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April 3, 2025

Avi Wigderson

The Institute of Advanced Studies, Princeton

The value of errors in proofs.


The fascinating journey from Turing's 1936 R ≠ RE to the 2020 breakthrough of MIP*=RE.

In the year 2020, a group of theoretical computer scientists posted a paper on the ArXiv with the strange-looking title "MIP* = RE", impacting and surprising not only complexity theory but also some areas of math and physics. Specifically, it resolved, in the negative, the "Connes' embedding conjecture" in the area of von Neumann algebras, and the "Tsirelson problem" in quantum information theory. You can find the paper here. As it happens, both acronyms MIP* and RE represent proof systems, of a very different nature. To explain them, we'll take a meandering journey through the classical and modern definitions of proof. I hope to explain how the methodology of computational complexity theory, especially modeling and classification (both problems and proofs) by algorithmic efficiency, naturally leads to the generation of new such notions and results (and more acronyms, like NP). A special focus will be on notions of proof which allow interaction, randomness, and errors, and their surprising power and magical properties. This talk requires no special background.

At 16:30 in AulaC03 of the Università degli Studi di Milano, via Mangiagalli 25

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About the speaker

Avi Wigderson is justifiably credited with deepening the connections between mathematics and theoretical computer science. First and foremost, it is difficult to overestimate the impact of Wigderson's work in the field of computational complexity, the branch of theoretical computer science concerned with the efficiency of algorithms. He has conducted research on every major open problem in this field, and in some real sense the field has grown around him. One key aspect of this work concerns the role of randomness in algorithms. Wigderson and collaborators N. Nisan and R. Impagliazza have shown that for every fast random algorithm (involving coin flips that can introduce errors), under natural conditions, there exists a deterministic algorithm that is almost as fast.

Wigderson has also given fundamental contributions to applications of internet security through his study of zero-knowledge proofs, which concern proofs of a claim which furnish no additional information than the validity of the claim. In addition, Widgerson’s work on the zig-zag product has shown deep connections between computational complexity, graph theory and group theory and has become a widely used tool in cryptography. Wigderson’s fundamental and lasting contributions have been recognized in the awarding of numerous prizes and awards, including the Nevanlinna Prize (1994), Gödel Prize (2009), Knuth Prize (2019), Abel Prize (2021) and the Turing Award (2023).

Wigderson received his Ph.D. in Computer Science from Princeton University in 1983 under the direction of Richard Lipton with a dissertation entitled "Studies in Computational Complexity". Following three temporary positions at distinguished institutions, Wigderson joined the faculty of Hebrew University in 1986, rising to Full Professor in 1991. In 1999 he accepted a joint appointment at the Institute for Advanced Study, Princeton. As of 2003, this is full time appointment and Wigderson is currently the Herbert H. Maass Professor in the School of Mathematics of the IAS, Princeton.

May 12, 2025

Simon Brendle

Columbia University, New York

Minimal Surfaces and the Isoperimetric Inequality.


The isoperimetric inequality has a long history in mathematics. In this lecture, we will discuss how the isoperimetric inequality can be generalized to submanifolds in Euclidean space. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds of codimension at most 2, answering a question going back to work of Carleman. The proof of that inequality is inspired by, but does not actually use, optimal transport.

At 16:30 in Aula T.1.1 (first floor) in the Edificio 13 (Trifoglio) of the Politecnico di Milano, via Bonardi 9

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About the speaker

Simon Brendle is a distinguished German-American mathematician renowned for his profound contributions to differential geometry, geometric analysis and partial differential equations. He earned his doctorate from the University of Tübingen in 2001 under the supervision of Gerhard Huisken. Brendle has held academic positions at Stanford University and is currently a professor at Columbia University.

Throughout his career, Brendle has achieved significant breakthroughs in geometry, including results on the Yamabe compactness conjecture, the differentiable sphere theorem (jointly with Richard Schoen), the Lawson conjecture, and the Perelman conjecture. His work also encompasses studies on singularity formation in the mean curvature flow, the Yamabe flow and the Ricci flow.

In recognition of his outstanding research, Brendle has received numerous honors, such as the EMS Prize in 2012, the Bôcher Prize from the American Mathematical Society in 2014, the Simons Investigator Award in 2017, and the Fermat Prize in 2017. Most recently, he was awarded the Breakthrough Prize in Mathematics in 2024 for "a series of remarkable leaps in differential geometry".

June 5, 2025

Jacob Tsimerman

University of Toronto

Finiteness Results in Shimura Varieties.


Shimura Varieties are higher dimensional analogues of modular curves, and they play a foundational role in modern number theory. The most familiar Shimura varieties are the moduli spaces of Abelian varieties, and in this context we have a wealth of diophantine results, both in the number field and function field setting: Finiteness of S-rational points, the Tate conjecture, the Shafarevich conjecture, semisimplicity of Galois representations, and others. We focus on the exceptional setting for Shimura varieties, where the lack of a moduli interpretation makes matters more difficult. We explain some analogues of the aforementioned results. Crucial to this is the existence of canonical integral models, which we construct at almost all primes. This is joint work with Ben Bakker and Ananth Shankar.

At 16:30 in Aula Sironi of the Università di Milano-Bicocca, Edificio U4-Tellus, Piazza della Scienza 4

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About the speaker

Jacob Tsimerman is a brilliant young mathematician who, despite his tender age, is widely and increasingly recognized for his outstanding contributions to number theory and arithmetic geometry. His undeniable mathematical potential was demonstrated by having received twice a gold medal at the International Mathematics Olympiad in 2003 and in 2004 with a perfect score. He completed his Ph.D. at Princeton University in 2011 under the supervision of Peter Sarnak and he is currently a Full Professor of Mathematics at the University of Toronto.

Among Tsimerman's most significant achievements are the proof of the Ax–Lindemann theorem for moduli spaces, developed in collaboration with Jonathan Pila, and his joint work with Benjamin Bakker on the geometric torsion conjecture for abelian varieties with real multiplication. He is particularly celebrated for his central role in the resolution of the André–Oort conjecture, a longstanding open problem and a major milestone at the intersection of arithmetic geometry and model theory.

Tsimerman's exceptional work has been recognized with numerous awards, including the SASTRA Ramanujan Prize (2015), the Ribenboim Prize (2016), the Coxeter-James Prize (2019), the New Horizons in Mathematics Prize (2022), the ICBS Frontiers of Science Award (2023), the Ostrowski Prize (2023) and the John L. Synge Award (2024). In 2025, he was elected a Fellow of the Royal Society, reflecting his significant impact on the mathematical community.