Thu, 18 Jan 2024
16:00
Lecture Room 4, Mathematical Institute

Traces of random matrices over F_q, and short character sums

Ofir Gorodetsky
(University of Oxford)
Abstract
Let g be a matrix chosen uniformly at random from the GL_n(F_q), where F_q is the field with q elements. We consider two questions:
1. For fixed k and growing n, how fast does Tr(g^k) converge to the uniform distribution on F_q?
2. How large can k be taken, as a function of n, while still ensuring that Tr(g^k) converges to the uniform distribution on F_q?
We will answer these two questions (as well as various variants) optimally. The questions turn out to be strongly related to the study of particular character sums in function fields.
Based on joint works with Brad Rodgers (arXiv:1909.03666) and Valeriya Kovaleva (arXiv:2307.01344).
Wed, 14 Feb 2024
17:00
Lecture Theatre 1

Logging the World - Oliver Johnson

Oliver Johnson
(University of Bristol)
Further Information

During the pandemic, you may have seen graphs of data plotted on strange-looking (logarithmic) scales. Oliver will explain some of the basics and history of logarithms, and show why they are a natural tool to represent numbers ranging from COVID data to Instagram followers. In fact, we’ll see how logarithms can even help us understand information itself in a mathematical way.

Oliver Johnson is Professor of Information Theory in the School of Mathematics at the University of Bristol. His research involves randomness and uncertainty, and includes collaborations with engineers, biologists and computer scientists. During the pandemic he became a commentator on the daily COVID numbers, through his Twitter account and through appearances on Radio 4 and articles for the Spectator. He is the author of the book Numbercrunch (2023), which is designed to help a general audience understand the value of maths as a toolkit for making sense of the world.

Please email @email to register.

The lecture will be broadcast on the Oxford Mathematics YouTube Channel on Wednesday 06 March at 5-6pm and any time after (no need to register for the online version).

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

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Search for 10--1000 GeV neutrinos from Gamma Ray Bursts with IceCube
 Collaboration, I Abbasi, R Ackermann, M Adams, J Agarwalla, S Aguilar, J Ahlers, M Alameddine, J Amin, N Andeen, K Anton, G Argüelles, C Ashida, Y Athanasiadou, S Ausborm, L Axani, S Bai, X V, A Baricevic, M Barwick, S Basu, V Bay, R Beatty, J Tjus, J Beise, J Bellenghi, C Benning, C BenZvi, S Berley, D Bernardini, E Besson, D Blaufuss, E Blot, S Bontempo, F Book, J Meneguolo, C Böser, S Botner, O Böttcher, J Braun, J Brinson, B Brostean-Kaiser, J Brusa, L Burley, R Busse, R Butterfield, D Campana, M Carloni, K Carnie-Bronca, E Chattopadhyay, S Chau, N Chen, C Chen, Z Chirkin, D Choi, S Clark, B Coleman, A Collin, G Connolly, A Conrad, J Coppin, P Correa, P Cowen, D Dave, P Clercq, C DeLaunay, J Delgado, D Deng, S Deoskar, K Desai, A Desiati, P Vries, K Wasseige, G DeYoung, T Diaz, A Díaz-Vélez, J Dittmer, M Domi, A Dujmovic, H DuVernois, M Ehrhardt, T Eimer, A Eller, P Ellinger, E Mentawi, S Elsässer, D Engel, R Erpenbeck, H Evans, J Evenson, P Fan, K Fang, K Farrag, K Fazely, A Fedynitch, A Feigl, N Fiedlschuster, S Finley, C Fischer, L Fox, D Franckowiak, A Fürst, P Gallagher, J Ganster, E Garcia, A Gerhardt, L Ghadimi, A Glaser, C Glauch, T Glüsenkamp, T Gonzalez, J Grant, D Gray, S Gries, O Griffin, S Griswold, S Groth, K Günther, C Gutjahr, P Ha, C Haack, C Hallgren, A Halliday, R Halve, L Halzen, F Hamdaoui, H Minh, M Handt, M Hanson, K Hardin, J Harnisch, A Hatch, P Haungs, A Häußler, J Helbing, K Hellrung, J Hermannsgabner, J Heuermann, L Heyer, N Hickford, S Hidvegi, A Hill, C Hill, G Hoffman, K Hori, S Hoshina, K Hou, W Huber, T Hultqvist, K Hünnefeld, M Hussain, R Hymon, K In, S Ishihara, A Jacquart, M Janik, O Jansson, M Japaridze, G Jeong, M Jin, M Jones, B Kamp, N Kang, D Kang, W Kang, X Kappes, A Kappesser, D Kardum, L Karg, T Karl, M Karle, A Katil, A Katz, U Kauer, M Kelley, J Zathul, A Kheirandish, A Kiryluk, J Klein, S Kochocki, A Koirala, R Kolanoski, H Kontrimas, T Köpke, L Kopper, C Koskinen, D Koundal, P Kovacevich, M Kowalski, M Kozynets, T Krishnamoorthi, J Kruiswijk, K Krupczak, E Kumar, A Kun, E Kurahashi, N Lad, N Gualda, C Lamoureux, M Larson, M Latseva, S Lauber, F Lazar, J Lee, J DeHolton, K Leszczyńska, A Lincetto, M Liu, Y Liubarska, M Lohfink, E Love, C Mariscal, C Lu, L Lucarelli, F Luszczak, W Lyu, Y Madsen, J Magnus, E Mahn, K Makino, Y Manao, E Mancina, S Sainte, W Mariş, I Marka, S Marka, Z Marsee, M Martinez-Soler, I Maruyama, R Mayhew, F McElroy, T McNally, F Mead, J Meagher, K Mechbal, S Medina, A Meier, M Merckx, Y Merten, L Micallef, J Mitchell, J Montaruli, T Moore, R Morii, Y Morse, R Moulai, M Mukherjee, T Naab, R Nagai, R Nakos, M Naumann, U Necker, J Negi, A Neumann, M Niederhausen, H Nisa, M Noell, A Novikov, A Nowicki, S Pollmann, A O'Dell, V Oeyen, B Olivas, A Orsoe, R Osborn, J O'Sullivan, E Pandya, H Park, N Parker, G Paudel, E Paul, L Heros, C Peterson, J Philippen, S Pizzuto, A Plum, M Pontén, A Popovych, Y Rodriguez, M Pries, B Procter-Murphy, R Przybylski, G Raab, C Rack-Helleis, J Rawlins, K Rechav, Z Rehman, A Reichherzer, P Resconi, E Reusch, S Rhode, W Riedel, B Rifaie, A Roberts, E Robertson, S Rodan, S Roellinghoff, G Rongen, M Rosted, A Rott, C Ruhe, T Ruohan, L Ryckbosch, D Safa, I Saffer, J Salazar-Gallegos, D Sampathkumar, P Herrera, S Sandrock, A Santander, M Sarkar, S Savelberg, J Savina, P Schaufel, M Schieler, H Schindler, S Schlickmann, L Schlüter, B Schlüter, F Schmeisser, N Schmidt, T Schneider, J Schröder, F Schumacher, L Sclafani, S Seckel, D Seikh, M Seunarine, S Shah, R Shefali, S Shimizu, N Silva, C Silva, M Skrzypek, B Smithers, B Snihur, R Soedingrekso, J Søgaard, A Soldin, D Soldin, P Sommani, G Spannfellner, C Spiczak, G Spiering, C Stamatikos, M Stanev, T Stezelberger, T Stürwald, T Stuttard, T Sullivan, G Taboada, I Ter-Antonyan, S Thiesmeyer, M Thompson, W Thwaites, J Tilav, S Tollefson, K Tönnis, C Toscano, S Tosi, D Trettin, A Tung, C Turcotte, R Twagirayezu, J Elorrieta, M Upadhyay, A Upshaw, K Vaidyanathan, A Valtonen-Mattila, N Vandenbroucke, J Eijndhoven, N Vannerom, D Santen, J Vara, J Veitch-Michaelis, J Venugopal, M Vereecken, M Verpoest, S Veske, D Vijai, A Walck, C Wang, Y Weaver, C Weigel, P Weindl, A Weldert, J Wen, A Wendt, C Werthebach, J Weyrauch, M Whitehorn, N Wiebusch, C Williams, D Witthaus, L Wolf, A Wolf, M Wrede, G Xu, X Yanez, J Yildizci, E Yoshida, S Young, R Yu, S Yuan, T Zhang, Z Zhelnin, P Zilberman, P Zimmerman, M (12 Dec 2023) http://arxiv.org/abs/2312.11515v3
Morse theory for complexes of groups
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Thu, 29 Feb 2024
16:00
Lecture Room 4

A new approach to modularity

Andrew Wiles
(University of Oxford)
Abstract

In the 1960's Langlands proposed a generalisation of Class Field Theory. I will review this and describe a new approach using the trace formua as well as some analytic arguments reminiscent of those used in the classical case. In more concrete terms the problem is to prove general modularity theorems, and I will explain the progress I have made on this problem.

Thu, 22 Feb 2024
16:00
Lecture Room 4

Tangent spaces of Schubert varieties

Rong Zhou
(University of Cambridge)
Abstract

Schubert varieties in (twisted) affine Grassmannians and their singularities are of interest to arithmetic geometers because they model the étale local structure of the special fiber of Shimura varieties. In this talk, I will discuss a proof of a conjecture of Haines-Richarz classifying the smooth locus of Schubert varieties, generalizing a classical result of Evens-Mirkovic. The main input is to obtain a lower bound for the tangent space at a point of the Schubert variety which arises from considering certain smooth curves passing through it. In the second part of the talk, I will explain how in many cases, we can prove this bound is actually sharp, and discuss some applications to Shimura varieties. This is based on joint work with Pappas and Kisin-Pappas.

Thu, 01 Feb 2024
16:00
Lecture Room 4, Mathematical Institute

Tame Triple Product Periods

Alice Pozzi
(University of Bristol )
Abstract

A recent conjecture proposed by Harris and Venkatesh relates the action of derived Hecke operators on the space of weight one modular forms to certain Stark units. In this talk, I will explain how this can be rephrased as a conjecture about "tame" analogues of triple product periods for a triple of mod p eigenforms of weights (2,1,1). I will then present an elliptic counterpart to this conjecture relating a tame triple product period to a regulator for global points of elliptic curves in rank 2. This conjecture can be proved in some special cases for CM weight 1 forms, with techniques resonating with the so-called Jochnowitz congruences. This is joint work in preparation with Henri Darmon. 

Thu, 15 Feb 2024
16:00
Lecture Room 4, Mathematical Institute

Strong Bounds for 3-Progressions

Zander Kelley
(UIUC)
Abstract
Suppose you have a set $A$ of integers from $\{1, 2,\ldots, N\}$ that contains at least $N / C$ elements.
Then for large enough $N$, must $A$ contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?
In 1953, Roth showed that this is indeed the case when $C \approx \log \log N$, while Behrend in 1946 showed that $C$ can be at most $2^{\sqrt{\log N}}$ by giving an explicit construction of a large set with no 3-term progressions.
Since then, the problem has been a cornerstone of the area of additive combinatorics.
Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on $C$ to $C = (\log N)^{1 + c}$, for some constant $c > 0$.
This talk will describe our work which shows that the same holds when $C \approx 2^{(\log N)^{1/12}}$, thus getting closer to Behrend's construction.
Based on a joint work with Raghu Meka.
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