RTG 2025 Workshop and Retreat
"Arithmetic Statistics"
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June 17-20, 2025 Mathematics Department The Ohio State University Columbus, OH, USA Organizers: |
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This event is funded by the Mathematics Research Institute at The Ohio State University and the National Science Foundation under RTG grant DMS 2231565.
Registration: For planning purposes, all participants (including invited speakers), are asked to register online. Applications received by May 15 will receive full consideration. Please fill out the following registration form.
| About | Speakers | Schedule | Titles and abstracts | Poster Session | Retreat |
The RTG Conferences and retreats
The graduate and postdoctoral training supported by the RTG award is anchored on five thematic years emphasizing different aspects of our combinatorial, arithmetic, and topological approaches to study algebraic varieties. Focused topics courses and research training seminars running each year will be complemented by an RTG Workshop, followed by a Group Retreat featuring a period of intensive mathematical collaboration, and promoting community-building through a goal-oriented activity.The YouTube channel of the RTG group has playlists containing a record of the talks delivered at each of the conferences.
Speakers
The RTG 2025 Workshop at The Ohio State University will feature two 4-lecture mini-courses by distinguished plenary speakers supported by some background talks from graduate student participants.
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| Alexander Smith (Clay Mathematics Institute - University of California Los Angeles) |
Jiuya Wang (University of Georgia) |
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Schedule
We expect participants to arrive on Monday June 16 and depart on Friday June 20 in the afternoon.
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Titles and abstracts
- Lecture 1: Goldfeld's conjecture, Selmer groups, and the Cassels-Tate pairing
In 1979, Goldfeld conjectured that, among the quadratic twists of a given elliptic curve over Q, 50% have rank 0, 50% have rank 1, and 0% have any higher rank. We will go through this conjecture, giving two different explanations for the source of the 50/50 distribution. The algebraic explanation for this behavior leads us to Selmer groups and the Cassels-Tate pairing, the arithmetic objects whose statistical behavior we are studying.
Pretalk: The Birch and Swinnerton-Dyer conjecture (possibly Wiles' writeup for the Millenium prize problems) - Lecture 2: Tamagawa ratios and the distribution of isogeny Selmer groups
If E has a rational isogeny of degree 2, the associated isogeny Selmer group of any quadratic twist of E may be considered as a subgroup of Q*/(Q*)^2. We find the distribution of these isogeny Selmer groups by taking advantage of this identification. This recovers a result of Kane and Klagsbrun. A key ingredient in our work is the Greenberg-Wiles' formula, which relates the size of a Selmer group to the size of the corresponding dual Selmer group.
Pretalk: Local Poitou-Tate duality - Lecture 3: The distribution of 2-Selmer groups in quadratic twist families
The work in the previous lecture relied on the isomorphism H1(Q, Z/2Z) = Q*/(Q*)^2. To study the distribution of 2-Selmer groups in the quadratic twist family of E, we similarly start by parameterizing the group H1(Q, E[2]). This approach relies on the fact that E[2] = E'[2] for any quadratic twist E' of E. The resulting distributions vary widely depending on the starting curve. We go through some of the behavior that can be seen.
Pretalk: Studying class groups of quadratic extensions using Selmer groups of quadratic twists.
Alternative pretalk: Fouvry Kluners results on 4-class ranks. - Lecture 4: Getting at higher Selmer groups
The previous lecture gave us a manner to find large families of quadratic twists with isomorphic 2-Selmer groups. We may use the isomorphisms between these 2-Selmer groups to compare the Cassels-Tate pairings over such families. With some care, we may force these pairings to be equidistributed among all reasonable possibilities. This allows us to control the distribution of higher Selmer ranks.
Pretalk: Weil pairing definition of the Cassels-Tate pairing;
Alternative pretalk: Proof of Stevenhagen's conjecture by Koymans and Pagano
- Lecture 1: Malle's Conjecture
We introduce Malle's Conjecture. Starting from the simplest example of quadratic extensions, we explore a couple approaches leading to Malle's Conjecture for abelian extensions. We introduce common tools en route. We then describe the general principle leading to Malle's Conjecture in its general form.
Pretalk: Tauberian Theorem. - Lecture 2: Genus Theory
We derive Gauss's genus theory for quadratic fields based on Lecture 1 and then generalize it to nilpotent extensions. This leads to the general dichotomy for nilpotent extensions and non-nilpotent extensions in arithmetic statistics.
Pretalk: Frattini subgroup/lower-central series of p-group - Lecture 3: Cohen-Lenstra Heuristics
We revisit Lecture 1 with general number fields and then we introduce the conjecture of Cohen-Lenstra heuristics and describe its moment distribution and probability distribution from the perspective of random matrix.
Pretalk: Idele class groups for number fields and Artin reciprocity. - Lecture 4: Function Field and Number Field Analogue
We introduce the basic dictionary between function field and number field. We then describe how Cohen-Lenstra heuristics is motivated and interpreted over function field. In some simple examples, we explain how roots of unity play a crucial role in the distribution of class groups.
Pretalk: Weil pairing and GSp-matrix.
Each of the lectures given by our lecturers will be preceded by pretalks given by some of the conference participants to cover background material.
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Poster Session
On Wednesday June 18, 4:15-5:15pm, we will have a poster session in MW 724.
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RTG Retreat
The retreat will take place at Burr Oak Lodge and Conference Cente, located in Burr Oak State Park, during June 23-26, 2025.
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Back to the main website of the RTG: "Arithmetic, Combinatorics, and Topology of Algebraic Varieties".
The Ohio State University, Department of Mathematics, 231 W. 18th Avenue, Columbus, OH, 43210, USA.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the National Science Foundation.