MAT 477H1-F: Seminar in Mathematics (Fall 2019)

Prof. Ila Varma

Course Information

Lectures: Thursdays from 5pm-8pm in Room 2159 of the Bahen Building

First class is meeting on Thursday, September 5th, 2019. Attendance is mandatory.

Class is cancelled on Thursday, October 24th, 2019. Please see the updated (and hopefully, finalized) schedule below.

Instructor: Ila Varma	Office: Bahen Building, Room 6108
Email: ila at math dot toronto dot edu	Office Hours: Monday afternoons, by appointment

Some Additional References:

M. Baker, Algebraic Number Theory, v. 2017 (Georgia Tech).
M. Bhargava, Higher Composition Laws, Ph.D. Thesis, 2001.
M. Bhargava, On the classification of rings of "small" rank, Arizona Winter School Lecture Notes, 2009.
M. Bhargava, The density of discriminants of quartic rings and fields. Annals of Mathematics 162 (2005), 1031-1063.
F. Bouyer, Composition and Bhargava's Cubes, Master's Thesis, 2012.
D. Cox, Primes of the form x² + ny², John Wiley & Sons, 1989.
H. Davenport, On the class numbers of binary cubic forms, I, J. London. Math. Soc. 26 (1951), 192-198.
P. Nielsen, An introduction to orders in number fields, Undergraduate Thesis, 2002. [see especially Theorem 3 for an exact sequence relating class groups of orders to class groups of maximal rings]
P. Olver, Classical Invariant Theory, Cambridge University Press, 1999.

References:

F. Seguin, Composition of binary quadratic forms: understanding the approaches of Gauss, Dirichlet, and Bhargava, Resonance Journal, to appear.
M. Bhargava, Higher Composition Laws I: A new view of Gauss composition, and quadratic generalizations, Annals of Mathematics 159 (2004), 217-250.
F. Levi, Kubische Zahlkorper und binare kubische Formenklassen., Berichte uber die Verhandlungen der Kniglich Schsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Klasse 66 (1914), no. I, 26-37.
B.N. Delone, D.K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, (1964).
W.T. Gan, B.H. Gross, G. Savin, Fourier coefficients of modular forms on G₂, Duke Mathematical Journal 115 (2002), no. 1, 105-169.
M. Bhargava, Higher Composition Laws II: On cubic analogues of Gauss composition, Annals of Mathematics 159 (2004), 865-886.
M. Bhargava, A. Shankar, J. Tsimerman, On the Davenport-Heilbronn theorems and second-order terms, Inventiones Mathematicae 193 (2013), no. 2, 439-499.
H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields, II, Proceedings of the Royal Society of London 322 (1971), 405-420.
M.M. Wood, Rings and ideals parameterized by binary n-ic forms, Journal of the London Mathematical Society 83 (2011), no. 2, 208-231.
J. Nakagawa, Binary forms and orders of algebraic number fields, Inventiones mathematicae 97 (1989), 219-235. erratum.
M.M. Wood, Parametrization of ideal classes in rings associated to binary forms, Crelle's Journal 689 (2014), 169-199.
W. Ho, A. Shankar, I. Varma, Odd degree number fields with odd class number, Duke Mathematical Journal 167 (2018) no. 5, 995-1047.
M. Bhargava, Higher Composition Laws III: The parametrization of quartic rings, Annals of Mathematics 159 (2004), 1329-1360.
M.M. Wood, Quartic rings associated to binary quartic forms, International Mathematics Research Notices 2012 (2012) no. 6, 1300-1320.
M. Bhargava, I. Varma, The mean number of 3-torsion elements in the class groups and ideal groups of quadratic orders, Proceedings of the London Mathematical Society 112 (2016), no. 3, 235-266.
M. Bhargava, The geometric sieve and the density of squarefree values of invariant polynomials, preprint.

Grading:

Presentation: 70%
Written Assignments: 30%

Please use Quercus to check grades and submit written assignments.

Written Assignments

Each student will be required to turn in 3 separate written summaries (minimum 1 page for each write-up) summarizing 3 of the other students' presentations. It is up to you which presentations you summarize in your write-ups. Each write-up will be worth 10% of your final grade.

It is strongly recommended that students use LaTeX for their written assignments. For those new to LaTeX, please see this link for a quick guide, and consider using this online editor with this template (or download here).

Submit your written assignments on Quercus anytime before December 5th, 2019 at 5pm.

Schedule of Presentations

Each student will be responsible for one 60-75 minute presentation during the semester.

Please make sure you cover background material (this includes but is not limited to notation, definitions of class groups, maximality, monogenicity, etc.) in addition to the main themes and proofs of the reference material described below.
The proposed punchline can be thought of as a possible goal for the lecture (where brackets [ ] denote a possible half-time goal).

Students are encouraged to work in groups to digest the material as well as to prepare their presentation. Students are also more than welcome to speak with me during office hours or between presentations.

	Date	Title	Reference Material	Proposed Punchline	Speaker
	9/5/19	Overview of Lectures and Speaker Scheduling			Varma
1	9/12/19	Gauss Composition	Section 2-3 of Seguin	Proposition 3.3	Liu
2	9/12/19	Dirichlet Composition	Section 4 of Seguin, Sections 3.1-3.2 of HCL1	Theorem 4.4-4.5 or Theorem 10	Bradford
3	9/19/19	Composition via the Bhargava Cube	Section 5 of Seguin, Sections 2.1-2.2 and Appendix of HCL1	after Theorem 5.6 or Theorem 1 + Appendix	Kohut
4	9/19/19	A first parametrization via the Bhargava Cube	Section 2.3, Section 3.3 of HCL1	Theorem 11-12	Davies
5	9/26/19	Parametrizing with binary cubic forms, a modern view	Section 2.4, Section 3.4 of HCL1	Corollary 14-15	Tenenbaum
6	9/26/19	Parametrizing with binary cubic forms, the classical view	Levi-Delone-Faddeev Correspondence: Section 4 of GGS, Section 2.2 of HCL2, Section 2 of BST Davenport-Heilbronn Correspondence: Section 6 of DH, Section 3 of BST	Theorem 9 and 14	Nunez Lon-wo
7	10/3/19	Rings and ideals parametrized by binary n-ic forms	Sections 2.1-2.2 of Wood11 Section 1 of Nakagawa	Corollary 2.9 (or Corollary of Proposition 1.2)	Fattori
8	10/3/19	Parametrizations over cubic rings: 2 x 3 x 3 boxes of integers	Section 1, Section 2.1, Section 2.3 of HCL2	Theorem 2	(S) Huang
9	10/10/19	Parametrizations over cubic rings, part 2: symmetrized boxes & resulting composition laws	Section 2.4, Sections 3.1-3.2 of HCL2	[Theorem 4] Corollary 10	Zhang
10	10/10/19	Parametrization of ideal classes in rings associated to binary n-ic forms	Wood14 (see also Section 2 of HSV)	Theorem 1.3 (Theorem 2.2)	Karapetyan
11	10/17/19	Resolvent rings	Section 2 of HCL3	Equation 10	Ghalayini
12	10/17/19	Parametrization of quartic rings with cubic resolvents	Sections 3.1-3.5 of HCL3	Theorem 1	Wang
13	10/31/19	Parametrization of quartic fields	Sections 3.6-3.9 of HCL3	Corollary 5 and Corollary 18	Rodriguez
14	10/31/19	Quartic rings associated to binary quartic forms	Wood12	Theorem 1.1	(YC) Huang
15	11/14/19	The number of cubic rings of bounded discriminant, part 1: geometry of numbers	Sections 5.1-5.3 of BST	Equation 26	Jackson
16	11/14/19	The number of cubic rings of bounded discriminant, part 2: computing the volume	Sections 5.4-5.5 of BST	Theorem 26	Pike
17	11/21/19	Density of discriminants of cubic fields	Lemma 19, Sections 8.2-8.4 of BST	Theorem 1 (or Theorem 8)	Zoghi
18	11/21/19	Counting in the cusp	Theorem 17, Section 4 of BV	Theorem 6	Zhu
19	11/28/19	Applications of class field theory to counting	Section 8.1, Section 8.5 of BST, Section 5 of BV	Theorem 2	Libman
20	11/28/19	Density of discriminant of S₄-quartic fields	Bhargava05	Theorem 1 or 2	Fattori