DOCSLIB.ORG

Modularity Lifting Theorems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modularity Lifting Theorems MODULARITY LIFTING THEOREMS RONG ZHOU 1. Introduction These are live-texed lecture notes from the MIT graduate class on Galois represen- tations given by Sug Woo Shin in Spring of 2014. The class expanded on the notes of Toby Gee's course from the Arizona Winter School in 2014, which it closely followed. The problem sets from the class contained some proofs of results which were stated in class and some of these were later added to the notes. Any mistakes in these notes are due to me and not the lecturer. 2. Lecture 1 Let F be a field and fix an algebraic closure F of F . We let Gal(F =F ) be the absolute Galois group of F . Definition 2.1. A Galois representation is a continuous group homomoprhism Gal(F =F ) ! GLn(R) where R is a topological ring. Most of the time we will take R to be Ql where l is a prime, such a representation will be a called an l-adic Galois representation. The first result we need about l-adic representations is that up to conjugation, the image lands in GLn(OL) where L is a finite extension of Qp. This follows from the next two lemma which actually hold in a more general context: Lemma 2.2. Let Γ be a compact topological group (in particular a Galois group is such) and let ρ :Γ ! GLn(Ql) be a continuous homomorphism. Then there exists a finite extension L=Ql such that ρ(Γ) ⊂ GLn(L). Proof. ρ(Γ) is compact Hausdorff, hence by the Baire Category theorem, the intersec- tion of a countable set of open dense subsets of ρ(Γ) is dense in ρ(Γ). Now since [ ρ(Γ) = (ρ(Γ) \ GLn(L)) L=Ql finite 1 2 RONG ZHOU and each set ρ(Γ) \ (GLn(L) is closed in ρ(Γ), it follows from the above, that some ρ(Γ) \ GLn(L) contains an open subset, hence ρ(Γ) \ GLn(L) itself is open. As Γ was profinite, we have that [ρ(Γ) : ρ(Γ) \ GLn(L)] is finite. If we choose coset representatives α1; :::; αr for ρ(Γ), there exists Li=Ql a finite ex- tension such that ρ(αi) 2 GLn(Li). Taking L to be the compositum of the Li we obtain the result. Lemma 2.3. Let Γ be a compact topological group, L=Ql a finite extension with ring of integers OL and ρ :Γ ! GLn(L) a continuous representation. Then there exists −1 g 2 GLn(L) such that gρg has image in GLn(OL) n Proof. It is enough to show that there exists a Γ invariant OL lattice Λ ⊂ L . Indeed n letting g be the change of basis matrix taking Λ to the standard basis of OL we get the result. n Define Λ0 := OL to be the standard lattice. Observe that GLOL (Λ0) ⊂ GLn(L) is −1 open, so that ρ (GLOL (Λ0)) ⊂ Γ has finite index. Let α1; :::; αr be a finite set of coset representatives in Γ. We then define r X Λ := ρ(Γ)Λ0 = αiΛ0 i=1 which is clearly a ρ(Γ) invariant lattice. Definition 2.4. Let R be a topological ring and Γ a topological group. Two continuous representations ρ1; ρ2 :Γ ! GLn(R) are said to be isomorphic if there exists and isomorphism φ : Rn ! Rn such that the following diagram commutes: ρ (γ) Rn 1 > Rn φ φ _ _ ρ (γ) Rn 2 > Rn for all γ 2 Γ As a consequence of the Lemma, a representation ρ :Γ ! GLn(Ql) with Γ compact is isomorphic to one with image in GLn(OL) Our next result will be the Brauer Nestbitt theorem, this gives a condition for when the semi simplification of two representations are isomorphic and which we will need for the study of the mod p reduction of a representation. We first need to define the notion of semisimple representation and semisimplification. As before Γ is a compact topological group. ∼ Definition 2.5. Let k be a field, and ρ :Γ ! GLn(K) = GLn(V ). Then ρ is semi simple if ∼ r ρ = ⊕i=1ρi MODULARITY LIFTING THEOREMS 3 for some irreducible ρi :Γ ! GLni (R). Definition 2.6. Let ρ :Γ ! GLn(V ) and choose 0 ⊂ V1 ⊂ ::: ⊂ Vr = V where the Vi are Γ invariant subgroups with Vi=Vi−1 irreducible. The semi simplification ss ss r ρ is the representation induced by ρ on the vector space V = ⊕i=1Vi=Vi+1 It can be shown that this is well defined and does not depend on the choice of the Vi's. Example 2.7. Consider ρ :Γ ! GL2(k) given by a(γ) b(γ) γ 7! 0 d(γ) Then ρss = a ⊕ d Theorem 2.8 (Brauer, Nesbitt). Let k be a field and Γ a (topological) group, n1; n2; ≥ 1. For i = 1; 2 and ρi :Γ ! GLni (k), assume either: i) 8γ 2 Γ we have det(1 − ρ1(γ)T ) = det(1 − ρ1(γ)T ) or ii) char k = 0(>> 0), and 8γ 2 Γ we have trρ1(γ) = trρ2(γ) ss ss Then ρ1 = ρ2 We will need the following lemma: Lemma 2.9. Let R be an associative k algebras (not necessarily commutative eg. R = k[Γ]) and let M1; :::; Mr be non-isomorphic R-simple modules which are finite dimensional over k. Then 9e1; :::; er 2 R such that eimi = mi 8i; 8mi 2 Mi eimj = 0 8j 6= i; 8mj 2 Mj r Proof. (Sketch) Upon replacing R by its image in Endk ⊕i=1 Mi wlog. we can assume, k is a finite dimensional semisimple k-algebra. ∼ Qs The Artin Wedderburn theorem then tells us that R = i=1 Mni (Di) where Di is a division algebra over k. One then deduces that r = s and R acts as Mni (Di) on Mi for th i = 1; :::; r (after reordering). Defining ei to be 1 on the i component and 0 elsewhere we obtain the result. Proof of Brauer Nesbitt: Let ρi :Γ ! GLn(V ); j = 1; 2. Since the conditions in the theorem don't change upon taking semisimplifications, we may assume the ρi are semisimple. 4 RONG ZHOU 1 2 Let M1; :::; Mr be the distinct irreducible subrepresentations of V1 ⊕ V2. Let mi ; mi 1 2 be the multiplicities of Mi in V1 and V2 respectively. It suffices to prove that mi = mi for i = 1; :::; r. Case i) Take ei as in Lemma 1.9, then the characteristic polynomial of ei on Mi is (t − 1)dim Mi and tdim Mj when j 6= i. From the equality of characteristic polynomials of 1 2 ρ1(ei) and ρ2(ei) one finds the dimensions mi and mi match. Case ii)For j = 1; 2, there exists a unique continuous map θj which makes the following diagram commute: ρ tr Γ >j GL (V ) > k n j > _ θj k[Γ] It can be shown that θ1(α) = θ2(α)8α 2 k[Γ]. Plugging in ei, we obtain θ1(ei) = 1 2 mi dim Mi and θ2(ei) = mi dim Mi. 3. Lecture 2 The Brauer Nesbitt Theorem allows us to define the reduction mod l of l-adic rep- resentations. As before L is a finite extension of Ql with ring of integers OL Definition 3.1. Let Γ be a compact group and ρ :Γ ! GLn(L) be a representation. 0 0 0 Choose a conjugate ρ of ρ such that ρ has image in GLn(OL) and let ρ be composition with GLn(OL) ! GLn(kL). The reduction of ρ mod l is then defined to be ρ := (ρ0)ss Lemma 3.2. ρ is well defined (up to isomorphism), i.e. it does not depends on the choice of ρ0 0 Proof. Let γ 2 Γ. Observe that char(ρ(γ)) = char(ρ (γ)) 2 kL[T ], the Lemma then follows by Theorem 1.8 We now recall some basic properties of number fields and local fields. In this course a number field will be a finite extension of Q and local field will be finite extension of Qp or R so we exclude the function field case. Definition 3.3. Let F be a number field. A place of F is an equivalence class of valuations on F (where two valuations are considered equivalent if they induce the same topology on F ). A place is finite if the valuation is non-archimedean, otherwise it is an infinite place. A finite place is p-adic if jpj < 1. MODULARITY LIFTING THEOREMS 5 We state some basic facts about places: 1)There are 1-1 correspondences between the following sets i) fprime ideals of OF dividing pg ii) fp-adic places of F g iii) Gal(Qp=Qp)nHomQ(F; Qp) The first bijection is given by sending a p-adic place of F to the set fa 2 OF : jaj < 1g. The bijection between ii) and iii) is given by sending a φ 2 Gal(Qp=Qp)nHomQ(F; Qp) to the place j:jp ◦ φ. 2) The infinite places of F are in one to one correspondence with Gal(C=R)nHomQ(F; C), and these are a disjoint union of of real places and complex places, where the real places are the 1 element orbits of Gal(C=R)nHomQ(F; C) and the complex places are the two elements orbits. Now for a place v of F , let Fv to be the completion of F with respect to the topology induced by v.
Load more

Recommended publications
  • Arxiv:2003.01675V1 [Math.NT] 3 Mar 2020 of Their Locations
    DIVISORS OF MODULAR PARAMETRIZATIONS OF ELLIPTIC CURVES MICHAEL GRIFFIN AND JONATHAN HALES Abstract. The modularity theorem implies that for every elliptic curve E=Q there exist rational maps from the modular curve X0(N) to E, where N is the conductor of E. These maps may be expressed in terms of pairs of modular functions X(z) and Y (z) where X(z) and Y (z) satisfy the Weierstrass equation for E as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the q - expansions of these parametrizations at any cusp. Using these functions, we determine the divisor of the parametrization and the preimage of rational points on E. We give a sufficient condition for when these preimages correspond to CM points on X0(N). We also examine a connection between the al- gebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the q-expansions of these objects. 1. Introduction and statement of results The modularity theorem [2, 12] guarantees that for every elliptic curve E of con- ductor N there exists a weight 2 newform fE of level N with Fourier coefficients in Z. The Eichler integral of fE (see (3)) and the Weierstrass }-function together give a rational map from the modular curve X0(N) to the coordinates of some model of E: This parametrization has singularities wherever the value of the Eichler integral is in the period lattice. Kodgis [6] showed computationally that many of the zeros of the Eichler integral occur at CM points.
    [Show full text]
  • Arxiv:1401.5507V2
    FAMILIES OF L-FUNCTIONS AND THEIR SYMMETRY PETER SARNAK, SUG WOO SHIN, AND NICOLAS TEMPLIER Abstract. In [100] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [32], [118], [73] [49], [72] and especially [108] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [68] for 1 the distribution of the zeros near s = 2 . In this note we carry this out after introducing some basic invariants associated to a family. 1. Definition of families and Conjectures The zoo of automorphic cusp forms π on G = GLn over Q correspond bijectively to their standard completed L-functions Λ(s, π) and they constitute a countable set containing species of different types. For example there are self-dual forms, ones corresponding to finite Galois representations, to Hasse–Weil zeta functions of varieties defined over Q, to Maass forms, etc. From a number of points of view (including the nontrivial problem of isolating special forms) one is led to study such Λ(s, π)’s in families in which the π’s have similar characteristics. Some applications demand the understanding of the behavior of the L-functions as π varies over a family. Other applications involve questions about an individual L-function. In practice a family is investigated as it arises. For example the density theorems of Bombieri [13] and Vinogradov [113] are concerned with showing that in a suitable sense most Dirichlet L-functions have few violations of the Riemann hypothesis, and as such it is a powerful substitute for the latter.
    [Show full text]
  • Counting Points on Igusa Varieties of Hodge Type by Thomas a Mack
    Counting Points on Igusa Varieties of Hodge Type by Thomas A Mack-Crane A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Sug Woo Shin, Chair Professor Xinyi Yuan Professor Richard Bamler Professor Ori Ganor Spring 2021 Abstract Counting Points on Igusa Varieties of Hodge Type by Thomas A Mack-Crane Doctor of Philosophy in Mathematics University of California, Berkeley Professor Sug Woo Shin, Chair The Langlands-Kottwitz method seeks to understand Shimura varieties in terms of automorphic forms by deriving a trace formula for the cohomology of Shimura varieties which can be compared to the automorphic trace formula. This method was pioneered by Langlands [Lan77, Lan79], and developed further by Kottwitz in [Kot90, Kot92] in the case of Shimura varieties of PEL type with good reduction. Igusa varieties were introduced in their modern form by Harris-Taylor [HT01] in the course of studying the bad reduction of certain simple Shimura varieties. The relation between Igusa varieties and Shimura varieties was expanded to PEL type by Mantovan [Man04, Man05], and their study of the cohomology of Igusa varieties was expanded to PEL type and streamlined in the Langlands-Kottwitz style by Shin [Shi09, Shi10]. Following the generalization of Mantovan’s work to Hodge type by Hamacher and Hamacher-Kim [Ham19, HK19], we carry out the Langlands-Kottwitz method for Igusa varieties of Hodge type, generalizing the work of [Shi09]. That is, we derive a trace formula for the cohomology of Igusa varieties suitable for eventual comparison with the automorphic trace formula.
    [Show full text]
  • Supersingular Zeros of Divisor Polynomials of Elliptic Curves of Prime Conductor
    Kazalicki and Kohen Res Math Sci(2017) 4:10 DOI 10.1186/s40687-017-0099-8 R E S E A R C H Open Access Supersingular zeros of divisor polynomials of elliptic curves of prime conductor Matija Kazalicki1* and Daniel Kohen2,3 *Correspondence: [email protected] Abstract 1Department of Mathematics, p p University of Zagreb, Bijeniˇcka For a prime number , we study the zeros modulo of divisor polynomials of rational cesta 30, 10000 Zagreb, Croatia elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, Full list of author information is 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of available at the end of the article supersingular elliptic curves over Fp. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form vE .This allows us to prove that if the root number of E is −1 then all supersingular j-invariants of elliptic curves defined over Fp are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in Fp seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of vE corresponding to supersingular elliptic curves defined over Fp are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest.
    [Show full text]
  • A Glimpse of the Laureate's Work
    A glimpse of the Laureate’s work Alex Bellos Fermat’s Last Theorem – the problem that captured planets moved along their elliptical paths. By the beginning Andrew Wiles’ imagination as a boy, and that he proved of the nineteenth century, however, they were of interest three decades later – states that: for their own properties, and the subject of work by Niels Henrik Abel among others. There are no whole number solutions to the Modular forms are a much more abstract kind of equation xn + yn = zn when n is greater than 2. mathematical object. They are a certain type of mapping on a certain type of graph that exhibit an extremely high The theorem got its name because the French amateur number of symmetries. mathematician Pierre de Fermat wrote these words in Elliptic curves and modular forms had no apparent the margin of a book around 1637, together with the connection with each other. They were different fields, words: “I have a truly marvelous demonstration of this arising from different questions, studied by different people proposition which this margin is too narrow to contain.” who used different terminology and techniques. Yet in the The tantalizing suggestion of a proof was fantastic bait to 1950s two Japanese mathematicians, Yutaka Taniyama the many generations of mathematicians who tried and and Goro Shimura, had an idea that seemed to come out failed to find one. By the time Wiles was a boy Fermat’s of the blue: that on a deep level the fields were equivalent. Last Theorem had become the most famous unsolved The Japanese suggested that every elliptic curve could be problem in mathematics, and proving it was considered, associated with its own modular form, a claim known as by consensus, well beyond the reaches of available the Taniyama-Shimura conjecture, a surprising and radical conceptual tools.
    [Show full text]
  • The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes By
    The Arbeitsgemeinschaft 2017: Higher Gross-Zagier Formulas Notes by Tony Feng1 Oberwolfach April 2-8, 2017 1version of April 11, 2017 Contents Note to the reader 5 Part 1. Day One 7 1. An overview of the Gross-Zagier and Waldspurger formulas (Yunqing Tang) 8 2. The stacks Bunn and Hecke (Timo Richarz) 13 3. (Moduli of Shtukas I) (Doug Ulmer) 17 4. Moduli of Shtukas II (Brian Smithling) 21 Part 2. Day Two 25 5. Automorphic forms over function fields (Ye Tian) 26 6. The work of Drinfeld (Arthur Cesar le Bras) 30 7. Analytic RTF: Geometric Side (Jingwei Xiao) 36 8. Analytic RTF: Spectral Side (Ilya Khayutin) 41 Part 3. Day Three 47 9. Geometric Interpretation of Orbital Integrals (Yihang Zhu) 48 10. Definition and properties of Md (Jochen Heinloth) 52 Part 4. Day Four 57 11. Intersection theory on stacks (Michael Rapoport) 58 12. LTF for Cohomological Correspondences (Davesh Maulik) 64 µ 13. Definition and description of HkM;d: expressing Ir(hD) as a trace (Liang Xiao) 71 3 4 CONTENTS 14. Alternative calculation of Ir(hD) (Yakov Varshavsky) 77 Part 5. Day Five 83 15. Comparison of Md and Nd; the weight factors (Ana Caraiani) 84 16. Horocycles (Lizao Ye) 89 17. Cohomological spectral decomposition and finishing the proof (Chao Li) 93 NOTE TO THE READER 5 Note to the reader This document consists of notes I live-TEXed during the Arbeitsgemeinschaft at Oberwolfach in April 2017. They should not be taken as a faithful transcription of the actual lectures; they represent only my personal perception of the talks.
    [Show full text]
  • Construction of Automorphic Galois Representations: the Self-Dual Case
    CONSTRUCTION OF AUTOMORPHIC GALOIS REPRESENTATIONS: THE SELF-DUAL CASE SUG WOO SHIN This survey paper grew out of the author's lecture notes for the Clay Summer School in 2009. Proofs are often omitted or sketched, but a number of references are given to help the reader find more details if he or she wishes. The article aims to be a guide to the outline of argument for constructing Galois representations from the so-called regular algebraic conju- gate self-dual automorphic representations, implementing recent improvements in the trace formula and endoscopy. Warning. Although it is important to keep track of various constants (e.g. sign matters!), twists by characters, etc in proving representation-theoretic identities, I often ignore them in favor of simpler notation and formulas. Thus many identities and statements in this article should be taken with a grain of salt and not cited in academic work. Correct and precise statements can be found in original papers. I should also point out that significant progress has been made, especially in the case of non-self-dual automorphic representations ([HLTT16], [Sch15]) since the original version of these notes was written. I have kept updates minimal (mostly taking place in introduction, adding references to some recent developments) and do not discuss these exciting new results here. The interested readers are referred to survey papers such as [Wei16, Mor16, Scha, Sch16, Car]. Acknowledgments. I heartily thank the Clay Mathematics Institute for its generous support toward the 2009 summer school and choosing an exciting location as Hawaii. I am grateful to the participants for their interest and wonderful questions.
    [Show full text]
  • Andrew Wiles's Marvelous Proof
    Andrew Wiles’s Marvelous Proof Henri Darmon ermat famously claimed to have discovered “a truly marvelous proof” of his Last Theorem, which the margin of his copy of Diophantus’s Arithmetica was too narrow to contain. While this proof (if it ever existed) is lost to posterity, AndrewF Wiles’s marvelous proof has been public for over two decades and has now earned him the Abel Prize. According to the prize citation, Wiles merits this recogni- tion “for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.” Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the mathematics of ancient Greece, simple enough to be understood It is also a centerpiece and appreciated Wiles giving his first lecture in Princeton about his by a novice of the “Langlands approach to proving the Modularity Conjecture in (like the ten- early 1994. year-old Andrew program,” the Wiles browsing of Theorem 2 of Karl Rubin’s contribution in this volume). the shelves of imposing, ambitious It is also a centerpiece of the “Langlands program,” the his local pub- imposing, ambitious edifice of results and conjectures lic library), yet edifice of results and which has come to dominate the number theorist’s view eluding the con- conjectures which of the world. This program has been described as a “grand certed efforts of unified theory” of mathematics. Taking a Norwegian per- the most brilliant has come to dominate spective, it connects the objects that occur in the works minds for well of Niels Hendrik Abel, such as elliptic curves and their over three cen- the number theorist’s associated Abelian integrals and Galois representations, turies, becoming with (frequently infinite-dimensional) linear representa- over its long his- view of the world.
    [Show full text]
  • Fermat's Last Theorem
    Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions.[1] The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death.[2] After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles's Abel Prize award in 2016.[3] It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.[4] Contents The 1670 edition of Diophantus's Arithmetica includes Fermat's Overview commentary, referred to as his "Last Pythagorean origins Theorem" (Observatio Domini Petri Subsequent developments and solution de Fermat), posthumously published Equivalent statements of the theorem by his son.
    [Show full text]
  • PETER SCHOLZE to RECEIVE 2013 SASTRA RAMANUJAN PRIZE the 2013 SASTRA Ramanujan Prize Will Be Awarded to Professor Peter Scholze
    PETER SCHOLZE TO RECEIVE 2013 SASTRA RAMANUJAN PRIZE The 2013 SASTRA Ramanujan Prize will be awarded to Professor Peter Scholze of the University of Bonn, Germany. The SASTRA Ramanujan Prize was established in 2005 and is awarded annually for outstanding contributions by young mathematicians to areas influenced by the genius Srinivasa Ramanujan. The age limit for the prize has been set at 32 because Ramanujan achieved so much in his brief life of 32 years. The prize will be awarded in late December at the International Conference on Number Theory and Galois Representations at SASTRA University in Kumbakonam (Ramanujan's hometown) where the prize has been given annually. Professor Scholze has made revolutionary contributions to several areas at the interface of arithmetic algebraic geometry and the theory of automorphic forms. Already in his master's thesis at the University of Bonn, Scholze gave a new proof of the Local Langlands Conjecture for general linear groups. There were two previous approaches to this problem, one by Langlands{Kottwitz, and another by Harris and Taylor. Scholze's new approach was striking for its efficiency and simplicity. Scholze's proof is based on a novel approach to calculate the zeta function of certain Shimura varieties. This work completed in 2010, appeared in two papers in Inventiones Mathematicae in 2013. Scholze has generalized his methods partly in collaboration with Sug Woo Shin to determine the `-adic Galois representations defined by a class of Shimura varieties. These results are contained in two papers published in 2013 in the Journal of the American Mathematical Society. While this work for his master's was groundbreaking, his PhD thesis written under the direction of Professor Michael Rapoport at the University of Bonn was a more mar- velous breakthrough and a step up in terms of originality and insight.
    [Show full text]
  • Junehyuk Jung –
    Junehyuk Jung B [email protected] • Í sites.google.com/brown.edu/junehyuk/ Positions Brown University, Department of Mathematics Providence, RI Assistant Professor 7/2020– Texas A&M University, Department of Mathematics College Station, TX Assistant Professor 8/2017–6/2020 KAIST, Department of Mathematical Science Daejeon, South Korea Researcher 6/2013–7/2016 Visiting Positions Rice University, Department of Mathematics Houston, TX Adjunct Assistant Professor 7/2019–3/2020 Yale University, Department of Mathematics New Haven, CT Visitor 4/2017–6/2017 University of California Berkeley, Department of Mathematics Berkeley, CA Visitor 2/2017–3/2017 Northwestern University, Mathematics Department Evanston, IL Visitor 10/2016–11/2016 Institute for Advanced Study, School of Mathematics Princeton, NJ Member 9/2014–12/2015 Northwestern University, Mathematics Department Evanston, IL Visiting Postdoctoral Fellow 9/2013–12/2013 Education Princeton University Princeton, NJ Doctor of Philosophy 9/2008–6/2013 Mathematics Advisor: Peter C. Sarnak Thesis Title: “On the zeros of automorphic forms” University of Chicago Chicago, IL Master of Science 9/2006–8/2008 Mathematics University of Chicago Chicago, IL Bachelor of Arts 9/2005–8/2008 Mathematics with Honors Honors and Awards National Science Foundation grant Awarded, DMS-1900993 7/2019–6/2022 Sloan Research Fellowship Awarded, $ 35,000 per year 7/2019–6/2021 Posco TJ Park Science Fellowship Awarded, $ 35,000 per year 3/2014–2/2016 Samsung Scholarship for Graduate Studies Awarded, $ 50,000 per
    [Show full text]
  • Arxiv:2003.08242V1 [Math.HO] 18 Mar 2020
    VIRTUES OF PRIORITY MICHAEL HARRIS In memory of Serge Lang INTRODUCTION:ORIGINALITY AND OTHER VIRTUES If hiring committees are arbiters of mathematical virtue, then letters of recom- mendation should give a good sense of the virtues most appreciated by mathemati- cians. You will not see “proves true theorems” among them. That’s merely part of the job description, and drawing attention to it would be analogous to saying an electrician won’t burn your house down, or a banker won’t steal from your ac- count. I don’t know how electricians or bankers recommend themselves to one another, but I have read a lot of letters for jobs and prizes in mathematics, and their language is revealing in its repetitiveness. Words like “innovative” or “original” are good, “influential” or “transformative” are better, and “breakthrough” or “deci- sive” carry more weight than “one of the best.” Best, of course, is “the best,” but it is only convincing when accompanied by some evidence of innovation or influence or decisiveness. When we try to answer the questions: what is being innovated or decided? who is being influenced? – we conclude that the virtues highlighted in reference letters point to mathematics as an undertaking relative to and within a community. This is hardly surprising, because those who write and read these letters do so in their capacity as representative members of this very community. Or I should say: mem- bers of overlapping communities, because the virtues of a branch of mathematics whose aims are defined by precisely formulated conjectures (like much of my own field of algebraic number theory) are very different from the virtues of an area that grows largely by exploring new phenomena in the hope of discovering simple This article was originally written in response to an invitation by a group of philosophers as part arXiv:2003.08242v1 [math.HO] 18 Mar 2020 of “a proposal for a special issue of the philosophy journal Synthese` on virtues and mathematics.” The invitation read, “We would be delighted to be able to list you as a prospective contributor.
    [Show full text]