Previous Talks

Uri Shapira, Distribution of conditional directional lattices
(Technion – Israel Institute of Technology)

Thursday, April 18, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 19, 2024 (1am AEST, 3am NZST)

Abstract: Given an integral vector v in Euclidean n-space we project the standard lattice Z^n into the hyperplane orthogonal to v and obtain in this manner a "lattice of rank n-1" in that hyperplane, which is called "The directional lattice D(Z^n,v)". 

In this talk I will discuss results about the limit distribution of directional lattices as we let the vector v vary in some natural sets from a number theoretic point of view. These include, balls, spheres, non-compact quadratic surfaces, and integral vectors approximating an irrational line.


Yiannis Petridis, Counting and equidistribution
(University College London)

Thursday, April 11, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 12, 2024 (1am AEST, 3am NZST)

Abstract: I will discuss how counting orbits in hyperbolic spaces lead to interesting number theoretic problems. The counting problems (and the associated equidistribution) can be studied with various methods, and I will emphasize automorphic form techniques, originating in the work of H. Huber and studied extensively by A. Good. My collaborators in various aspects of this project are Chatzakos, Lekkas, Risager, and Voskou.

Link to recording (YouTube)

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Dan Petersen, Moments of families of quadratic L-functions over function fields via homotopy theory
(Stockholm University)

Thursday, April 4, 2024 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 5, 2024 (2am AEDT, 5am NZDT)

Abstract: This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows for precise predictions for the asymptotics of moments of many different families of L-functions. Our work concerns the CFKRS predictions in the case of the quadratic family over function fields, i.e. the family of all L-functions attached to hyperelliptic curves over some fixed finite field. One can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove a uniform range for homological stability with these coefficients. Together, these results imply the CFKRS predictions for all moments in the function field case, for all sufficiently large (but fixed) q.

Link to recording (YouTube)

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Wouter Castryck, The isogeny interpolation problem
(KU Leuven)

Thursday, March 28, 2024 (10am PDT, 1pm EDT, 5pm GMT, 6pm CET, 7pm Israel Standard Time, 10:30pm Indian Standard Time)
Friday, March 29, 2024 (1am CST, 4am AEDT, 6am NZDT)

Abstract: It is easy to prove that a degree-d isogeny f between two elliptic curves E and E' is completely determined by the images of any 4d + 1 points. In this talk we will study the algorithmic problem of evaluating f at a given point P on E, merely upon input of such "interpolation data". In case the interpolation points generate a group containing E[N] such that N^2 > 4d is smooth and coprime to d and the field characteristic, this problem was solved in 2022 by Robert, in the context of breaking SIKE (= SIDH), a former candidate for post-quantum key exchange that had advanced to the final stage of a standardization effort run by the National Institute of Standards and Technology. We will discuss this solution, and then show how to address more general instances of the isogeny interpolation problem, while also publicizing some unsolved cases.

Link to recording (YouTube)

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Javier Fresán, E-functions and Geometry
(Sorbonne University)

Thursday, March 21, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 22, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: E-functions are power series which solve a differential equation and whose coefficients are algebraic numbers that satisfy certain growth conditions of arithmetic nature. They were introduced in Siegel's 1929 memoir on the applications of diophantine approximation with the goal of generalising the Hermite--Lindemann--Weierstrass theorem about the transcendence of the values of the exponential function at algebraic arguments. Besides the exponential, standard examples include the Bessel function and confluent hypergeometric series. After briefly surveying on the history of E-functions, I will present a joint work in progress with Peter Jossen where we prove that exponential period functions provide us with a rich geometric source of E-functions. The easiest examples, attached to polynomials of degree 4, already allowed us a couple of years ago to exhibit some E-functions which are not polynomial expressions in hypergeometric series, thus solving one of the problems in Siegel's original paper. 

Link to recording (YouTube)

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Marc Munsch, Two tales on quadratic character sums
(Jean Monnet University)

Thursday, March 14, 2024 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 15, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Link

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Vesselin Dimitrov, The next case after Apéry on mixed Tate periods
(Caltech)

Thursday, March 7, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 8, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: I will introduce a method, joint with Frank Calegari and Yunqing Tang, for proving linear independence results and effective bad approximability measures. It is an outgrowth of our previous joint work on the so-called "unbounded denominators conjecture," which was in some sense an application of transcendental number theory to modular forms theory, with the key step being to prove sufficiently sharp $\mathbb{Q}(x)$-linear dimension bounds on certain spaces of algebraic functions. This time, we step into the wilder realm of G-functions with infinite monodromy, and devise holonomy bounds fine enough to prove the linear independence of two certain Dirichlet L-function values, a result that, in the realm of mixed Tate periods, can be considered as the next-simplest case after Apery's proof of the irrationality of $\zeta(3)$ (excluding the cases that reduce to the Hermite--Lindemann theorem or the Gelfond--Baker theorem on linear forms in logarithms). One key input turns out to be the classical Shidlovsky lemma on functional bad approximability, the point Siegel missed for three decades to complete his theory of algebraic relations among special values of E-functions. 

This is all a joint work with Frank Calegari and Yunqing Tang. 

Paul Pollack, Stretching, the truth about nonunique factorization
(University of Georgia)

Thursday, February 29, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 1, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Number theorists learn at their mother's knee that unique factorization fails in \Z[\sqrt{-5}]. Less well-known is that \Z[\sqrt{-5}] exhibits only a "half-failure" of unique factorization: while two factorizations into irreducibles of the same element need not agree up to unit factors, their lengths (number of factors) does always agree. This is a special case of a 1960 result of Leonard Carlitz. I will discuss offshoots of Carlitz's theorem. Particular attention will be paid to certain questions of Coykendall regarding "elasticity" of orders in quadratic number fields.

Link to recording (YouTube)

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Arul Shankar, Secondary terms in the first moment of the 2-Selmer groups of elliptic curves
(University of Toronto)

Thursday, February 22, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 23, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: A web of interrelated conjectures (due to work of Goldfeld, Katz--Sarnak, Poonen-Rains, Bhargava--Kane--Lenstra--Poonen--Rains) predict the distributions of ranks and Selmer groups of elliptic curves over Q. These conjectures predict that the average rank of elliptic curves is 1/2. Furthermore, it is known (due to Bhargava and myself) that the average size of the 2-Selmer group of elliptic curves is 3 (when the family of all elliptic curves is ordered by (naive) height). 

On the computational side, Balakrishnan, Ho, Kaplan, Spicer, Stein, and Weigand collect and analyze data on ranks, 2-Selmer groups, and other arithmetic invariants of elliptic curves, when ordered by height. Interestingly, they find both a larger average rank as well as a smaller average size of the 2-Selmer group in the data.  In this talk, we will discuss joint work with Takashi Taniguchi, in which we give a possible theoretical explanation for deviation of the data on 2-Selmer groups from the predicted distribution, namely, the existence of a secondary term.

Link to recording (YouTube)

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Damaris Schindler, Density of rational points near manifolds
(University of Göttingen)

Thursday, February 15, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 16, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Given a bounded submanifold M in R^n, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre's dimension growth conjecture as well as applications in Diophantine approximation. For this I'll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau.

Link to slides

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Victor Y. Wang, Sums of three cubes over a function field
(IST Austria)

Thursday, February 8, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 9, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: I will talk about joint work with Tim Browning and Jakob Glas on producing sums of three cubes over a function field, assuming a q-restricted form of the Ratios Conjecture for a geometric family of L-functions. If time permits, I may also discuss some recent developments in homological stability that could help to resolve this q-restricted Ratios Conjecture. 

Link to slides (Note by the organizers: the final slide was added after the talk following a comment of Trevor Wooley.)

Link to recording (YouTube)

Akshat Mudgal,  Recent progress towards the sum–product conjecture and related problems
(University of Oxford)

Thursday, February 1, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 2, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: An important open problem in combinatorial number theory is the Erdös–Szemerédi sum–product conjecture, which suggests that for any positive integers s, N, and for any set A of N integers, either there are many s-fold sums of the form a_1 + … + a_s or there are many s-fold products of the form a_1…a_s. While this remains wide open, various generalisations of this problem have been considered more recently, including the question of finding large additive and multiplicative Sidon sets in arbitrary sets of integers as well as studying the so-called low energy decompositions.

In this talk, I will outline some recent progress towards the above questions, as well as highlight how these connect very naturally to other key conjectures in additive combinatorics.

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Thomas Tucker, Tits and Borel type theorems for preperiodic points of finite morphisms
(University of Rochester)

Thursday, January 25, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 26, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: We pose a general question: Given a finitely generated semigroup S of finite morphisms from a variety to itself, what can one say about how the structure of the semigroup is connected to the relationship between the preperiodic points of the elements of S? When S consists of polarized morphisms, we can give a fairly simple answer to this question using Tate's limiting procedure for Weil and Moriwaki heights.  We formulate some conjectures that generalize this answer and prove some results relating to these conjectures.

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Oded Regev, An Efficient Quantum Factoring Algorithm
(Courant Institute of Mathematical Sciences)

Thursday, January 18, 2024 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 19, 2024 (12am CST, 3am AEDT, 5am NZDT)

Abstract: We show that n-bit integers can be factorized by independently running a quantum circuit with \tilde{O}(n^{3/2}) gates for \sqrt{n}+4 times, and then using polynomial-time classical post-processing. In contrast, Shor's algorithm requires circuits with \tilde{O}(n^2) gates. The

correctness of our algorithm relies on a number-theoretic heuristic assumption reminiscent of those used in subexponential classical factorization algorithms. It is currently not clear if the algorithm can lead to improved physical implementations in practice.

No background in quantum computation will be assumed.

Based on the arXiv preprint: https://arxiv.org/abs/2308.06572

Link to slides

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Zeev Rudnick, A talk in honor of Peter Sarnak's 70th birthday
(Tel Aviv University)

Celebrating Peter Sarnak's 70th birthday

Special Chair: Alex Kontorovich (Rutgers University)


Thursday, December 21, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 22, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: I will give selected highlights of Peter Sarnak's works on automorphic forms and some of the outstanding problems remaining.

Link to slides

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Djordje Milićević, Beyond the spherical sup-norm problem
(Bryn Mawr College)

Thursday, December 14, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 15, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: The sup-norm problem on arithmetic Riemannian manifolds occupies a prominent place at the intersection of harmonic analysis, number theory, and quantum mechanics. It asks about the sup-norm of L^2-normalized joint eigenfunctions of invariant differential operators and Hecke operators — that is, automorphic forms — most classically in terms of their Laplace eigenvalues (as in the QUE problem for high-energy eigenstates), but also in terms of the volume of the manifold and other parameters.

In this talk, we will motivate the sup-norm problem and then describe our results, joint with Blomer, Harcos, and Maga, which for the first time solve it for non-spherical Maass forms of an increasing dimension of the associated K-type, on an arithmetic quotient of G=SL(2,C), with K=SU(2). We combine representation theory, spectral analysis, and Diophantine arguments, developing new Paley-Wiener theory for G and sharp estimates on spherical trace functions of arbitrary K-type on the way to a novel counting problem of Hecke correspondences close to various special submanifolds of G.

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Misha Rudnev, The sum-product problem for integers with few prime factors
(University of Bristol)

Thursday, December 7, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 8, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: It was asked by Szemerédi if the known sum-product estimates can be improved for a set of N integers under the constraint that each integer has a small number of prime factors. We prove, if the maximum number of prime factors for each integer is sub-logarithmic in N, the sum-product exponent 5/3-o(1)

This becomes a corollary of an additive energy versus the product set cardinality estimate, which turns out to be the best possible. 

It is based on a scheme of Burkholder-Gundy-Davis martingale square function inequalities in p-adic scales, followed by an application of a variant of the Schmidt subspace theorem.

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Wei Zhang, Diagonal cycles: some results and conjectures
(MIT)

Thursday, November 30, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 1, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Algebraic cycles are among the most fundamental mathematical objects. I will discuss a class of special algebraic cycles related to the diagonal cycle, including the Gross-Schoen cycle (the small diagonal) on the triple product of a curve, the arithmetic diagonal cycle appearing in the Gan-Gross-Prasad conjecture, as well as the Fourier-Jacobi cycle defined by Yifeng Liu. 

Link to slides

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Anke Pohl, Period functions for vector-valued automorphic functions via dynamics and cohomology
(University of Bremen)

Thursday, November 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 24, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Vector-valued automorphic functions, or generalized automorphic functions, occur naturally in many areas, most notably in spectral theory, number theory and mathematical physics. Already Selberg promoted the idea to investigate vector-valued automorphic functions alongside their classical relatives and to exploit their interaction in order to understand their properties. While during the last decades the focus has been on automorphic functions equivariant with regard to unitary representations, the investigations recently turned to non-unitary representations as well. I will report on the status of an ongoing project to investigate simultaneously unitarily and non-unitarily equivariant automorphic functions with a view towards period functions and a classical-quantum correspondence by means of dynamics (transfer operator methods) and cohomology theory. This is joint work with R. Bruggeman.

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Henri Darmon, Explicit class field theory and orthogonal groups
(McGill University)

Thursday, November 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 17, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Essentially all abelian extensions of the rational numbers or of a quadratic imaginary field can be generated by special values of the exponential function  or of the modular j-function at explicit arguments in the ground field. Describing the mathematical objects which could play the role of trigonometric and modular functions in generating class fields of more general base fields is the stated goal of explicit class field theory. Around 5 years ago Jan Vonk and I proposed  a framework in which class fields of real quadratic fields can  be generated from the special values of certain “rigid meromorphic cocycles” at real quadratic arguments. Without delving into the details of this framework, I will present some simple concrete consequences of it in settings where the base field is totally real, and explain how they can be proved. The more general statements rest on (but do not require the full force of) the notion of rigid meromorphic cocycles for orthogonal groups of signature (r,r) described in joint work with  Lennart Gehrmann and Mike Lipnowski, and are also inspired by the  calculations in Romain Branchereau’s PhD thesis. (Joint with Jan Vonk)

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Jens Marklof, Smallest denominators
(University of Bristol)

Thursday, November 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 10, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: If we partition the unit interval into 3000 equal subintervals and take the smallest denominator amongst all rational points in each subinterval, what can we say about the distribution of those 3000 denominators? I will discuss this and related questions, its connection with Farey statistics and random lattices. In particular, I will report on higher dimensional versions of a recent proof of the 1977 Kruyswijk-Meijer conjecture by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305] on the convergence of the expectation value of the above distribution, as well as closely related work by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405--1413]. In fact, we will uncover the full distribution and prove convergence of more moments than just the expectation value. (This I believe was previously not known even in one dimension.)  We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130--149] as well as new results on pigeonhole statistics.

Link to slides

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Joseph H. Silverman, Field of Moduli and Fields of Definition in Arithmetic Geometry and Arithmetic Dynamics
(Brown University)

Thursday, November 2, 2023 (8am PDT, 11am EDT, 3pm GMT, 4pm CET, 5pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, November 3, 2023 (2am AEDT, 4am NZDT)

Abstract: Let X/Qbar be an algebraic variety defined over the field of algebraic numbers. We say that a number field K is a field of definition (FOD) for X if there is a variety Y/K such that Y is Qbar-isomorphic to X.

The field of moduli (FOM) of X is the fixed field of

          { s \in G_Q : s(X) is Qbar-isomorphic to X }.

It is easy to check that every FOD for X contains the FOM of X, but there are many situations where the FOM of X is not a FOD. I will briefly discuss the FOM versus FOD problem in the classical case of abelian varieties, and then turn to the the analogous question for morphisms f : P^N --> P^N defined over Qbar, where two maps are (dynamically) isomorphic if they are conjugate by a linear fractional transformation. I will describe what is known for N=1, including examples of maps for which the FOM is not an FOD. I will then discuss recent results for higher dimensional projective spaces in which we show that every map f has a FOD whose degree over its FOM is bounded by a function depending only on N and deg(f).  (Joint work with John Doyle.)

Link to slides

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Ananth Shankar, Canonical heights on Shimura varieties and the Andre-Oort conjecture
(University of Wisconsin, Madison)

Thursday, October 26, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 27, 2023 (2am AEDT, 4am NZDT)

Abstract: Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by Jacob Tsimerman. However, this conjecture was unknown for Shimura varieties without a moduli interpretation. Binyamini-Schmidt-Yafaev build on work of Binyamini to reduce the Andre-Oort conjecture to establishing height bounds on special points. I will describe joint work with Jonathan Pila and Jacob Tsimerman where we establish these height bounds, and therefore prove the Andre Oort conjecture in full generality. 

Link to recording (YouTube)

Timothy Browning, When is a random Diophantine equation soluble over $\mathbb{Q}_p$ for all $p$?
(IST Austria)

Thursday, October 19, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 20, 2023 (2am AEDT, 4am NZDT)

Abstract: The question in the title is of growing importance in number theory and represents a more tractable staging post than the question of solubility over $\mathbb{Q}$. I'll describe the landscape for various families of varieties, which can be interpreted as a more delicate version of Manin's conjecture, in which one counts rational points of bounded height which lie in the image of adelic points under a  morphism. This leads to more subtle asymptotic behaviours and depends intimately on the geometry of the morphism. This is joint work with Julian Lyczak, Roman Sarapin and Arne Smeets.

Link to recording (YouTube)

David Masser, Some new elliptic integrals
(University of Basel)

Thursday, October 12, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 13, 2023 (2am AEDT, 4am NZDT)

Link to recording (YouTube)

Isabel Vogt, Brauer--Manin obstructions requiring arbitrarily many Brauer classes
(Brown University)

Thursday, October 5, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 6, 2023 (2am AEDT, 4am NZDT)

Abstract: A fundamental problem in the arithmetic of varieties over global fields is to determine whether they have a rational point.  As a first effective step, one can check that a variety has local points for each place.  However, this is not enough, as many classes of varieties are known to fail this local-global principle.  The Brauer–Manin obstruction to the local-global principle for rational points is captured by elements of the Brauer group. On a projective variety, any Brauer–Manin obstruction is captured by a finite subgroup of the Brauer group.  I will explain joint work that shows that this subgroup can require arbitrarily many generators.  This is joint with J. Berg, C. Pagano, B. Poonen, M. Stoll, N. Triantafillou and B. Viray.

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Holly Krieger, A transcendental birational dynamical degree
(University of Cambridge)

Thursday, September 28, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 29, 2023 (1am AEST, 4am NZDT)

Abstract: In the study of a discrete dynamical system defined by polynomials, we wish to understand the integer sequence formed by the degrees of the iterates of the map: examples of such a sequence include the Fibonacci and other integer linear recurrence sequences, but not all examples satisfy a finite recurrence.  The growth of this sequence is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for a birational map of projective space, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.

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Kaisa Matomäki, Detecting primes in multiplicatively structured sequences
(University of Turku)

Thursday, September 21, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 22, 2023 (1am AEST, 4am NZDT)

Abstract: I will discuss a new sieve set-up which allows one to find prime numbers in sequences that have a suitable multiplicative structure and good "type I information". Among other things, the method gives a new L-function free proof of Linnik's theorem concerning the least prime in an arithmetic progression. The talk is based on on-going joint work with Jori Merikoski and Joni Teräväinen.

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Andrew Sutherland, Murmurations of arithmetic L-functions
(MIT)

Thursday, September 14, 2023 (9am PDT, 12pm EDT, 5pm BST, 6pm CEST, 7pm Israel Daylight Time, 9:30pm Indian Standard Time)
Friday, September 15, 2023 (12am CST, 2am AEST, 4am NZST)

Abstract: While conducting a series of number-theoretic machine learning experiments last year, He, Lee, Oliver, and Pozdnyakov noticed a curious oscillation in the averages of Frobenius traces of elliptic curves over Q.  If one computes the average value of a_p(E) for E/Q of fixed rank with conductor in a short interval, as p increases the average oscillates with a decaying frequency determined by the conductor.  That the rank influences the distribution of Frobenius traces has long been known (indeed, this was the impetus for the experiments that led to the conjecture of Birch and Swinnerton-Dyer), but these oscillations do not appear to have been noticed previously.  This may be due in part to the critical role played by the conductor; in arithmetic statistics it is common to order elliptic curves E/Q by naive height rather than conductor, but doing so obscures these oscillations.

I will present results from an ongoing investigation of this phenomenon, which is remarkably robust and not specific to elliptic curves. One finds similar oscillations in the averages of Dirichlet coefficients of many types of L-functions when organized by conductor and root number, including those associated to modular forms and abelian varieties.  The source of these murmurations in the case of weight-2 newforms with trivial nebentypus is now understood, thanks to recent work of Zubrilina, but all other cases remain open.

This is based on joint work with Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov.

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Alexander Mangerel, Correlations, sign patterns and rigidity theorems for multiplicative functions
(Durham University)

Thursday, September 7, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 8, 2023 (1am AEST, 3am NZST)

Abstract: 

The Liouville function \lambda(n), defined to be +1 for n having an even number of prime factors (counted with multiplicity) and -1 otherwise, is a multiplicative function with deep connections to the distribution of primes. Inspired by the prime k-tuples conjecture of Hardy and Littlewood, Chowla conjectured that for every k each of the 2^k distinct sign patterns, i.e., tuples in \{-1,+1\}^k are assumed by the tuples (\lambda(n+1),...,\lambda(n+k)), n \in \mathbb{N}, with the same asymptotic frequency.

The underlying phenomenon at hand is that the prime factorisations of n+1,\ldots,n+k are expected to be (in a precise sense) statistically independent as n varies. As conjectured by Elliott, the same equidistribution of sign patterns is expected to hold for other \pm 1-valued multiplicative functions, provided they are ``far from being periodic''. To the best of our knowledge, until recently no explicit constructions of multiplicative functions with this behaviour were known.

In this talk we will discuss precisely what Chowla's and Elliott's conjectures say, survey some of the literature on correlations, and discuss some related problems about sign patterns. Specifically, we will address:

i) the construction of ``Liouville-like'' functions f: \mathbb{N} \rightarrow \{-1,+1\} whose k-tuples (f(n+1),...,f(n+k)) equidistribute in \{-1,+1\}^k, answering a question of de la Rue from 2018, and

ii) in the case k = 4, the classification of all \pm 1-valued completely multiplicative functions f with the (rigid) property that the sequence of tuples (f(n+1),f(n+2),f(n+3),f(n+4)) omits the pattern (+1,+1,+1,+1), solving a 50-year old problem of R.H. Hudson.

Key to these developments is a new result about the vanishing of correlations of ``moderately aperiodic'' multiplicative functions along a dense sequence of scales.

Based on joint work with O. Klurman and J. Teräväinen.

Link to recording (YouTube)

Link to slides

Alex Iosevich, Some number theoretic aspects of finite point configurations
(University of Rochester)

Thursday, June 29, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 30, 2023 (1am AEST, 3am NZST)

Abstract: We are going to survey some recent and less recent results pertaining to the study of finite point configurations in Euclidean space and vector spaces over finite fields, centered around the Erdos/Falconer distance problems. We shall place particular emphasis on number-theoretic ideas and obstructions that arise in this area. 

Link to slides

Εfthymios Sofos, The second moment method for rational points
(University of Glasgow)

Thursday, June 22, 2023 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 23, 2023 (1am AEST, 3am NZST)

Abstract: In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel's Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen [arXiv:2212.10373] that develops the method and establishes asymptotics for averages of an arithmetic function over the values of typical polynomials. Part of the new ideas come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer--Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties.

Link to slides

Youness Lamzouri, A walk on Legendre paths
(Institut Elie Cartan de Lorraine)

Thursday, June 15, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 16, 2023 (1am AEST, 3am NZST)

Abstract: In this talk, we shall explore certain polygonal paths, that we call ''Legendre paths'', which encode important information about the values of the Legendre symbol. More precisely, the Legendre path modulo a prime number p is defined as the polygonal path in the plane whose vertices are the points (j, S_p(j)) for 0≤j≤p-1, where S_p(j) is the (normalized) sum of Legendre symbols (n/p) for n up to j.  In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed?  how do their maximums behave? when does a Legendre path decreases for the first time? what is the typical number of x-intercepts of such paths? and what proportion of a Legendre path is above the real axis?  We will see that some of these questions correspond to important and longstanding problems in analytic number theory, including understanding the size of the least quadratic non-residue, and improving the Pólya-Vinogradov inequality for character sums. Among our results, we prove that as we average over the primes, the Legendre paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademacher random multiplicative functions.  

Part of this work is joint with Ayesha Hussain and with Oleksiy Klurman and Marc Munsch. 

Link to slides

Carlo Pagano, On Chowla's non-vanishing conjecture over function fields
(Concordia University)

Thursday, June 8, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 9, 2023 (1am AEST, 3am NZST)

Abstract: A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz--Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui--Florea, David--Florea--Lalin, Ellenberg--Li--Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields. 

Link to slides

Lior Bary-Soroker, Random additive polynomials
(Tel Aviv University)

Thursday, June 1, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 2, 2023 (1am AEST, 3am NZST)

Abstract: Random polynomials with integer coefficients tend to be irreducible and to have a large Galois group with high probability. This was shown more than a century ago in the large box model, where we choose the coefficients uniformly from a box and let its size go to infinity, while only recently there are results in the restricted box model, when the size of the box is bounded and its dimension (i.e. the degree of the polynomial) goes to infinity. 

In this talk, we will discuss an important class of random polynomials — additive polynomials, which have coefficients in the polynomial ring over a finite field. In this case, the roots form a vector space, hence the Galois group is naturally a subgroup of GLn. 

While we prove that the Galois group is the full matrix group both in the large box model, and in the large finite field limit, our main result is in the restricted box model: under some necessary condition the Galois group is large (in the sense that it contains SLn) asymptotically almost surely, as the degree goes to infinity.

The proof relies crucially on deep results on subgroups of GLn by Fulman and Guralnick, combined with tools from algebra and number theory. 

Based on a joint work with Alexei Entin and Eilidh McKemmie

Link to slides

Barak Weiss, New bounds on lattice covering volumes, and nearly uniform covers
(Tel Aviv University)

Thursday, May 25, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 26, 2023 (1am AEST, 3am NZST)

Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem in the so-called “Geometry of Numbers”, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.

of the stated result. 

Link to slides

Mark Shusterman, Counting Minimally Ramified Global Field Extensions
(Harvard University)

Thursday, May 18, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 19, 2023 (1am AEST, 3am NZST)

Abstract: Given a finite group G, one is interested in the number of Galois extensions of a global field with Galois group G and bounded discriminant. We consider a refinement of this problem where the discriminant is required to have the smallest possible number of (distinct) prime factors. We will discuss existing results and conjectures over number fields, and present some recent results over function fields. 

Link to slides

Ben Green, On Sarkozy's theorem for shifted primes
(University of Oxford)

Thursday, May 11, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 12, 2023 (1am AEST, 3am NZST)

Abstract: Suppose that N is large and that A is a subset of {1,..,N} which does not contain two elements x, y with x - y equal to p-1, p a prime. Then A has cardinality at most N^{1 - c}, for some absolute c > 0. I will discuss the history of this kind of question as well as some aspects of the proof of the stated result. 

Link to slides

Peter Koymans, Counting nilpotent extensions
(University of Michigan)

Thursday, May 4, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, May 5, 2023 (1am AEST, 3am NZST)

Abstract: We discuss some recent progress towards the strong form of Malle’s conjecture. Even for nilpotent extensions, only very few cases of this conjecture are currently known. We show how equidistribution of Frobenius elements plays an essential role in this problem and how this can be used to make further progress towards Malle’s conjecture. We will also discuss applications to the Massey vanishing conjecture and to lifting problems. This is joint work with Carlo Pagano.

Link to slides

Pierre Le Boudec, 2-torsion in class groups of number fields
(University of Basel)

Thursday, April 27, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 28, 2023 (1am AEST, 3am NZST)

Abstract: It is well-known that the class number of a number field K of fixed degree n is roughly bounded by the square root of the absolute value of the discriminant of K. However, given a prime number p, the cardinality of the p-torsion subgroup of the class group of K is expected to be much smaller. Unfortunately, beating the trivial bound mentioned above is a hard problem. Indeed, this task had only been achieved for a handful of pairs (n,p) until Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao managed to do so for any degree n in the case p=2. In this talk we will go through their proof and we will present new bounds which depend on the geometry of the lattice underlying the ring of integers of K. This is joint work with Dante Bonolis. 

Link to slides

Bjorn Poonen, Integral points on curves via Baker's method and finite étale covers
(MIT)

Thursday, April 20, 2023 (6am PDT, 9am EDT, 2pm GMT, 3pm CEST, 4pm Israel Daylight Time, 6:30pm Indian Standard Time, 9pm CST, 11pm AEST)
Friday, April 21, 2023 (1am NZST)

Abstract: We prove results in the direction of showing that for some affine curves, Baker's method applied to finite étale covers is insufficient to determine the integral points.

Hélène Esnault, Integrality Properties of the Betti Moduli Space
(Freie Universität Berlin)

Thursday, April 13, 2023 (8am PDT, 11am EDT, 4pm GMT, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 14, 2023 (1am AEST, 3am NZST)

Abstract: We use de Jong’s conjecture and the existence of $\ell$-adic companions to single out integrality properties of the Betti moduli space. The first such instance was in joint work with Michael Groechenig on Simpson’s integrality conjecture for (cohomologically) rigid local systems. This integrality property yields an obstruction for a finitely presented group to be the fundamental group of a sooth quasi-projective complex variety. (joint with Johan de Jong)

Kannan Soundararajan, Covering integers using quadratic forms
(Stanford University)

Thursday, April 6, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, April 7, 2023 (1am AEST, 3am NZST)

Abstract: How large must $\Delta$ be so that we can cover a substantial proportion of the integers below $X$ using the binary quadratic forms $x^2 +dy^2$ with $d$ below $\Delta$?  Problems involving representations by binary quadratic forms have a long history, going back to Fermat.  The particular problem mentioned here was recently considered by Hanson and Vaughan, and Y. Diao.  In ongoing work with Ben Green, we resolve this problem, and identify a sharp phase transition:  If $\Delta$ is below $(\log X)^{\log 2-\epsilon}$ then zero percent of the integers below $X$ are represented, whereas if $\Delta$ is above $(\log X)^{\log 2 +\epsilon}$ then 100 percent of the integers below $X$ are represented.  

Link to recording

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Ziyang Gao, Sparsity of rational and algebraic points
(Leibniz University Hannover)

Thursday, March 30, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, March 31, 2023 (2am AEDT, 4am NZDT)

Abstract: It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.

In my talk, I will explain the historical and recent developments of this problem according to the different grades.

Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.

If time permits, I will mention the corresponding result in high dimensions.

Link to recording

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Ofir Gorodetsky, How many smooth numbers and smooth polynomials are there?
(University of Oxford)

Thursday, March 23, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 24, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Smooth numbers are integers whose prime factors are smaller than a threshold y. In the 80s they became important outside of pure math, as Pomerance's quadratic sieve for factoring integers relied on their distribution. The density of smooth numbers up to x can be approximated, in some range, using a peculiar function ρ called Dickman's function, defined via a delay-differential equation. All of the above is also true for smooth polynomials over finite fields.

We'll survey these topics and discuss recent results concerning the range of validity of the approximation of the density of smooth numbers by ρ, whose proofs rely on relating the counting function of smooth numbers to the Riemann zeta function and the counting function of primes. In particular, we uncover phase transitions in the behavior of the density at the points y=(log x)^2 (as conjectured by Hildebrand) and y=(log x)^(3/2), when previously only a transition at y=log x was known and understood. These transitions also occur in the polynomial setting. We'll also show that a standard conjecture on the error in the Prime Number Theorem implies ρ is always a lower bound for the density, addressing a conjecture of Pomerance.

Link to recording

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Florian Luca, Recent progress on the Skolem problem
(University of the Witwatersrand)

Thursday, March 16, 2023 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 17, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. One of the main contributions of the talk is to present an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct -- a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the p-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool SKOLEM point to the practical applicability of this method. 

 In the second part of the talk, we present the notion of an Universal Skolem Set, which is a subset of the positive integers on which the Skolem is decidable regardless of the linear recurrence. We give two examples of such sets, one of which is of positive density (that is, contains a positive proportion of all the positive integers).

Link to recording

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Alex Wilkie, Integer points on analytic sets
(University of Oxford and University of Manchester)

Thursday, March 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 10, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 2004 I proved an O(loglogH) bound for the number of integer points of height at most H lying on a globally subanalytic curve. (The paper was published in the Journal of Symbolic Logic and so probably escaped the notice of most of you reading this.) Recently, Gareth Jones and Gal Binyamini proposed a generalization of the result to higher dimensions (where the obvious statement is almost certainly false) and I shall report on our joint work: one obtains the (hoped for) (loglogH)^n  bound for (not globally subanalytic but) globally analytic sets of dimension n.  

Link to recording

Alexandra Shlapentokh, Defining integers using unit groups
(East Carolina University)

Thursday, March 2, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, March 3, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: We discuss some problems of definability and decidability over rings of integers of algebraic extensions of $\Q$.  In particular, we show that for a large class of fields $K$ there is a simple formula defining rational integers over $O_K$.  Below $U_K$ is the group of units of $O_K$. 

$\Z=\{x| \forall \varepsilon \in U_K\setminus \{1\}\ \exists \delta \in U_K: x \equiv \frac{\delta-1}{\varepsilon-1} \bmod (\varepsilon-1)\}$. This talk is based on a joint paper with Barry Mazur and Karl Rubin.

Link to recording

Link to slides

Terence Tao, Infinite Partial Sumsets in the Primes
(UCLA)

Thursday, February 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 24, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture).  Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a_1 < a_2 < ... and b_1 < b_2 < ... such that a_i + b_j is prime for all i<j.  Equivalently, the primes are not "translation-finite" in the sense of Ruppert.  As an application of these methods we show that the orbit closure of the primes is uncountable.

Link to recording

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Wanlin Li, Ordinary and Basic Reductions of Abelian Varieties
(Université de Montréal)

Thursday, February 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 17, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc). 

In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including some cases of A with almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang.

Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries, and Tang.

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Sarah Peluse, Divisibility of character values of the symmetric group
(Institute for Advanced Study and Princeton University)

Thursday, February 9, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 10, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 2017, Miller computed the character tables of $S_n$ for all $n$ up to $38$ and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and, based on his computations, Miller conjectured that almost all entries of the character table of $S_n$ are divisible by any fixed prime power as $n$ tends to infinity. In this talk, I will discuss joint work with K. Soundararajan that resolves this conjecture, and mention some related open problems.

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Matilde Lalín, Distributions of sums of the divisor function over function fields
(Université de Montréal)

Thursday, February 2, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 3, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick studied the mean-square of sums of the divisor function $d_k(f)$  over short intervals and over arithmetic progressions for the function field  $\mathbb{F}_q[T]$. By results from the Katz and Sarnak philosophy, they were able to relate these problems to certain integrals over the ensemble of unitary matrices when  $q$ goes to infinity. We study similar problems leading to integrals over the ensembles of symplectic and orthogonal matrices when $q$ goes to infinity. This is joint work with Vivian Kuperberg.

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Robert Wilms, On equidistribution in Arakelov theory
(University of Basel)

Thursday, January 26, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 27, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: As a motivating example of its own interest I will first discuss a new equidistribution result for the zero sets of integer polynomials. More precisely, I will give a condition such that the zero sets tends to equidistribute with respect to the Fubini-Study measure and I will show that this condition is generically satisfied in sets of polynomials of bounded Bombieri norm. In the second part, I will embed this example in a much more general framework about the distribution of the divisors of small sections of arithmetically ample hermitian line bundles in Arakelov theory.

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Cécile Dartyge, On the largest prime factor of quartic polynomial values: the cyclic and dihedral cases
(Institut  Élie Cartan, Université de Lorraine)

Thursday, January 19, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 20, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: Let P(X) be a monic, quartic, irreducible polynomial of Z[X] with cyclic or dihedral Galois group. We prove that there exists c_P >0, such that for a positive proportion of integers n, P(n) has a prime factor bigger than n^{1+c_P}.

This is a joint work with James Maynard.

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Régis de la Bretèche, Higher moments of primes  in arithmetic progressions
(Institut de Mathématiques de Jussieu-Paris Rive Gauche)

Thursday, January 12, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, January 13, 2023 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In a joint work with Daniel Fiorilli, we develop a new method to prove lower bounds of some moments related to the distribution of primes in arithmetic progressions. We shall present  main results and explain some aspects of the proofs. To prove our results, we assume GRH but we succeed to avoid linearly independence on zeroes hypothesis.

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Umberto Zannier, Bounded generation in linear groups and exponential parametrizations
(Scuola Normale Superiore Pisa)

Thursday, December 22, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 23, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of “bounded generation” in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on “anisotropic groups” (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields)  are “sparse”. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.

William Duke, On the analytic theory of isotropic ternary quadratic forms
(UCLA)

Thursday, December 15, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 16, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is explicitly computed in terms of local data determined by the orbit.  This is compared  with the well-known asymptotic for the  count of all primitive zeros. Together with an extension  of results of Kneser by R. Schulze-Pillot on the classes  in a genus of representations, this yields a formula for the number of  orbits,  summed over a genus of forms,  in terms of the number of local orbits. For a certain special class of forms a simple explicit formula is given for this number.

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Laura DeMarco, Lattès maps, bifurcations, and arithmetic
(Harvard University)

Thursday, December 8, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 9, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In the field of holomorphic dynamics, we learn that the Lattès maps -- the rational functions on P^1 that are quotients of maps on elliptic curves -- are rather boring.  We can understand their dynamics completely.  But viewed arithmetically, there are still unanswered questions.  I'll begin the talk with some history of these maps.  Then I'll describe one of the recent questions and how it has led to interesting complex-dynamical questions about other families of maps on P^1 and, in turn, new perspectives on the arithmetic side.  The new material is a joint project with Myrto Mavraki.

Robert J. Lemke Oliver, Uniform exponent bounds on the number of primitive extensions of number fields
(Tufts University)

Thursday, December 1, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 2, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: A folklore conjecture asserts the existence of a constant $c_n > 0$ such that $N_n(X) \sim c_n X$ as $X\to \infty$, where $N_n(X)$ is the number of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by $X$.  This conjecture is known if $n \leq 5$, but even the weaker conjecture that there exists an absolute constant $C\geq 1$ such that $N_n(X) \ll_n X^C$ remains unknown and apparently out of reach.

Here, we make progress on this weaker conjecture (which we term the ``uniform exponent conjecture'') in two ways.  First, we reduce the general problem to that of studying relative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation.  Second, for almost all such groups, we prove the strongest known upper bound on the number of such extensions.  These bounds have the effect of resolving the uniform exponent conjecture for solvable groups, sporadic groups, exceptional groups, and classical groups of bounded rank.  This is forthcoming work that grew out of conversations with M. Bhargava.

Link to slides

Jared Duker Lichtman, A proof of the Erdős primitive set conjecture
(University of Oxford)

Thursday, November 24, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 25, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the sum of 1/(a log a), ranging over a in A, is uniformly bounded over all choices of primitive sets A. In 1986 he asked if this bound is attained for the set of prime numbers. In this talk we describe recent work which answers Erdős’ conjecture in the affirmative. We will also discuss applications to old questions of Erdős, Sárközy, and Szemerédi from the 1960s.

Link to slides

Trevor Wooley, Waring’s Problem
(Purdue University)

Thursday, November 17, 2022 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 18, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In 1770, E. Waring made an assertion these days interpreted as conjecturing that when $k$ is a natural number, all positive integers may be written as the sum of a number $g(k)$ of positive integral $k$-th powers, with $g(k)$ finite. Since the work of Hardy and Littlewood a century ago, attention has largely shifted to the problem of bounding $G(k)$, the least number $s$ having the property that all sufficiently large integers can be written as the sum of $s$ positive integral $k$-th powers. It is known that $G(2)=4$ (Lagrange), $G(3)\le 7$ (Linnik), $G(4)=16$ (Davenport), and $G(5)\le 17$, $G(6)\le 24$, ..., $G(20)\le 142$ (Vaughan and Wooley). For large $k$ one has $G(k)\le k(\log k+\log \log k+2+o(1))$ (Wooley). We report on very recent progress joint with Joerg Bruedern. One or two new world records will be on display.

Link to slides

Emanuel Carneiro, Hilbert spaces and low-lying zeros of L-functions
(ICTP)


Thursday, November 10, 2022 (8am PST, 11am EST, 4pm BST, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 11, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: In this talk I would like to present some ideas behind a general Hilbert space framework for solving certain optimization problems that arise when studying the distribution of the low-lying zeros of families of $L$-functions. For instance, in connection to previous work of Iwaniec, Luo, and Sarnak (2000), we will discuss how to use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. We will also discuss the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). This is based on joint work with M. Milinovich and A. Chirre.

Link to slides

Shai Evra, Optimal strong approximation and the Sarnak-Xue density hypothesis
(Hebrew University of Jerusalem)


Thursday, November 3, 2022 (9am PDT, 12pm EDT, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, November 4, 2022 (12am CST, 3am AEDT, 5am NZDT)

Abstract: It is a classical result that the modulo map from SL_2(Z) to SL_2(Z/qZ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL_2(Z/qZ) has a lift in SL_2(Z) with coefficients of size at most q^e? A simple pigeonhole principle shows that e > 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL_2(Z). The proof relies on a density theorem of the Ramanujan conjecture for SL_2(Z).

In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands program.

This talk is based on ongoing joint works with B. Feigon, M. Gerbelli-Gauthier, H. Gustafssun, K. Maurischat and O. Parzanchevski.

Link to slides

Evelina Viada, Rational points on curves in a product of elliptic curves
(University of Göttingen)


Thursday, October 27, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 28, 2022 (2am AEDT, 4am NZST)

Abstract: The Mordell-Conjecture (Faltings Theorem) states that an algebraic curve of genus at least 2 has only finitely many rational points. The Torsion Anomalous Conjecture (TAC) generalises Faltings Theorem. In some cases the proofs of the TAC are effective, implying effective cases of the Mordell-Conjecture. I would like to explain an effective method to determine the K-rational points on certain families of curves and to present some new specific examples. I will give an overview of the methods used in the context of the TAC  presenting some general theorems and applications.

Link to slides

Jack Thorne, Symmetric power functoriality for GL(2)
(University of Cambridge)


Thursday, October 20, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 21, 2022 (2am AEDT, 4am NZST)

Abstract: Langlands’s functoriality conjectures predict the existence of “liftings” of automorphic representations along morphisms of L-groups. A basic case of interest comes from the irreducible algebraic representations of GL(2) – the associated symmetric power L-functions are then the ones identified by Serre in the 1960’s in relation to the Sato—Tate conjecture.

I will describe the background to these ideas and then discuss the proof, joint with James Newton, of the existence of these symmetric power liftings for Hilbert modular forms. One arithmetic consequence is that if E is a (non-CM) elliptic curve over a real quadratic field, then all of its symmetric power L-functions admit analytic continuation to the whole complex plane.

Link to slides

Thomas Gauthier, A complex analytic approach to sparsity, rigidity and uniformity in arithmetic dynamics
(Université Paris-Saclay)


Thursday, October 13, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 14, 2022 (2am AEDT, 4am NZST)

Abstract: This talk is concerned with connections between arithmetic dynamics and complex dynamics. The first aim of the talk is to discuss several open problems from arithmetic dynamics and to explain how these problems are related to complex dynamical tool: bifurcation measures. If time allows, I will give a strategy to tackle several of those problems at the same time. This is based on a joint work in progress with Gabriel Vigny and Johan Taflin.

Link to slides

Jeffrey C. Lagarias, The Alternative Hypothesis and Point Processes
(University of Michigan)


Thursday, October 6, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, October 7, 2022 (2am AEDT, 4am NZST)

Abstract: The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced. It asks that  nearly all  normalized zero spacings be  near half-integers. This possible zero distribution is incompatible with the GUE distribution of zero spacings. Ruling it out arose as an obstacle to  the long-standing problem of proving there are no  exceptional zeros of Dirichlet L-functions. The talk describes joint work with Brad Rodgers, that constructs a point process realizing  Alternative Hypothesis type statistics, which is  consistent  with the known results on correlation functions for spacings of zeta zeros. (A similar result was independently obtained by  Tao with slightly different methods.) The talk  reviews point process models and presents further results on the general problem of to what extent two point processes, a continuous one on the real line, the other a discrete one on a lattice aZ, can  mimic each other in the sense of having perfect agreement of all their correlation functions when convolved with bandlimited test functions of a given bandwidth B.

Link to slides

Paul Nelson, The sup norm problem in the level aspect
(Aarhus University)


Thursday, September 29, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 30, 2022 (1am AEST, 4am NZDT)

Abstract: The sup norm problem concerns the size of L^2-normalized eigenfunctions of manifolds.  In many situations, one expects to be able to improve upon the general bound following from local considerations.  The pioneering result in that direction is due to Iwaniec and Sarnak, who in 1995 established an improvement upon the local bound for Hecke-Maass forms of large eigenvalue on the modular surface.  Their method has since been extended and applied by many authors, notably to the "level aspect" variant of the problem, where one varies the underlying manifold rather than the eigenvalue.  Recently, Raphael Steiner introduced a new method for attacking the sup norm problem.  I will describe joint work with Raphael Steiner and Ilya Khayutin in which we apply that method to improve upon the best known bounds in the level aspect.

Link to slides

Alexandra Florea, Negative moments of the Riemann zeta function
(University of California Irvine)


Thursday, September 22, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 23, 2022 (1am AEST, 3am NZST)

Abstract: I will talk about work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how one can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.

Link to slides

Danny Neftin, Reducible fibers of polynomial maps
(Technion-Israel Institute of Technology)


Thursday, September 15, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 16, 2022 (1am AEST, 3am NZST)

Abstract: For a polynomial $f\in \mathbb Q[x]$, the fiber $f^{-1}(a)$ is irreducible over $\mathbb Q$ for all values $a\in \mathbb Q$ outside a ``thin" set of exceptions $R_f$ whose explicit description is unknown in general. The problem of describing $R_f$ is closely related to reducibility and arboreal representations in arithmetic dynamics, as well as to Kronecker and arithmetic equivalence for polynomial maps, that is, polynomial versions of the question: "can you hear the shape of the drum?". We shall discuss recent progress on the above problem and topics.

Link to slides

Ping Xi, Analytic approaches towards Katz’s problems on Kloosterman sums
(Xi'an Jiaotong University)


Thursday, September 8, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 9, 2022 (1am AEST, 3am NZST)

Abstract: Motivated by deep observations on elliptic curves/modular forms, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory (e.g., sieve methods and automorphic forms) combining certain tools from $\ell$-adic cohomology.

Link to slides

Yann Bugeaud, B'
(University of Strasbourg)


Thursday, September 1, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, September 2, 2022 (1am AEST, 3am NZST)

Abstract: Link

Alexander Gamburd, Arithmetic and dynamics on varieties of Markoff type
(CUNY Graduate Center)


Thursday, June 30, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, July 1, 2022 (1am AEST, 3am NZST)

Abstract: The Markoff equation $x^2+y^2+z^2=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as  ensuing implications, diophantine and dynamical.


Amir Shpilka, Points, lines and polynomial identities
(Tel Aviv University)


Thursday, June 23, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 24, 2022 (1am AEST, 3am NZST)

Abstract: The Sylvester-Gallai (SG) theorem in discrete geometry asserts that if a finite set of points P has the property that every line through any two of its points intersects the set at a third point, then P must lie on a line. Surprisingly, this theorem, and some variants of it, appear in the analysis of locally correctable codes and, more noticeably, in algebraic program testing (polynomial identity testing). For these questions one often has to study extensions of the original SG problem: the case where there are several sets, or with a robust version of the condition (many "special" lines through each point) or with a higher degree analog of the problem, etc.

In this talk I will present the SG theorem and some of its variants, show its relation to the above mentioned computational problems and discuss recent developments regarding higher degree analogs and their applications.

Link to slides

John Voight, Counting elliptic curves with level structure
(Dartmouth College)


Thursday, June 16, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 17, 2022 (1am AEST, 3am NZST)

Abstract: Following work of Harron and Snowden, we provide an asymptotic answer to questions like: how many elliptic curves of bounded height have a cyclic isogeny of degree N? We'll begin with a survey the recent spate of work on this topic, and then we will report on joint work with Carl Pomerance and Maggie Pizzo, with John Cullinan and Meagan Kenney, and finally with Grant Molnar.

Link to slides

Nicole Looper, The Uniform Boundedness Principle for polynomials over number fields
(Brown University)


Thursday, June 9, 2022 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm Israel Daylight Time, 8:30pm Indian Standard Time, 11pm CST)
Friday, June 10, 2022 (1am AEST, 3am NZST)

Abstract: This talk is about uniform bounds on the number of K-rational preperiodic points across families of endomorphisms of projective space defined over various fields K. We will focus on the case where K is a number field, and the morphisms are polynomial maps on P^1. Along the way, I will highlight the more challenging aspects behind the known approaches, and discuss the obstacles to be addressed in future research.

Link to slides

Elon Lindenstrauss, Effective equidistribution of some unipotent flows with polynomial rates
(Hebrew University of Jerusalem)