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.

As it is well-known by epoch-making work of Franke, the cohomology of arithmetic (congruence) subgroups of a reductive group G can be expressed as the relative Lie algebra cohomology of a space of automorphic forms A(G). In this talk we will show how to use Franke’s filtration of A(G) in order to provide a description of automorphic cohomology in low degrees. These results of ours improve certain bounds of vanishing, established by Borel and also by Zucker, and strengthen a non-vanishing result of Rohlfs-Speh.  

The following phenomena have been observed for hyperbolic 3-manifolds M : in the first homology group H_1(M) (with ZZ coefficients) the free part tends to have a small rank while the torsion subgroup tends to be quite large. In arithmetci setting Bergeron and Venkatesh give a precise quantitative statement about the asymptotic size of the torsion part in terms of the hyperbolic volume of the manifold, as well as some more tetative heuristics for its finer structure. In fact they provide such statements for arithmetic lattices in all symmetric spaces. Proofs remain elusive but there have been a number of efforts to numerically verify the first conjecture, in particular in the setting of arithmetic lattices in hyperbolic 3-space (by Şengün, Calegari--Dunfield and others). I will spend most of the talk giving details for all the above, and i will finish by reporting on difficulties arinsing when numerically testing the conjecture for higher-dimensional hyperbolic spaces.

We discuss an analogue of the period polynomial we have associated with values of derivatives of L-functions. We state a conjecture for the location of its zeros and provide evidence for its validity, including some proved special cases. This is joint work with L. Rolen.

I will discuss recent progress towards establishing modularity of elliptic curves over CM number fields, particularly imaginary quadratic fields. One way of phrasing "modularity" in this context is that the L-function of the elliptic curve can be described in terms of eigenvalues of Hecke operators on the cohomology of arithmetic subgroups of SL(2,C). The new results I will talk about are joint work with Ana Caraiani.

First, we will introduce cohomology of GL(2,Z) and its relations to modular forms of the group SL(2,Z).

Then, we will present explicit result of our computations of the (Eisenstein) cohomology of the GL(3,Z) with coefficients in any irreducible finite dimensional highest weight representation. When the presentation is not self dual, this is the entire group cohomology. It is a joint result with Harder, Bajpai and Moya Guisti. It is based on the Borel-Serre compactification, Kostant formula, Euler characteristics of arithmetic groups and Poincare duality. We have applied similar techniques for the computation for the cohomology of Sp(4,Z) with coefficients in irreducible highest weight representations (a joint result with Bajpai and Moya Giusti). I will mention it briefly.

After that, I will present an older result  of mine on cohomology of GL(4,Z) with coefficients in the standard representation twisted by the determinant, based on the same ideas. It has a current continuation that has surprising consequences for the cohomology of GL(3,Z). From the current computations, it follows that there is a ghost class in H^2(GL(3,Z), M) where M is the dual of the standard representation of GL(3,Z). Having a ghost class means that the cohomological class in GL(3,Z) is not generated by a maximal parabolic subgroup. In this case, it is generated by a minimal parabolic subgroup - the Borel subgroup. 

Given a number field F with ring of integers O, one can associate to any torsion free subgroup of SL(2,O) of finite index a complete Riemannian manifold of finite volume with fibered cusp ends.  For natural choices of flat vector bundles on such a manifold, we will explain how analytic torsion can be related to Reidemeister torsion.  As an application, we will indicate how, in some arithmetic settings, this relation can be used to derive exponential growth of torsion in cohomology for various sequences of congruence subgroups.  This is an ongoing joint work with Werner Mueller.  

For n>1, congruence covers X(m) of the locally symmetric space SL(n+1, Z) \ SL(n+1, R) / SO(n+1) encode the information of all finite covering spaces. We will use geometric and arithmetic methods to determine lower bounds on the growth, as a function of m, of the dimension of a subspace rational homology groups H_n(X(m); Q) spanned by cycles represented by flat submanifolds. This builds on work of, and addresses a question of, Avramidi and Nguyen-Phan, who showed that the homology of such covers arising from flat cycles grows arbitrarily large. The proof of our result combines their techniques with perspective of Millson--Raghunathan and a topological argument of Xue, along with concrete number theoretic constructions. We will also mention similar results about orthogonal groups and Hilbert modular groups, following work of Tshishiku and Zschumme. This work is joint with Bena Tshishiku.

While the rational cohomology of arithmetic groups such as SL(n, Z) and Sp(2n, Z) can often be completely computed if the cohomological degree is small compared to n, little is known about it in high cohomological degrees. In this talk, I will discuss vanishing results that have recently been obtained for the high-dimensional rational cohomology of SL(n, Z), Sp(2n, Z) and other arithmetic Chevalley groups. This is related to a conjecture of Church--Farb--Putman and based on joint works with Brück--Miller--Patzt--Wilson, Brück--Patzt and Brück--Santos Rego.

How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain both general upper (and in a few cases, show that they are sharp) for lattices in unitary groups using Arthur’s stable trace formula. This is joint work with Rahul Dalal. 

I will report on rationality results on the ratios of critical values for Rankin-Selberg L-functions for GL(n) x GL(m) over a totally imaginary base field. In contrast to a totally real base field, when the base field is totally imaginary, some delicate signatures enter the reciprocity laws for these special values. These signatures depend on whether or not the totally imaginary base field contains a CM subfield. The proof depends on a generalization of my work with Günter Harder on rank-one Eisenstein cohomology for GL(N) where N = n+m. The rationality result comes from interpreting Langlands’s constant term theorem in terms of an arithmetically defined intertwining operator between Hecke summands in the cohomology of the Borel-Serre boundary of a locally symmetric space for GL(N). The signatures arise from Galois action on certain local systems that intervene in boundary cohomology.

We show some of the similarities and some of the differences between irreducible lattices in product of semisimple Lie groups and their siblings in product of locally compact groups.  In the case of product of trees, we give a concrete example with interesting properties, among which some in terms of bounded cohomology and quasimorphisms. 

Joint work with Yasuda (Hokkaido U).  Let F be a global field of positive characteristic and v a place of F. We study automorphic forms for GL(d) over F such that the v-component of the associated automorphic representation is isomorphic to the Steinberg representation. We introduce modular symbols in this context and show that the modular symbols generate the space of such automorphic forms with Q-coefficients.   We also have some results with Z-coefficients.

(joint with Jared Weinstein) The ideas of Wiles on the modularity of elliptic curves over Q, and subsequent extensions and adaptations, have had a great influence on the study of Diophantine equations through the modular method.  There is an analogous concept for elliptic curves over function fields over finite fields, known as Drinfeld modularity: an elliptic curve over F_q(t) with split multiplicative reduction at infinity is covered by a Drinfeld modular curve, which parametrizes Drinfeld modules of rank 2 with a suitable level structure.  More generally, let E be an elliptic curve over F_q(t), and let E_i be the elliptic curve over F_q(t_1,...,t_n)$ obtained by replacing t by t_i.  Then there is an n-dimensional moduli space of "shtukas" over F_q(t_1,...,t_n) that is conjectured to be in correspondence with E_1 x .... x E_n.  We describe how to construct these moduli spaces concretely as the sets of 2 x2 matrices of polynomials satisfying certain specialization conditions and prove the conjecture in a few special cases by means of computations on K3 surfaces.

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A rigid meromorphic cocycle is a class in the first cohomology of the group SL(2, Z[1/p]) acting on the non-zero rigid meromorphic functions on the Drinfeld p-adic upper half plane by Moebius transformation. Rigid meromorphic cocycles can be evaluated at points of "real multiplication'', and their values conjecturally lie in composita of abelian extensions of real quadratic fields, suggesting striking analogies with the classical theory of complex multiplication.

In this talk, we discuss the proof of this conjecture for a special class of rigid meromorphic cocycles. Our proof connects the values of rigid meromorphic cocycles to the study of certain p-adic variations of Hilbert modular forms. This is joint work with Henri Darmon and Jan Vonk.

The monodromy groups of hypergeometric differential equations of type nFn-1 are often called hypergeometric groups. These are subgroups of GL(n). Recently, the arithmeticity and thinness of these groups have caught a lot of attention. In the talk, a gentle introduction and recent progress in the theory of hypergeometric groups will be presented. 

I will discuss several older and more recent results about relationships between arithmetic groups and diffeomorphism groups of high-dimensional manifolds, which in turn relate their cohomology groups. This includes joint work with Oscar Randal-Williams, Mauricio Bustamente, and Manuel Krannich.

Hilbert modular varieties are Shimura varieties attached to GL(2)

over a totally real field, generalizing modular curves. I will discuss

on-going work with Matteo Tamiozzo, whose aim is to understand the

cohomology of Hilbert modular varieties with torsion coefficients. I will

focus on a result known as Ihara's lemma, which leads to a

representation-theoretic description of the cohomology. I will explain a

phenomenon known as geometric Jacquet-Langlands which plays a key role in

our proof.

A lattice is called virtually rectangular if it contains an orthogonal sublattice of finite index. We establish necessary and sufficient conditions for a lattice to be virtually rectangular and determine the smallest index of an orthogonal sublattice. This investigation is closely connected to the study of sparsity and a certain sparse analogue of Minkowski’s successive minima theorem. In the 2-dimensional case, our results imply certain isogeny properties of elliptic curves.

In this seminar I will discuss recent work studying the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology. We show that the decision version of this problem is QMA1-hard - where QMA1 is a quantum version of the classical complexity class NP. Moreover, we show that a version of the problem satisfying a suitable promise and certain constraints is contained in QMA (a slightly different quantum analogue of NP). This suggests that the seemingly classical problem may in fact be quantum mechanical. In fact, we are able to significantly strengthen this by showing that the problem remains QMA1-hard in the case of clique complexes, a family of simplicial complexes specified by a graph which is relevant to the problem of topological data analysis. The proof combines a number of techniques from Hamiltonian complexity and homological algebra. I will discuss potential implications for the problem of quantum advantage in topological data analysis. 

Let A be an abelian variety over a number field. The connected monodromy field of A is the minimal field over which the images of all the l-adic torsion representations have connected Zariski closure. During this talk, we will show that for all even g >3, there exist infinitely many geometrically nonisogenous abelian varieties A over the rationals of dimension g where the connected monodromy field is strictly larger than the field of definition of the endomorphisms of A. Our construction arises from explicit families of hyperelliptic Jacobians with definite quaternionic multiplication. This is a joint work with Lombardo and Voight.

A general conjecture of Harder relates the denominator of the Eisenstein cohomology of certain locally symmetric spaces to special values of L-functions. In this talk, we consider the locally symmetric space associated with SL(2,K) where K is an imaginary quadratic field. I will explain how results of Ito and Sczech can be used to prove an upper bound on the denominator in terms of a special value of a Hecke L-function. When the class number of K is one, we can combine this result with a lower bound obtained by Berger to get the exact denominator.