We give necessary and sufficient conditions for an integral polynomial to be the characteristic polynomial of an isometry of some even, unimodular lattice of given signature. This result has applications in knot theory (existence of knots with given Alexander polynomial and Milnor signatures) as well as to K3 surfaces (existence of K3 surfaces having an automorphism with given dynamical degree and determinant).

We will discuss the interactions of the two fields in the title, with a focus on algorithmic elements.

In this talk, I will explain how to classify the isometry classes of unimodular integral euclidean lattices in rank up to 28. In particular, there are respectively 2566, 17059 and 374062 such lattices in rank 26, 27 and 28 (this last and most difficult computation is a joint work with Bill Allombert). As a general new ingredient, for any two lattices L and L' in a same (and arbitrary) genus, we prove an asymptotic formula for the proportion of Kneser p-neighbors of L which are isometric to L', when the prime p goes to infinity.

In this talk I will describe some current efforts to understand the high-degree rational cohomology of SL_n(Z), or more generally the cohomology of SL_n(O) when O is a number ring. Although the groups SL_n(O) do not satisfy Poincare duality, they do satisfy a twisted form of duality, called Bieri--Eckmann duality. Consequently, their high-degree rational cohomology groups are governed by an SL_n(O)-representation called the Steinberg module. I will overview some results, conjectures, and ongoing work concerning these representations.

There are several recent constructions by many authors of Eisenstein cocycles of arithmetic groups. I will discuss a point of view on these constructions using equivariant cohomology and differential forms. The resulting objects behave like theta kernels relating the homology of arithmetic groups to algebraic objects. I will also discuss an application to conjectures of Sczech and Colmez on critical values of Hecke L-functions. The talk is based on work-in-progress with Nicolas Bergeron, Pierre Charollois and Akshay Venkatesh.

Jim Arthur's conjectures from the 1980s predict a fascinating family of automorphic forms, connected to exotic unitary representations. I'll describe some recent examples from work with Joseph Hundley, as well as more recent results on the real group aspects with Jeffrey Adams, Marc van Leeuwen, and David Vogan. Together this proves the unitary aspect of Arthur's conjectures for all real forms of exceptional groups. The talk will include a discussion of parallel computing techniques (such as SLURM) which were used to speed up some computational parts of the proof.

The Rademacher symbol is algebraically expressed as a conjugacy class invariant quasimorphism PSL(2,Z) -> Z yielding the bounded Euler class. I will explain (1) how, using continued fractions, it is realized as the winding number for closed curves on the modular surface around the cusp; (2) how, using Eisenstein series, one can naturally construct a Rademacher symbol for any cusp of a general noncocompact Fuchsian group; (3) and discuss some connections to arithmetic geometry.

Given a real quadratic field, there is a naturally defined Hecke-stable subspace of the cohomology of a congruence subgroup of SL(3,Z). We investigate this subspace and make conjectures about its dependence on the real quadratic field and the relationship to boundary cohomology. This is joint work with Avner Ash.

In 2016, Mazur and Rubin conjectured that modular symbols should be normally distributed. This conjecture was resolved (on average) independently, by Petridis--Risager and Lee--Sun using two completely different approaches (resp. spectral and dynamical methods).

In this talk, I will give an introduction to the conjectures of Mazur and Rubin and talk about a number of different generalizations of the modular symbols conjecture (including higher weight holomorphic forms, Maass forms, groups different from GL2, and residual distribution) and how they can be tackled. With the topic of the seminar in mind, I will put special emphasis on the cohomological perspective.

The talk will feature joint work with Petru Constantinescu and Sary Drappeau (in progress).

In the talk, I will present an efficient algorithm to compute the decomposition of the Jacobians of modular curves, using modular symbols.

This is obtained by working intrinsically with the curve, unlike previous methods. I will also discuss the possible consequences for deriving equations of modular curves.

We can decide whether two elements T and S of GL(n,Z) are conjugate under GL(n,Q) by computing their rational canonical forms. However, the problem of whether they are conjugate under GL(n,Z) is much harder. In 1980 it was shown by Fritz Grunewald, that the conjugacy problem in GL(n,Z) is decidable. More recently, in a joint work with Tommy Hofmann and Eamonn O'Brien, we developed a first practical method to solve this problem. This talk reports on this algorithm and its applications.

I will discuss briefly the concept of denominators of Eisenstein classes and the resulting congruences. I will speak in very general terms about the conjectural relationship between the denominators and special values of L-functions. I will also mention the experimental aspects. If time permits I will discuss in a special example the influence of the denominator (or the special value of the L-function) on the structure of the Galois group.

In joint work with Gerard van der Geer we have studied the cohomology of local systems on the Picard modular surface of Eisenstein type and the related modular forms. Our main technique is to use computer counts of the points over finite fields of small cardinality. This is done via the interpretation of this surface as a moduli space of degree three covers of the projective line.

In this talk, we investigate the action of Hecke operators on automorphic forms through some graphs, known as graphs of Hecke operators. Geometric tools are raised to connect the problem of describe these graphs to calculate some products in the associated Hall algebra. In the case of elliptic function fields, we present an algorithm which describes the graphs.

In joint work with Parker and Paupert, we gave a construction of several non-arithmetic lattices in the isometry group of the complex hyperbolic plane, that produces all examples known to this day. Our original proof, which is based on the construction of explicit fundamental domains, relies heavily on computational tools. If time allows, I will sketch methods to get alternative proofs that no longer rely on the computer.

Together with Simon Drewitz, we showed recently that the 1-cusped quotient of the (real) hyperbolic 3-space by the tetrahedral Coxeter group Γ = [5, 3, 6] has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice Γ is incommensurable to any Gromov-Piatetski-Shapiro type lattice.

Our methods have their origin in the work of Colin Adams. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp. I shall present and provide a borad outline of the proof.

Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD), that assigns a topological signature, called a barcode, to any geometric tree (i.e, any finite binary tree embedded in R^3). We showed that the TMD effectively determines the reliability of clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.

We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the dearth of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized neurons are statistically indistinguishable from real neurons of the same type.

In this talk I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties, in which symmetric groups and Coxeter complexes play a key role.

This talk is based on joint work with Adélie Garin and Lida Kanari, as well as with Justin Curry, Jordan Desha, and Brendan Mallery, and on work of Adélie Garin and Benjamin Brück, building on earlier collaborations led by Lida Kanari.

We review how historically the problem of string phenomenology lead theoretical physics first to algebraic/differential geometry, and then to computational geometry, and now to data science and AI.

With the concrete playground of the Calabi-Yau landscape, accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades, we show how the latest techniques in machine-learning can help explore problems of physical and mathematical interest, from geometry, to group theory, to combinatorics and number theory.

Proof assistants, such as Coq, Isabelle, or Lean, are software tools which assist in rigorously expressing mathematical statements and their proofs in a formal logical language. The mathematics that has been formalized this way, ranges through many different fields. In this talk, after some general introduction, we will take a pragmatic "working number theorist" point of view and discuss some past, present, and potential future formalization work, focusing mostly (but not exclusively) on the Lean proof assistant.

Mathematical logic studies proofs as mathematical objects: existence, shape, classification, etc. Yet these formal proofs are very far from the demonstrations that constitute the contemporary mathematical writing, as rigorous as they might be. On the other hand, formal proofs provide data structures that can be processed by computers, so that they can be constructed, observed, verified, by mechanical means. Proof assistants are pieces of software designed for performing this nature of operations, in practice and in the large. In this talk we will try to provide some hints of the mathematics that can be done with the help of a proof assistant, and of the benefits one can expect from this activity.

In this talk, which is based on joint work with Luca Notarnicola, I will present the Hidden Lattice Problem (HLP), which is the task of recovering a "small" lattice from the knowledge of only one or a few of its vectors. This problem can be traced back at least to the work on the Hidden Subset Sum Problem by Nguyen and Stern, who also came up with the "orthogonal lattice attack" for solving this kind of problem. The main novelty that I am going to discuss and illustrate is an alternative algorithm for the HLP.

Given a group acting on a topological space it often useful to have a “nice” set of representatives, a so-called fundamental domain, for this action. In practice it is also useful to not only know that such a domain exists, but also to know exactly how to reduce a given point to its representative.

For the modular group, PSL(2,Z), a number of fundamental domains and associated reduction algorithms have been known for a long time and are relatively simple to describe.

In the case of the Hilbert modular group PSL(2,O), where O is the ring of integers of a totally real number field, the fundamental domain is harder to describe geometrically but an algorithmic description has been known in principle since works of Blumenthal, Maass and others. Until recently, however, no explicit (finite-time) reduction algorithm has been known in the case of class number greater than one.

The aim of this talk is to present some of the motivations and the recent development and implementation of a new reduction algorithm for Hilbert modular groups, valid for any class number and degree.

We describe an algorithm that we used to compute the q-expansions of all weight 2 cusp forms of prime level at most 2,000,000 and dimension at most 6, and to verify that these are all but one form per Atkin-Lehner eigenspace. Our algorithm is based on Mestre's Méthode des Graphes, and involves supersingular isogeny graphs and Wiedemann's algorithm for finding the minimal polynomial of sparse matrices over finite fields.

In this talk, I will establish a sharp bound on the growth of cuspidal Bianchi modular forms. By the Eichler-Shimura isomorphism, we actually give a sharp bound of the second cohomology of a hyperbolic three manifold (Bianchi manifold) with local system rising from the representation Sym^k \otimes \overline{Sym^k} of SL_2(C). I will explain how a p-adic algebraic method is used for deriving our result.

For a commutative ring R, the integral homology groups of SL_2(R) are naturally modules over the group ring of the group of units modulo squares. We will explain how this action can be understood and exploited to calculate the third homology of SL_2(Q) with half-integer coefficients. We will discuss connections with K-theory, scissors congruence groups and homology stability questions.

Instigated by work of Borel and Bloch, Zagier formulated his Polylogarithm Conjecture in the late eighties and proved it for weight 2. After a flurry of activity and advances at the time, notably by Goncharov who provided not only a proof for weight 3 but set out a vast program with a plethora of conjectural statements for attacking it, progress seemed to be stalled for a number of years. More recently, a solution to one of Goncharov's central conjectures in weight 4 has been found. Moreover, by adopting a new point of view, work by Goncharov and Rudenko gave a proof of the original conjecture in weight 4.

In this impressionist talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and, time permitting, also hint at a number of recent results for higher weight with new formulas for Grassmannian and Aomoto polylogarithms in terms of iterated integrals (joint with S.Charlton and D.Radchenko).

I will explain how it is possible to prove various congruences of Hecke eigenvalues, between Siegel cusp forms of genus 2 and paramodular level, and genus 1 cusp forms, including some of a type conjectured by Harder, for which Fretwell obtained computational evidence, and some of a type discovered by Buzzard and Golyshev. Exploiting the recent proof by Roesner and Weissauer of Ibukiyama's genus 2 Jacquet-Langlands correspondence, and my joint work with Pacetti, Rama and Tornaria, relating algebraic modular forms for GU2 of a definite quaternion algebra and for O(5), we can prove several examples using linear algebra computations.

Let S be a Shimura variety (e.g. the moduli space Ag of principally polarized abelian varieties of dimension g) and let V be an irreducible algebraic subvariety of S contained in no proper Shimura subvariety. The Zilber-Pink conjecture predicts that the intersection Y of V with the Shimura subvarieties (e.g. the loci of abelian varieties with additional endomorphisms) of codimension less than dim V is contained in a proper subvariety of V (in other words, it is non-Zariski dense in V) — it is known as a problem in unlikely intersections.

The Zilber-Pink conjecture is, so to speak, wide open. Primarily, this is because of its arithmetic complexity — in some sense, the geometric aspect of the problem is now resolved. Indeed, when V is a curve, the conjecture follows from two arithmetic hypotheses: (1) the large Galois orbits conjecture, and (2) the parametrization problem. The large Galois orbits conjecture calls for a lower bound on the Galois orbits of the points in Y. The parametrization problem calls for an upper bound on the complexity of data parametrizing Shimura subvarieties.

In this talk, I will survey ongoing programmes with Martin Orr (University of Manchester) aimed at problems (1) and (2), respectively, which have yielded unconditional cases of the Zilber-Pink conjecture in Ag.

A classical result of Deligne shows that nontrivial central extensions of integral symplectic groups are not residually finite. We explore the case of mapping class groups and compute the Schur multiplier of finite symplectic groups.

I will discuss recent large-scale computations, which utilize numerical linear algebra and highly distributed, high-performance computing to generate data about the syzygies of various algebraic surfaces. Further, I will discuss how this data has led to several new conjectures

I will discuss joint work with Carel Faber, Soren Galatius, and Sam Payne in which we prove a formula, conjectured by Zagier in 2008, for the S_n equivariant top-weight Euler characteristics of the moduli spaces of n-marked, genus g algebraic curves. Our techniques involve tropical geometry and graph complexes.

Deciding whether two lattices over orders of finite-dimensional algebras over number fields is a classical problem in algorithmic number theory. We present a new algorithm for this problem, assuming that the Wedderburn decomposition of the algebra is "nice". As an application we discuss the connection to the similarity problem for integral matrices (the conjugacy problem in GL(n,Z)).

The resulting algorithm for the latter problem is the first with proven complexity and performs very well in practice. This is joint work with Werner Bley and Henri Johnston.

Locally symmetric spaces are generalizations of modular curves, and their cohomology plays an important role in the Langlands program. In this talk, I will first speak about vanishing conjectures and known results about the cohomology of locally symmetric spaces of a reductive group G with mod p coefficient after localizing at a maximal ideal of spherical Hecke algebra of G and after that, I will explain a sketch of my proof for the case G = GL_2(F), where F is a CM field.

In a previous paper with Ichino, we showed that the Jacquet-Langlands correspondence for Hilbert modular forms, all of whose weights are at least two, preserves rational Hodge structures. In this talk, I will discuss some work in progress (with Ichino) on the case of weight one forms. Since weight one forms are not cohomological, it is not clear how to formulate an analogous result. I will explain the formulation, which is suggested by another recent development, namely the conjectural connection between the motivic cohomology of adjoint motives and the cohomology of locally symmetric spaces.

I will explain how to complete and extend an argument proposed by F. Calegari for determining the F_p-cohomology of SL_n(Z, p^m) in a certain range (namely in cohomological degrees * < p and for all large enough n). The result has a uniform description at regular primes, but at irregular primes has interesting correction terms, controlled by torsion in K_*(Z) and by special values of the p-adic L-function. The argument for m>1 turns out to be almost trivial, but for m=1 it involves a delicate analysis of the cohomology of the finite groups SL_n(Z/p) with coefficients in certain modular representations. The talk is based on the preprint arXiv:2203.01697.

I will make a partial survey of what is expected and known about the degeneration loci of l-adic local systems over varieties over number fields. For l-adic local systems arising from geometry, understanding these degeneration loci is closely related to describing the variation of certain algebraic-geometric invariants (those encapsulated in l-adic cohomology) in algebraic families of smooth proper varieties.

In the 70s Deligne gave a topological formula for the local epsilon factors attached to an orthogonal representation. We consider the case of a symplectic representation and present a conjecture giving a topological formula for a finer invariant, the square class of its central value. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds, and give a sketch of the proofs. This is joint work with Akshay Venkatesh.

In this joint work with Michael Lipnowski, we describe an algorithm that computes the cohomology of a given compact arithmetic manifold together with the action of Hecke operators.

It has been known since the 1930s that for all positive rational numbers p/q, there exist holomorphic modular forms on SL(2,R) with weight p/q. This contrasts with the situation for Sp(2n,R) with n >1, where one has only integral and half-integral weight forms. Until recently, it was an open question whether there is any other Lie group (other than SL_2(R)) with holomorphic modular forms whose weight is neither integral nor half-integral. In this talk I will describe how we recently found examples of holomorphic modular forms of weight 1/3 on the group SU(2,1).

This is joint work with Eberhard Freitag.

In the early 2000s, Darmon initiated a fruitful study of analogies between Hilbert modular surfaces and quotients Y := SL_2(ZZ[1/p]) \ H x H_p, where H is the complex upper half plane and H_p is Drinfeld's p-adic upper half plane. As Y mixes complex and p-adic topologies, making direct sense of Y as an analytic space seems difficult. Nonetheless, Y supports a large collection of exotic special points - corresponding to the units of real quadratic fields which are inert at p - and Darmon-Vonk have described an incarnation of meromorphic functions on Y, so called rigid meromorphic cocycles.

This talk describes joint work with Henri Darmon and Lennart Gehrmann, in which we study generalizations Y' of the space Y to orthogonal groups G for quadratic spaces over QQ of arbitrary real signature. The spaces Y' support large collections of exotic special points - corresponding to subtori of G of maximal real rank - and we define explicit rigid meromorphic cocycles on Y'; these RMCs are analogous to meromorphic functions on orthogonal Shimura varieties with prescribed special divisors first studied by Borcherds, and they generalize the RMCs constructed by Darmon-Vonk. We will also discuss some computations suggesting that values of our RMCs at special points might realize new instances of explicit class field theory.

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