Organisers: Aditya Ghosh, Alan Zhao, Austin Lei
Date: Spring 2026
Time: 4:30 PM - 5:30 PM
Location: Room 528, Mathematics Hall
| Serial No. | Date | Speaker | Title and Abstract | Notes |
|---|---|---|---|---|
| 1 | 20 Jan | Aditya Ghosh | Recap of Fall semester seminar. We shall go over the key prerequisites of the Orbit method covered in the last semester. These includes passing from a Lie group representation to the corresponding coadjoint orbit, Kirillov's orbit formula and the construction of bump functions using the orbit method | link |
| 2 | 27 Jan | Aditya Ghosh | Application of the orbit method to lower bounds of integrals of matrix coefficients. Having set up the prerequisites for the orbit method, we look at how it is used to obtain test functions that give lower bounds of integrals of matrix coefficients. This is used as one of the main components in the proof of subconvexity or the supnorm problem. | |
| 2 | 3 Feb | Alan Zhao | Kloosterman Sums on $GSp_4$. In this talk, we will introduce the density hypothesis on $GSp_4$ and introduce prerequisites for understanding the proof of “one half” of the problem. | |
| 3 | 10 Feb | Alan Zhao | Kloosterman Sums on $GSp_4$, Part II. We will continue where we left off last time and finish off introducing higher rank Kloosterman sums. Then, we will continue with exposing the proof of a density hypothesis for $GSp_4$. | |
| 4 | 17 Feb | Aditya Ghosh | Supnorm problem for $PGL(2,\mathbb{R})$ We continue with the estimates of integrals of matrix coefficients from last time. Using the $Op(a)$ operator defined earlier, we will find nice test vectors that can be applied to the supnorm problem. | |
| 24 Feb | Aditya Ghosh | Cancelled due to snow | ||
| 5 | 3 Mar | Aditya Ghosh | Supnorm problem for $PGL(2,\mathbb{R})$: Part II We establish the Relative Trace formula used in the supnorm problem and relate the lower bounds of the local periods with the relative trace estimates of $Op(a)$ obtained using the Orbit Method. | |
| 6 | 10 Mar | Aditya Ghosh | Supnorm problem for $PGL(2,\mathbb{R})$: Part III Having established the local period estimate in the previous talk, we look at bounding the geometric side of the relative trace formula. This gives us the convexity bound for the supnorm. This sets up the use of amplification to obtain subconvex bounds. | |
| 7 | 24 Mar | Alan Zhao | Triple Products of Fourier Coefficients: Part I I will introduce the Kuznetsov trace formula on GL(2) following Goldfeld’s formulation for higher rank and reconcile it with Motohashi’s derivation, the purpose being conceptual. Then, I will explain how one might generalize this formula to handle products of many Fourier coefficients. | |
| 8 | 31 Mar | Alan Zhao | Triple Products of Fourier Coefficients: Part II We continue where we left off last time with the analysis of the error term. | |
| 9 | 7 Apr | Aditya Ghosh | Long sums of triple product periods using analytic newvectors. Let $f$ be an $ SL(2,\mathbb{Z})$ Maass form. Consider the sum of $\langle |f|^2, f_i \rangle$ where $f_i$ ranges over $ SL(2,\mathbb{Z})$ Maass forms of Langlands parameters in the range $[T,2T]$. A classic result of Bernstein-Reznikov states that this sum is bounded by an absolute constant. They also obtained a generalisation to Maass forms over congruence subgroups $\Gamma_0(N)$, but the bound was implicit on the level $N$. Using the idea of analytic newvectors developed by Jana-Nelson, we provide an alternate proof of the Bernstein-Reznikov bound with an explicit dependence on $N$. This has applications to shifted convolution sums, generalising the result of Goldfeld-Hinkle-Hoffstein (https://arxiv.org/abs/2311.06587). |
Aditya Ghosh: ag4794 (at) columbia (dot) edu
Alan Zhao: asz2115 (at) columbia (dot) edu
Austin Lei: ayl2158 (at) columbia (dot) edu