Andrei Seymour Howell, and Nina Zubrilina, exploring the murmurations of Maass forms. We prove the existence of such murmurations within families of L-functions associated with Maass forms of weight 0 and level 1, as their Laplace eigenvalue parameter tends to infinity.
Title: Yang–Mills theory from a topological theory
Speaker: Alberto Cattaneo
Abstract: Four-dimensional Yang–Mills theory can be obtained from a topological field theory of Schwarz type via a procedure known as BV pushforward, which amounts in integrating out some gauge-fixed fields. This turns out to be an equivalence in the sense that it establishes an isomorphism between the observables of the two theories, which in turns implies that their expectation values can be computed in either theory with the same outcome. I will give an introduction to the basic concept, including the BV pushforward, and discuss in details an easier case.
Title: Towards a stable homotopy refinement of Legendrian contact homology
Speaker: Robert Lipshitz [University of Oregon]
Abstract: The Chekanov-Eliashberg dga, or Legendrian contact homology, was the first modern invariant of Legendrian knots in R^3. This talk is a progress report on a project to give a stable homotopy refinement of Legendrian contact homology, inducing operations like Steenrod squares on linearized Legendrian contact homology. In general, contact homology is defined by counting J-holomorphic curves, but in this case those counts reduce to the Riemann Mapping Theorem and the invariant is described purely combinatorially, from an appropriate knot diagram, and our refinement has a similarly combinatorial flavor. After recalling the basics of Legendrian knot theory and Legendrian contact homology, we will outline our program to refine it, sketch its status, and perhaps describe some examples. This is joint with Lenhard Ng and Sucharit Sarkar.
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Title: Numerical Investigation of Lower Order Biases in Moment Expansions of One-Parameter Families of Elliptic Curves
Speaker: Steven Miller and Tomothy Cheek
Abstract: For a fixed elliptic curve $E$ without complex multiplication, $a_p := p+1 - \#E(\mathbb{F}_p)$ is $O(\sqrt{p})$ and $a_p/2\sqrt{p}$ converges to a semicircular distribution. Michel proved that for a one-parameter family of elliptic curves $y^2 = x^3 + A(T)x + B(T)$ with $A(T), B(T) \in \mathbb{Z}[T]$ and non-constant $j$-invariant, the second moment of $a_p(t)$ is $p^2 + O(p^{{3}/ {2}})$. The size and sign of the lower order terms have applications to the distribution of the zeros near the central point of the associated $L$-functions (i.e., the Birch and Swinnerton-Dyer Conjecture). Based on data from special families where the Legendre sum calculations can be done in closed form to compute the second moment, S. J. Miller conjectured that the highest order term of the lower order terms of the second moment that does not average to zero is on average negative; this is now known for many families where $A(T), B(T)$ have small degree, and in many cases interesting arithmetic emerges in these lower order terms. We create a database and a framework to quickly and systematically investigate biases in the second moment of \emph{any} one- parameter family, justifying the large start-up cost needed. When looking at families which have so far been beyond current theory, we find several potential violations of the conjecture for $p \leq 250,000$ and discuss new conjectures motivated by the data, especially for higher moments (which are not theoretically tractable as they involve at least cubic Legendre sums).
Title: Root Number Correlation Bias of Fourier Coefficients of Modular Forms
Speaker: Nina Zubrilina
Abstract: In a recent study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the p-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number. In my talk, I will discuss this root number correlation in families of holomorphic and Maass forms.
Title: Distributions of local signs and murmurations
Speaker: Kimball Martin
Abstract: Murmurations describe a correlation between Fourier coefficients and global root numbers of modular forms. I will describe analogous phenomena for local root numbers, including the degenerate case of no root numbers.
Title: Computing L-functions
Speaker: Edgar Costa
Abstract: We overview several methods to compute the Dirichlet coefficient of L-functions for several families, including Calabi-Yau varieties (curves to 3folds) and hypergeometric motives.
Title: Random Geometries from Supersymmetric Yang-Mills theory
Speaker: Nikita Nekrasov
Title: TBA
Speaker: Jonathan Bober
Abstract: TBA
Title: Random matrix theory and distributions of multiplicative functions
Speaker: Matilde Lalin
Abstract: We will discuss some connections between the variance of multiplicative functions (particularly the divisor function) for the function field $\mathbb{F}_q[T]$ and certain integrals over the ensembles of unitary and symplectic matrices. We will report on recent advances on the formulation of conjectures over the number field case for problems connected to symplectic matrices. This is joint work with Vivian Kuperberg.
Title: Murmurations of Hecke L-functions of imaginary quadratic fields
Speaker: Zeyu (Steven) Wang
Abstract: We compute the murmuration density for the family of Hecke L-functions associated to non-trivial Hecke characters on the class group of imaginary quadratic fields with 1-mod-4 discriminant. One interesting phenomenon about this family is that after averaging the Frobenius traces $a_p(f)$ over the family, the murmuration density has a mild dependence on the arithmetics of $p$, and a second average over primes is required. We also analyze the asymptotic behavior of the murmuration.
Title: Explicit point counting for Delsarte K3 quartic surface pencils
Speaker: Ursula Whitcher
Abstract: We study ten pencils of Delsarte K3 quartic surface pencils that arise naturally in the context of mirror symmetry. We use finite field hypergeometric equations and hypergeometric Picard-Fuchs differential equations to describe their point counts over finite fields, together with associated L-functions and modular forms, and explore the implications for related families of Calabi-Yau threefolds.
Title: Tangent cones of Kahler-Ricci flow singularity models
Speaker: Max Hallgren [Rutgers University]
Abstract: Abstract: By the compactness theory of Bamler, any finite time singularity of the Kahler-Ricci flow is modeled on a singular Kahler-Ricci soliton, and such solitons are infinitesimally metric cones. In this talk, we will see that these cones are normal affine algebraic varieties, using a new method for proving Hormander-type L^2 estimates on singular shrinking solitons.
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Title: Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves
Speaker: Lin Weng
Abstract: For elliptic curves over rationals, there are a well-known conjecture of Sato-Tate and a new computational guided murmuration phenomenon, for which the abelian Hasse-Weil zeta functions are used. In this talk, we show that both the murmurations and the Sato-Tate conjecture stand equally well for non-abelian high rank zeta functions of elliptic curves over rationals. We establish our results by carefully examining the asymptotic behaviors of the $p$-reduction invariants $a_{E/\mathbb F_q,n}\ (n\geq 1)$, the rank $n$ analogous of the rank one $a$-invariant $a_{E/\mathbb F_q}=1+q-N_{E/\mathbb F_q}$ of elliptic curve $E/\mathbb F_{q}$. Such asymptotic results are based on the counting miracle of the so-called $\alpha_{E/\mathbb F_q,n}$- and $\beta_{E/\mathbb F_q,n}$-invariants of $E/\mathbb F_q$ in rank $n$, and a remarkable recursive relation on the $\beta_{E/\mathbb F_q,n}$-invariants, established by Weng and Zagier, previously. This is a joint work with Zhan Shi.
Title: Murmurations and trace formulas
Speaker: David Lowry-Duda
Abstract: In work with Bober, Booker, Lee, Seymour-Howell, and Zubrilina, we proved murmuration behavior for Maass forms in the eigenvalue aspect and for modular forms in the weight aspect. For both, we used an approach based on the Selberg trace formula. But different trace formulas, such as the Kuznetsov or Petersson trace, offer weighted variants. In this talk we examine murmuration behavior from the perspective of different trace formulas.
Title: Regularity Structures
Speaker: Ilya Chevyrev
Title: Improving Mestre-Nagao heuristics with data science
Speaker: Matija Kazalicki
Abstract: The Mestre-Nagao sums, based on the Birch and Swinnerton-Dyer conjecture, offer heuristics for identifying elliptic curves of high rank. In this talk, we present data science experiments—some drawing inspiration from murmurations of elliptic curves—aimed at improving the performance of these sums and developing new heuristics. Our focus is on both classifying high-rank curves and distinguishing between curves of rank 0 and 1. This is a joint work with Zvonimir Bujanović, Lukas Novak, and Domagoj Vlah.
Title: Einstein gravity from matrix integral
Abstract: We formulate and test holography between a supersymmetric mass deformation of the Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) matrix model and type IIB supergravity backgrounds with exceptional F4 supersymmetry. This is arguably the simplest example of holography in which the dual description contains Einstein gravity. We conjecture a one-to-one correspondence between saddle points of matrix integral and supergravity backgrounds, and test it through the supersymmetric localization of the matrix integral.
Title: Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves
Speaker: Angelica Babei, Barinder S. Banwait and Xiaoyu Huang
Abstract: We describe machine learning classifiers to predict the order of the Shafarevich-Tate group of an elliptic curve over $\Q$, building on earlier work of He, Lee, and Oliver. We show that a feed-forward neural network trained on invariants arising in the Birch-Swinnerton-Dyer conjectural formula yields higher accuracies ($> 0.9$) than any model previously studied.
Title: Conway's "Life" perturbed
Speaker: Raimundas Vidunas
Abstract: This recreational talk will concern John Conway's famous game ``Life'', which is by far the best known cellular automaton. Its simple rules generate often unforeseeably complicated evolution of live and dead cells in a rectangular lattice. We will consider a perturbed variant of ``Life'' where errors (of following the rules) are possible with a small probability. The error probability at each step is assumed to be so small that the errors would only affect stabilized patterns, including cyclical or moving patterns. This gives a Markovian process between the stabilized patterns. Such a stochastic model should approximate emergence of complexity and live processes more interestingly than Conway's original game. Concrete results will be presented for this new game on small toruses, of size up to 10x10 cells.
Title: Ball quotients and moduli spaces
Speaker: Klaus Hulek [Leibniz U. Hannover]
Abstract: A number of moduli problems are, via Hodge theory, closely related to ball quotients. In this situation there is often a choice of possible compactifications such as the GIT compactification and its Kirwan blow-up or the Baily-Borel compactification and the toroidal compactification. The relationship between these compactifications is subtle and often geometrically interesting. In this talk I will discuss several cases, mostly concentrating on Deligne-Mostow varieties, but also briefly touching on cubic surfaces and cubic threefolds. This discussion links several areas such as birational geometry, moduli spaces of pointed curves, modular forms and derived geometry. This talk is based on joint work with S. Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.
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Title: Computing paramodular forms I
Speaker: Ariel Pacetti
Abstract: The goal of this talk is to explain how cohomological degree two Paramodular forms can be computed. We will start recalling the basic definitions and main properties of degree two Siegel modular forms, and explain the relation between paramodular forms and definite orthogonal modular forms on quinary lattices. The later forms are well suited for computations, as will be explained in Tornaria's talk.
Title: Computing paramodular forms II
Speaker: Gonzalo Tornaria
Abstract: The second part of the talk will be how to compute the quinary forms, and the Hecke operators acting on them (as in the tables he computed with Gustavo Rama).
Title: Regularity Structures
Speaker: Hao Shen
Title: Murmurations and ratios conjectures
Speaker: Alex Cowan
Abstract: Subject to GRH, we prove that murmurations arise for primitive quadratic Dirichlet characters, and for holomorphic modular forms of prime level tending to infinity with sign and weight fixed. Moreover, subject to ratios conjectures, we prove that murmurations arise for elliptic curves ordered by height, and for quadratic twists of a fixed elliptic curve. We demonstrate the existence of murmurations for these arithmetic families using results from random matrix theory.
Title: Machine Learning Cluster Algebras
Speaker: Edward Hirst
Abstract: Cluster Algebras have shown recent popularity due to their various applications within physics. They are defined via mutation process acting on a seed of generators, where physically this mutation connects equivalent gauge theories under Seiberg duality. The mutation process leads to rich combinatorics, here studied through the lens of machine learning, where classification methods learn to distinguish algebras, and network science methods promote the raising of new conjectures. For applicability of many cluster algebra theorems one is interested in an algebras mutation-acyclicity, a property we examine with machine learning and use interpretable methods to hint at the structure of new relevant mutation invariants.
Title: Recent progress in mean curvature flow
Speaker: Bruce Kleiner [NYU Courant]
Abstract: Abstract: An evolving surface is a mean curvature flow if the normal component of its velocity field is given by the mean curvature. First introduced in the physics literature in the 1950s, the mean curvature flow equation has been studied intensely by mathematicians since the 1970s with the aim of understanding singularity formation and developing a rigorous mathematical treatment of flow through singularities. I will discuss progress in the last few years which has led to the solution of several longstanding conjectures, including the Multiplicity One Conjecture.
This is based on joint work with Richard Bamler.
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Title: Paramodular forms from Calabi-Yau operators
Speaker: Nutsa Gegelia
Abstract: We report on the conjectural identification of paramodular forms from Calabi-Yau motives of Hodge type (1,1,1,1) of moderately low conductor. The identifications are done by calculating Euler factors from Calabi-Yau operators from the AESZ list, seeking a match with Hecke eigenvalues provided in the paramodular forms database and checking the approximate functional equation for the Euler product numerically.
Title: Murmurations of quadratic characters
Speaker: Alexey Pozdnyakov
Abstract: We calculate murmuration densities for several families of Dirichlet L-functions, focusing on the case of L-functions arising from quadratic characters. We also prove that the resulting murmuration function interpolates the phase transition in the 1-level density for a symplectic family of L-functions.
Title: TBA
Speaker: Willie Rush Lim [Brown University]
Abstract: TBA
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