I began and continued to organize the Langlands Program Seminar at the CUNY Graduate Center between the spring of 2014 and the summer of 2016, consisting of a mixture of introductory and research level talks. Unless otherwise indicated, I have given the talks listed below. At various points in time, our seminar has been financially supported by the CUNY Doctoral Students Council, and the Number Theory Foundation.

Sep 2 Speaker: Edmund Karasiewicz (Rutgers New Brunswick) Title: TBA Abstract: TBA

Jul 29 Speaker: Joe Kramer-Miller (GC/UCL) Title: The monodromy of F-isocrystals with log-growth singularities Abstract: Let U be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification X. Following Dwork and Katz, a $p$-adic representation $\rho$ of $\pi_1(U)$ corresponds to an F-isocrystal. By work of Tsuzuki and Crew an F-isocrystal is overconvergent precisely when $\rho$ has finite monodromy. However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities. We give a Galois-theoretic interpretation of these log growth F-isocrystals in terms of asymptotic properties of higher ramification groups.

Jul 22 Speaker: Joe Kramer-Miller (GC/UCL) Title: p-adic Galois representations and p-adic differential equations Abstract: I'll explain Katz' analogue of the Reimann-Hilbert correspondence for schemes in positive characteristic. In the case of curves, I will explain how this locally reduces to Fontaine's theory of $\phi$-modules over a certain ring of Laurent series. I'll explain the notion of overconvergence and Tsuzuki's work on overconvergent F-isocrystals. I'll give a few simple examples and relate everything to modular forms. I will try and keep things as hands on as possible! References: Katz - Travoux de Dwork, Katz - p-adic properties of modular forms and modular schemes, Deligne - Weil I paper, Tsuzuki - Unit root F-isocrystals with finite monodromy, Kedlaya - p-adic Local Monodromy expository paper.

Jun 24 Speaker: Joseph Kramer-Miller (GC/UCL) Title: Elliptic curves in positive characteristic Abstract: I will discuss the classical theory of supersingular and ordinary elliptic curves in characteristic p. Using Riemann-Roch and Artin-Shreier theory I will explain how the Hasse invariant relates to the phenomenon of being supersingular.

May 12 Title: Chromatic homotopy theory for number theorists Abstract: Chromatic homotopy theory is powerful method in the study of stable homotopy groups of spheres. Surprisingly, arithmetic data has arisen in this context, for example in Adams' conjecture on the image of the J-homomorphism one finds Bernoulli numbers. On the other hand, the connection between chromatic homotopy theory and complex cobordism, which, as we saw, is related to modular forms and carries a formal group law. This in turn leads to a certain p-adic division algebra. In this talk we will introduce the basic definitions and discuss these connections. References: Ravenel, Complex cobordism theory for number theorists; Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres.

May 5 Time: 12 to 1pm (note time change!). Speaker: Ken Kramer (Queens College and GC) Title: On semistable abelian surfaces of given conductor: Existence, non-existence, uniqueness of Isogeny Class Abstract: This is joint work with Armand Brumer. We use arithmetic information from division fields of a possible abelian surface A over Q of prime conductorN for the following objectives: (1) We show that if A exists, it is the Jacobian of a curve (of genus 2). (2)For certain conductors and certain Galois module structures of division points, we show that there is no abelian surface. We develop criteria under which there is exactly one isogeny class of abelian surfaces with a given 2-division field and prime conductor N. By our previous non-existence results, the first prime conductor of an abelian surface is N = 277 and we know an example. Our criteria now imply that it represents the unique isogeny class of abelian surfaces with N = 277. According to our Paramodular Conjecture, there should be a unique paramodular eigenform (up to scalar multiple) of suitable type for N = 277, as Chris Poor and David Yuen have demonstrated. Moreover, Brumer, Poor and Yuen, Eric Urban, John Voight have combined to show that the Galois representation associated to the paramodular form agrees with that of the abelian surface. This is the first full verification of the Paramodular Conjecture in a non-endoscopic case.

Apr 26 Title: Manifolds and modular forms Abstract: A basic problem in algebraic topology is to study invariants of manifolds. In particular, one may study manifolds up to (co)bordism, where under this equivalence these manifolds form a ring. Now, invariants of manifolds can be viewed as functions on this ring, and it was observed that such functions are in fact modular forms. We discuss this phenomenon, and the appearance of other objects of number theoretic interest, for example formal groups and Bernoulli numbers. Reference: Ravenel, Complex Cobordism Theory for Number Theorists. Hopkins, Algebraic Topology and Modular Forms. Hirzebruch, Manifolds and Modular Forms.

Apr 7 Speaker: Heekyoung Hahn (Duke University) Title: Langlands' beyond endoscopy proposal and the Littlewood-Richardson semigroup Abstract: Langlands' beyond endoscopy proposal for establishing functoriality motivates the study of irreducible subgroups of GL(n) that stabilize a line in a given representation of GL(n). Such subgroups are said to be detected by the representation. In this talk we study the important special case where the representation of GL(n) is the triple tensor product representation. We prove a family of results describing when subgroups isomorphic to classical groups of type B_n, C_n, D_2n are detected.

Mar 31 Title: Survey of the Stark conjectures Abstract: Dirichlet's class number formula expresses the leading term of the Taylor expansion of the zeta function of a number field at s = 0. In the 70s Stark formulated certain conjectures regarding the shape of the leading term of the Artin L-function associated to a finite extension of numbers fields, refining the class number formula. We survey the conjectures of Stark, and as time permits, the reformulation by Tate and idea of Brumer.

Mar 24 No meeting today

Mar 17 No meeting today

Mar 10 Speaker: Jayce Getz (Duke University) Title: Limiting forms of trace formulae and triple product $L$-functions Abstract: Langlands has proposed studying limits of trace formulae as an approach to proving Langlands functoriality in general. The proposal has only been carried out in very special cases corresponding more or less to the standard representation and symmetric square representation of GL(2) and the tensor product of GL(2) with itself. We explain how the analytic part of the proposal can be carried out in the case of the tensor product of three copies of GL(2). Time permitting, we discuss the prospects of generalizing the approach to the tensor product of three copies of general linear groups of arbitrary rank.

Mar 3 Speaker: Saul Glasman (IAS) Title: A survey of the relationship between algebraic K-theory and L-functions Abstract: I'll give a brief introduction to algebraic K-theory, and I'll endeavor to explain, with reference to the examples provided finite fields and smooth projective curves over them, why one might expect K-groups to be related to special values of L-functions. I'll draw some analogies between topological properties of K-theory and algebraic and analytic properties of L-functions. If time permits, I'll mention recent work of Connes-Consani and Hesselholt which gives new perspectives using related invariants such as cyclic homology.

Feb 25 Title: Remarks on higher reciprocity laws Abstract: We follow up Karpanov's 1995 paper discussed last week with Parshin's 2012 paper, which revisits Karpranov's framework for a 2-dimensional Langlands correspondence, i.e., for 2-dimensional local fields. Parshin discusses the various local and global fields in consideration, functorial properties that follow from Kapranov's framework, and connections with the geometric Langlands correspondence. We will also comment on higher reciprocity laws, in the spirit of Kato and also Milnor. Reference: Parshin, Questions and remarks to the Langlands Program.

Feb 18 Title: Excerpts from Analogies between The Langlands Correspondence and TQFT Abstract: We discuss some aspects of M. Kapranov's paper Analogies between the Langlands Correspondence and Topological Quantum Field Theory (TQFT). In particular, we will lightly touch on the TQFT, and focus on the perspectives on the Langlands Correspondence, for example possible formulations for an n-dmiensional correspondence for higher local fields, using higher categories. No background will be assumed for this talk. References: M. Kapranov, Analogies between the Langlands Correspondence and Topological Quantum Field Theory

Feb 11 Speaker: Ivan Horozov (Bronx Community College) Title: Non-abelian reciprocity laws on Riemann surfaces Abstract: Abelian reciprocity laws in an arithmetic setting such as the quadratic reciprocity law have geometric analogues. Weil reciprocity on a curve (or on a Riemann surface) is such an analogue. In this talk we present a generalization of Weil reciprocity to non-abelian reciprocity laws. A key role in this construction is played by the notion of Chen's iterated integrals. Double iteration recovers Weil reciprocity. Higher order iterations lead to new non-abelian reciprocity laws on Riemann surfaces, which will be the heart of the talk. As far as I know there are no arithmetic analogues to these reciprocity laws. There are two candidates for 3 and for 4 primes which are guided by solvable Galois groups.

Feb 4 Speaker: Jorge Florez (Graduate Center) Title: Explicit reciprocity laws and the BSD conjecture. Abstract: In this talk we will continue with Tian An's chronological exposition of the reciprocity laws and explain the role of these in the development of the BSD and Iwasawa Main conjectures. In particular, this will lead us to the work of Kato and Kolyvagin on Euler systems, which will be mentioned briefly at the end of the talk.

Jan 28 Title: A concise history of reciprocity Abstract: We provide an informal introduction to reciprocity laws, beginning with the quadratic reciprocity law of proved by Gauss, followed by higher reciprocity laws on the way to class field theory, i.e., Artin reciprocity. From here the search for nonabelian reciprocity laws has found several generalizations, the ones which we will consider are those of Langlands and Kato. Time permitting, we will outline the suggestion of Kapranov unifying these two. This talk is intended to be informal and accessible to graduate students, with an emphasis on the development of ideas rather than proofs. References: Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein; Kapranov, Analogies between the Langlands Correspondence and Topological Quantum Field Theory

Dec 8 Title: Motivic homotopy theory for dummies Abstract: The topology of algebraic varieties is traditionally understood either with the Euclidean topology of their real or complex points, or with the Zariski and étale topologies. In the latter, Grothendieck has given us motives which ought to be universal objects for various cohomology theories. The former is more recent: Voevodsky has taught us how to do homotopy theory for algebraic varieties, which relate to algebraic cycles and K-theory. In this informal talk I will present a non-technical introduction to Voevodsky and Morel's motivic or A1-homotopy theory. References: Voevodsky's Nordfjordeid Lectures on Motivic Homotopy Theory; Motivic Homotopy Theory, Marc Levine.

Dec 16 Title: Singularities and Transfer, II Abstract: We discuss paper of Langlands, Singularités et Transfert, which is the sequel to the paper of Frenkel, Langlands, and Ngô. The principal purpose of these papers is to introduce the use of the Poisson formula in combination with the stable transfer as a central tool in the development of the stable trace formula and its applications to global functoriality. In this second paper, Langlands examines the transfer from an elliptic torus to SL(2), which reduces to questions for harmonic analysis on reductive groups over local fields. Reference: Langlands, Singularités et Transfert, 2013.

Dec 9 Title: Singularities and Transfer Abstract: We discuss paper of Langlands, Singularités et Transfert, which is the sequel to the paper of Frenkel, Langlands, and Ngô. The principal purpose of these papers is to introduce the use of the Poisson formula in combination with the stable transfer as a central tool in the development of the stable trace formula and its applications to global functoriality. In this second paper, Langlands examines the transfer from an elliptic torus to SL(2), which reduces to questions for harmonic analysis on reductive groups over local fields. Reference: Langlands, Singularités et Transfert, 2013.

Dec 2 No Meeting today

Nov 25 Title: The Arthur trace formula beyond endoscopy Abstract: Following Ali's overview of the story so far, we present Arthur's interpretation of the Beyond Endoscopy proposal, in particular a primitive trace formula and "r-trace formula" that refines the stable trace formula, which was established by the Fundamental Lemma. Time permitting, we will outline some problems suggested by Arthur, related to establishing this new trace formula.

Nov 18 Speaker: S. Ali Altuǧ Title: An overview of beyond endoscopy and related problems Abstract: In the first part of the talk I will give an overview of beyond endoscopy from a global perspective. This will more or less amount to a summary of the discussions of the seminar for this term. If time permits I will also discuss some more recent results about elliptic curves which can be obtained along similar lines.

Nov 11 No meeting today

Nov 4 Title: Remarks on Poisson Summation Abstract: We explore current ideas on applying Poisson summation formula on the geometric side of the trace formula, and on the way review its relationship to functional equations and integral representations of L-functions. As time permits we discuss connections to monoids.

Oct 28 Title: Trace Formula and Functoriality III: Trivial representation Abstract: We conclude the FLN paper with the removal of the contribution of the trivial representation to the trace formula using the dominant term in the Poisson summation formula. References: Frenkel, Langlands, Ngô, Formule des Traces et Fonctorialité: le Début d'un Programme, 2010.

Oct 14 Title: Trace Formula and Functoriality II: Poisson summation Abstract: Continuing with the paper of Frenkel, Langlands and Ngô (FLN), we examine the geometric side of the trace formula, particularly, the geometric setup for the Poisson summation formula. As in FLN, we will aim to isolate the dominant term in the summation formula. References: Frenkel, Langlands, Ngô, Formule des Traces et Fonctorialité: le Début d'un Programme, 2010.

Oct 7 Title: Trace Formula and Functoriality I: The setup Abstract: Having surveyed the Beyond Endoscopy proposal and Altuǧ's analysis of the simplest case of GL(2), we embark on the paper of Frenkel, Langlands and Ngô (FLN) initiating a geometric approach to Beyond Endoscopy for a general reductive G. In this talk we will summarize the setup in FLN required to perform Poisson summation, in order to again analyze the dominant term of the summation formula, and also the contribution of the trivial representation. References: Frenkel, Langlands, Ngô, Formule des Traces et Fonctorialité: le Début d'un Programme, 2010.

Sep 30 No meeting today

Sep 23 Title: GL(2) elliptic terms: Approximate equation and Poisson summation Abstract: Last week we placed the elliptic terms in the GL(2) trace formula into an explicit form, mainly following the exposition of Langlands. This is the starting point of Altug's analysis of the elliptic part, in particular, using an approximate equation and Poisson summation formula. We summarize this method, with an eye towards the dominant term in the Poisson summation. Reference: S. Altuǧ, Beyond Endoscopy via the Trace Formula I, 2015.

Sep 16 Title: GL(2) elliptic terms: the setup Abstract: Langlands' idea of approaching functoriality is to use the order of the poles of L-functions, and summing over all automorphic representations, one arrives at a family of trace formulas. Using GL(2) for the most basic example, through a limit process one shows that most of the terms in the trace formula tend to zero. The most difficult piece to analyze is the elliptic contribution, which is addressed in Ali Altuǧ's thesis. In this talk I will give an overview of his analytic method, using an approximate equation and Poisson summation formula. Reference: S. Altuǧ, Beyond Endoscopy via the Trace Formula I, 2015.

Sep 9 Title: Beyond Endoscopy, II Abstract: Last week we surveyed Langlands' idea of approaching functoriality using the order of the poles of L-functions and summing over all automorphic representations. At the end of this process one arrives at a family of trace formulas. In this talk I will give the highlights of the second half of Langlands' Beyond Endoscopy proposal, where he studies the geometric side of this trace formula. Reference: R. P. Langlands, Beyond Endoscopy, 2004.

Sep 2 Title: Introduction to Beyond Endoscopy Abstract: Abstract: After Ngô's 2009 proof of the fundamental lemma, Langlands' Functoriality conjecture is now within reach for so-called endoscopic groups. In 2004, Langlands proposed a method to address cases that are 'Beyond Endoscopy'. I will present the main ideas of this proposal. Very little background will be assumed. Reference: R. P. Langlands, Beyond Endoscopy, 2004.

May 14 Title: The eigencurve Abstract: I will sketch the construction of the Coleman-Mazur eigencurve, and along the way describe the historical development various families of p-adic modular forms (Hida, Coleman, overconvergent) developed in the process.

May 7 No meeting today

Apr 30 Title: An infinite fern in the space of Galois representations Abstract: This is the 1997 paper of Mazur titled "An infinite fern in the universal deformation space of Galois representations". The article is described as a quick overview of how modular representations fit into the theory of deformations of Galois representations, as the image of an "infinite fern". The goal of this digression is to give an overview of some tools from the modularity theorem, find a relation back to motives.

Apr 23 Title: The conjectural Langlands group Abstract: Last week, we described a class of automorphic forms on GL(n) that are believed to behave like representations of some algebraic group. This was suggested by Langlands in 1979, and today remains mostly conjectural. We'll discuss some expected properties and consequences of this hypothetical group. Reference: Ramakrishnan, Motives and automorphic forms; Arthur, A note on the automorphic Langlands group.

Apr 20 Special seminar co-hosted with Dennis Sullivan's Einstein Chair Seminar Speaker: Mark Andrea de Cataldo (SUNY Stony Brook) Title: Perverse sheaves and some applications Time: 10:45–12:45 (class) and 2–3:30pm (seminar); Location: Room 6417 Abstract: I will first give a motivated introduction to perverse sheaves. I will then discuss a couple of geometric applications to representation theory and combinatorics.

Apr 16 Title: Motives and automorphic forms Abstract: Having developed the basic theory of motives, we now explore the conjectural connections between motives and automorphic forms. In particular, I'll describe the class of automorphic representations expected to satisfy the Tannakian formalism, leading to the so-called hypothetical automorphic Langlands group.References: Clozel, Motifs et formes automorphes (1989); Langlands, Automorphic forms, shimura varieties, and motives (1979).

Apr 9 No meeting — Spring break

Apr 2 Title: The Taniyama group Abstract: In this talk we'll explore the main properties of the Taniyama group. In 1979 Langlands introduced the Taniyama group as an extension of the so-called Serre group, parametrizing algebraic Hecke characters. On the other hand, Delgine showed using his theory of Hodge cycles that Taniyama group is the one given by the Tannakian duality for potentially abelian motives, arising from CM abelian varieties. A key result is that the L-function for such a motive is 'virtually automorphic'. References: Fargues, Motives and automorphic forms: the (potentially) abelian case

Mar 26 Speaker: Ray Hoobler (City College and CUNY Graduate Center)) Title: The Standard Conjectures Abstract: I will briefly define a Weil cohomology theory and then, using Hodge cohomology as a guide, state the standard conjectures. I will sketch how they would have proved the Weil conjectures if they had been demonstrated, but I will not venture a guess as to why so little is known about them, even today, other than Jannsen’s semi-simplicity result.

Mar 19 Title: The motivic Galois group Abstract: Assuming Grothendieck's standard conjectures, the category of pure motives is a Tannakian category, hence equivalent to the representations of a (pro)algebraic group, which Grothendieck called the motivic Galois group. We will make some cursory remarks about these; the talk will focus on giving examples of such groups. To motivate this from the point of view of automorphic forms, one expects a homomorphism from the motivic Galois group to the conjectural 'automorphic Langlands group', parametrizing automorphic representations. References: André, Une Introduction aux Motifs; Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques

Mar 4 No meeting this week — 29th Automorphic Forms Workshop

Feb 26 Title: What is a motive? Abstract: Having introduced briefly Galois representations, we now make our way towards Grothendieck's theory of motives. They make up 'pieces' of the cohomology of algebraic varieties, and thus carry a Galois action. I will start with some general remarks about what motives are and why they are interesting to us, then begin the selective journey through Y. André's book. References: Milne, Motives: Grothendieck's dream; André, Une Introduction aux Motifs.

Feb 19 Title: Introduction to Galois Representations, III Abstract: The final sections of Taylor's ICM talk introduces the automorphy of Galois represenations. I will give an informal introduction to automorphic representations, then discuss what we can say about Galois representations with them.

Feb 12 Title: Introduction to Galois Representations, II Abstract: Based on R. Taylor's ICM talk for non-experts, I will continue last week's introduction to Galois representations, stating the Fontaine-Mazur conjecture and how it relates to Langlands' reciprocity conjecture. Then we will start on to L-functions associated to Galois representations.

Feb 5 Title: Introduction to Galois Representations, I Abstract: Based on R. Taylor's ICM talk for non-experts, I will give an introduction to Galois representations with some basic examples. Time permitting we will state the Fontaine-Mazur conjecture. Reference: Taylor, Galois Representations.

Dec 3 Speaker: Ali Altuǧ (Columbia Univesity) Title: Beyond Endoscopy Abstract: I will talk about Beyond Endoscopy and explain some things related to it. The talk will be self contained so it should be easily accessible to more or less everyone in the audience.

Nov 26 Title: Towards the proof of the fundamental lemma II: geometric objects Abstract: We will give an idea of the proof of the fundamental lemma for Lie algebras in positive characteristic. In this talk we survey the key geometric objects used. In particular, the affine Springer fiber and Hitchin fibration have been shown to be the correct local and global analogues of orbital integrals. Several other objects arising from their study will also be important to the discussion. References: Chaudouard, The geometry of the fundamental lemma; Dat T. and Ngô, D.T. Lemme fondamentale pour les algèbres de Lie; Hales, The fundamental lemma and the HItchin fibration; Nadler, The geometric nature of the fundamental lemma; Ngô, B.C., Endoscopy theory of automorphic forms.

Nov 19 No meeting today

Nov 12 Title: Towards the proof of the fundamental lemma I Abstract: First encountered in the theory of endoscopy, the fundamental lemma is a more general obstruction in the comparison of trace formulas. We will give an idea of the proof of the fundamental lemma. In this talk I will describe the transfer of the lemma to Lie algebras in characteristic p, then define the local objects, particularly the affine Springer fiber and Picard stack. References: Dat T. and Ngô, D.T. Lemme fondamentale pour les algèbres de Lie; Ngô, B.C., Endoscopy theory of automorphic forms

Nov 5 Title: Introduction to the fundamental lemma II Abstract: We continue our introduction of the fundamental lemma via the theory of endoscopy.

Oct 29 Speaker: Brooke Feigon (CCNY/Graduate Center) Title: An introduction to the relative trace formula Abstract: We will give an introduction to the relative trace formula with various examples and applications.

Oct 22 Title: Introduction to the fundamental lemma I Abstract: The fundamental lemma has proven itself to be the key obstruction to the stabilization of the trace formula. Conjectured in the 1980s by Langlands and Shelstad, it was finally proven by the hands of many experts, culminating in Ngô's completion of it in 2008. He was awarded the Fields medal in 2010. The lemma belongs to what is now called the theory of endoscopy, which I will explain. In this talk I'll only motivate the problem and give the background; we will not discuss the proof in any depth. References: Arthur, An introduction to the trace formula; Labesse, Introduction to endoscopy; Ngô, Endoscopy theory of automorphic forms

Oct 15 Title: Noninvariant base change for GL(n) Abstract: Having outlined Langlands' proof of GL(2) base change, we continue on to GL(n). While Arthur and Clozel first proved the result using the invariant trace formula, I will outline a shorter proof by Labesse using the noninvariant trace formula. In particular, Labesse uses ideas from stability and endoscopy, which will be useful to us on the way to the stable trace formula. References: Labesse, Noninvariant base change identities (1995)

Oct 8 Title: Base Change for GL(2) Abstract: Given a cyclic extension of prime degree E/F of number fields, 'base change' is a lifting of automorphic representations from GL(2,F) to the group obtained by restriction of scalars of GL(2,E). Building on work of Saito and Shintani, Langlands proved the existence of such a lift for GL(2), followed by Arthur and Clozel for GL(n). These results leads to special cases of the Artin Conjecture. In this talk I will outline Langlands' proof of base change for GL(2), requiring the invariant trace formula. References: Arthur, An Introduction to the Trace Fromula, 2004; Arthur-Clozel, Simple algebras, base change, and the advanced theory of the trace formula (1990); Langlands, Base Change for GL(2), 1980.

Oct 1 Title: The (Refined) Invariant Trace Formula Abstract: Having achieved the coarse expansion of the Arthur-Selberg trace formula, we will forge ahead to refine the expansion and place it in invariant form, known as the Invariant Trace Formula. In particular, the trace formula is an identity of distributions that vary under conjugation, while for comparing trace formulas one would like to match invariant orbital integrals, as occurred in the Jacquet-Langlands correspondence. References: Arthur, An Introduction to the Trace Fromula, 2004; Gelbart, S., Lectures on the Arthur-Selberg trace formula, 1995.

Sep 24 Title: The Jacquet-Langlands correspondence for GL(2) Abstract: The Jacquet-Langlands correspondence describes certain representations of GL(2) and its inner forms. Specifically, it gives a bijection between automorphic representations with dim > 1 of a division algebra and cuspidal representations. While generalizations to GL(n) are known, the GL(2) case is the setting in which the method of comparing trace formulas was first used. In this talk we will give the main ideas of the proof, using the trace formula developed previously. References: Gelbart, S., and Jacquet, H., Forms of GL(2) from the analytic point of view, 1979; Gelbart, S., Lectures on the Arthur-Selberg trace formula, 1995.

Sep 17 Title: The coarse Arthur trace formula Abstract: Arthur's trace formula generalizes Selberg's trace formula to any reductive group. The first step in its development is known as the coarse expansion. I will describe the general formula by way of the example of GL(2) over the adele ring. If time permits I will give an idea of the Jacquet-Langlands correspondence, the first use of the adelic GL(2) trace formula. References: Gelbart, S., and Jacquet, H., Forms of GL(2) from the analytic point of view, 1979; Gelbart, S., Lectures on the Arthur-Selberg trace formula, 1995.

Sep 10 Title: The principle of functoriality and the role of trace formulas. Abstract: The principle of functoriality is the guiding light in all work surrounding what is now called the Langlands program, in particular the correspondence between Galois representations and automorphic representations. After stating the conjecture, I will focus on some key applications and outline the role of the trace formulas which we will survey.

Jun 25 Title: The Bruhat-Tits building of a p-adic reductive group. Abstract: We apply last week's discussion on root systems to the construction of the Bruhat-Tits building for a Chevalley group over a p-adic field, which generalizes to reductive groups. The construction will be illustrated by way of several explicit examples with classical groups. References: Rabinoff, The Bruhat-Tits building of a p-adic Chevalley group; Tits, Reductive groups over local fields.

Jun 18 Title: Roots systems and highest weight theory for complex reductive groups Speaker: Bart van Steirteghem (Medgar Evers) Abstract: We will continue our discussion of the root system of a complex reductive group (or a compact Lie group). I will then describe how root systems classify semisimple groups and how "root data" classify reductive groups. Finally, I will present the basic theorems of highest weight theory. I will include several explicit examples.

Jun 11 Title: Structure Theory for Compact Lie groups Speaker: Robert Donley Jr. (Medgar Evers) Abstract: We review the structure theory for Lie algebras associated to connected compact Lie groups. Topics include root systems and Dynkin diagrams.

Jun 4 Title: Structure theory of reductive groups Abstract: We review some basic structure theory of reductive groups. We discuss roots, tori, parabolic, Levi subgroups; and the behaviour of G over local and global fields. Reference: T.A. Springer, Reductive groups.

May 21

No meeting — David Vogan birthday conference at MIT.

May 14 Title: Tempered Representations and the Langlands Classification Speaker: Robert Donley Jr. (Medgar Evers) Abstract: We introduce further tools needed for the Langlands Classification. With tempered representations, one may generalize real parabolic induction and describe the classification of irreducible admissible representations in terms of the discrete series and their limits.

May 7 Title: Irreducible Admissible Representations Speaker: Robert Donley Jr. (Medgar Evers) Abstract: With the SL(2, R) example as a model for the Langlands classification, we expand the nonunitary principal series to general reductive Lie groups. To utilize, we review matrix decompositions, compact groups, and matrix coefficients.

Apr 30 Title: Three pictures for SL(2, R) Speaker: Robert Donley Jr. (Medgar Evers) Abstract: We continue our study of the irreducible unitary representations of SL(2, R). Where the group action was the unifying theme in the previous part, we now consider a uniform construction of the Hilbert spaces based on matrix factorizations. In particular, the induced picture gives the proper context for Langlands quotients and the Langlands Classification. References: Donley, Irreducible Representations of SL(2, R), Edinburgh conference (now with reprints); Knapp, Representation Theory: An Overview Based on Examples; Bump, Representation Theory and Automorphic Forms

Apr 23 Title: The Langlands Classification for SL(2, R) Speaker: Robert Donley Jr. (Medgar Evers) Abstract: As a model for the Langlands Classification, we first review the irreducible unitary representations of SL(2, R) and widen this class to the irreducible admissible representations. We view these representations using three pictures, which then lead to the integral intertwining operators that play a key role in the general classification. References: Donley, Irreducible Representations of SL(2, R), Edinburgh conference; Knapp, Representation Theory: An Overview Based on Examples; Bump, Representation Theory and Automorphic Forms

Apr 9 Title: Automorphic L-functions Abstract: We outline the construction of the L-function attached to an automorphic representation of a reductive algebraic group. These L-functions appear on the 'automorphic side' of the Langlads correspondence, which are expected to match with Artin's L-functions on the 'Galois side.' If time permits we will also mention the 'standard' L-functions for GLn, and its relation to the general theory. Main References: A. Borel, Automorphic L-functions; Cogdell, Murty and Kim, Lectures on Automorphic L-functions

Apr 2 Title: The Local Langlands Correspondence Abstract: Abstract: Having set up the basic machinery, besides the automorphic L-function which we will introduce briefly, we are now ready to state the local Langlands conjecture, relating admissible homomorphisms of the Weil(-Deligne) group to the L-group with irreducible admissible automorphic representations of a reductive group. The plan is to motivate the discussion by taking Artin reciprocity as a starting point, formulate the conjecture, then discuss progress towards the conjecture since its formulation. References: A. Borel, Automorphic L-functions; J. Cogdell, Langlands Conjectures for GLn and Dual Groups and Langlands Functoriality; A. Knapp, Introduction to the Langlands Program

Mar 26 Title: Weil-Deligne Group Abstract: We introduce the Weil-Deligne group and its connection with the L-group. References: J. Tate, Number theoretic Background; A. Borel, Automorphic L-functions; A. Knapp, Introduction to the Langlands Program.

Mar 19 No meeting today

Mar 12 Title: The Satake Isomorphism (cot'd) and the Weil Group Abstract: We continue the description of the Satake isomorphism, and if time permits we will also introduce the Weil group. References: Benedict Gross' On the Satake Isomorphism and Bill Casselman's The L-Group,

Mar 5 Title: The L-Group and the Satake Isomorphism Abstract: We outline the construction of the L-group, or the Langlands dual group of a connected reductive group, a key component of automorphic L-functions. Having outlined the construction of the Hecke algebra the previous week, we describe the Satake isomorphism which identifies algebra homomorphisms of the Hecke algebra with the coordinate ring of the dual torus, and its relation to the L-group. References: W. Casselman, The L-Group and A. Borel, Automorphic L-Functions.

Feb 26 Title: An Elementary Introduction to the Langlands Program Abstract: In this talk we give a rough outline of the Langlands Program, based on Gelbart's Elementary Introduction to the Langlands Program. As is the paper, the talk will be aimed at a general mathematical audience, giving a brief overview of the historical development, particularly artin reciprocity, hecke theory, and automorphic representations, leading to the formulation of the Langlands conjectures. Future direction for the seminar, including alternative meeting times, will be discussed at the beginning. References: S. Gelbart, Elementary Introduction to the Langlands Program and A. Knapp, Introduction to the Langlands Program.