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In joint work in progress with Francesco Campagna, we formulate a conjecture about unlikely intersections in split semiabelian schemes over some ring of S-integers in a number field. Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its "size" should not be too big compared to the "complexity" of the subgroup scheme. Focusing on the case of powers of the multiplicative group over the rational integers, I will show some evidence that we have acquired so far for our conjecture and discuss in some detail the example of a certain arithmetic surface inside the fibered cube of the multiplicative group over the integers, which leads to the sequence of integers from the title.

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We will present an example of adelic line bundles, explain the analytification of the adelic divisors and line bundles, define the volume and then prove the height inequality

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Moderator: Ellen Eischen

Panelists: Abbey Bourdon, Mirela Ciperiani, Wanlin Li and Jan Vonk

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In the first part of the talk, we explain how the study of the Coleman families passing through a critical p-stabilization of an Eisenstein series allows us to establish relations between different kinds of Euler systems. In the second part, we discuss some applications of this theory to the arithmetic of the adjoint of a weight one modular form. Beginning with the Gross-Stark conjecture for the adjoint p-adic L-function, which involves a unit and a p-unit, we discuss possible refinements in the spirit of the Harris-Venkatesh conjecture. Not surprisingly, a conceptual explanation for this phenomenon comes precisely for the degeneration of a triple product when one of the Coleman families specializes to a critical Eisenstein series. This circle of ideas contains the results of joint works with David Loeffler, Alice Pozzi and Victor Rotger.

Registration is required. Seating is first come, first served.

The scale of global societal problems looks daunting. One person, or even a small team, is minuscule relative to the number of people who need help. For example, since ChatGPT has exploded onto the scene, our children's future employment prospects (and current educational experience, with ChatGPT-powered cheating) are in existential danger. There is an area close to mathematics, however, which devises solutions in which problems solve themselves even through self-serving human behavior: game theory.
Po-Shen Loh is a math professor, researcher, and educator who transitioned to devise new solutions for large-scale real-world problems. He will talk about his experience going between the ivory tower of academia and the practicality of the real world, where he ultimately innovated fundamentally new approaches to pandemic control (https://novid.org) and scalable advanced live math education (https://live.poshenloh.com).

He will also discuss educational strategies that build relevant skills to survive this new era of Generative AI (e.g. ChatGPT). He has been working extensively on that problem, and draws from experience teaching across the entire spectrum, from underprivileged schools to the International Math Olympiad.

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The first talk will give an overview of the proof. We will also briefly discuss Tsimerman's proof of the Andre-Oort conjecture for Ag, and will introduce the average form of Colmez's conjecture. We will not define local heights, but we will set up the objects (etale local systems, flat bundles and automorphic vector bundles on Shimura varieties) required to define local heights in the next two lectures.

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Moderator: Alice Silverberg

Panelists: Ziyang Gao, Catherine Hsu, Tatiana Toro & Anthony Várilly-Alvarado

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Registration required to attend in-person or to receive Zoom link

The NP != P conjecture states that short proofs of mathematical theorems are hard to find efficiently in general. It is often speculated that the conjecture has a self-referential nature: even if there was a relatively short proof of the conjecture itself, this proof might be hard to find. In the first part of this talk, Rahul Santhanam will discuss various situations in which the self-referentiality phenomenon for proving complexity lower bounds can be established formally, and will speculate on the extent to which this is a fundamental obstacle to attacking the P vs. NP problem and others of its kind.

Known proof-theoretic barriers to attacking NP vs. P and similar problems apply mostly to non-uniform lower bounds. In the second part of the talk, Santhanam will discuss a new approach to showing uniform lower bounds, which has already led to some progress on lower bounds for weak circuit classes.

Rahul Santhanam is Professor in the Department of Computer Science at the University of Oxford, and Tutorial Fellow at Magdalen College. He received his PhD from the University of Chicago, and taught at the University of Edinburgh after postdoctoral stints in Vancouver and Toronto, before moving to Oxford in 2016. He works across computational complexity theory, including in structural complexity, circuit complexity, proof complexity, foundations of cryptography and learning theory. His recent research is mostly in the study of "meta-complexity", i.e., the complexity of complexity, and its applications to various areas of computer science. 

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Just as a current impacts the effort required by a swimmer, so too does the research atmosphere in a community or conference impact research output.  In this talk, I will discuss various examples of this, both long-standing programs of others, and many examples that I have experienced or witnessed.  In particular, I will discuss different branches of my research program and how their development was impacted by the atmosphere in conferences, seminars, and research communities, and also what I have learned from times when my actions have created counter currents for others. Content note: This talk will include some descriptions of harassment.

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In the second talk, we will define two versions of a local height associated to certain p-adic Galois representations of the Galois group of a local field. We will compare these heights in the unramified case. If time permits, we will also demonstrate examples of these two heights in certain ramified cases in the setting of CM abelian varieties using beautiful results of Colmez as an input.

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To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

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In the third talk, we will define local heights in the setting of families (using work of Liu-Zhu, Scholze, and Diao-Lan-Liu-Zhu), and will compare these heights. If time permits, we will also briefly describe the details involved in comparing heights on 0-dimensional Shimura varieties associated to different automorphic line bundles.

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To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

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To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

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To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

To receive a link to participate remotely, please subscribe to our weekly Math Lecture Announcements email list.

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