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11:00am to 12:20pm - 340P - Machine Learning Mingshuo Liu - (UC Irvine) Integrating Neural and Symbolic Reasoning: A Deep Dive into AlphaGeometry for Automated Geometry Theorem Proving Abstract: Mathematical theorem proving has long been a goal in artificial intelligence (AI), but progress has been slow due to the scarcity of human-generated proofs that can be translated into machine-readable formats. Geometry, in particular, poses unique challenges due to the difficulty of representing geometric figures and relations in a symbolic format that machines can easily process. Researchers at Google DeepMind have recently proposed a novel architecture, AlphaGeometry, which utilizes a neuro-symbolic approach. This system integrates a neural language model trained on a large corpus of synthetic theorems and proofs with a symbolic reasoning engine that ensures logical deductive correctness. On the IMO-AG-30 benchmark, AlphaGeometry achieved remarkable accuracy, solving 25 out of 30 problems, nearly matching the gold medalist benchmark of 25.9/30. In this talk, I will provide an overview of the construction and integration of the symbolic reasoning engine and the neural language model in AlphaGeometry. Specifically, I will detail how the symbolic reasoning engine, comprising the Deductive Database (DD) and Algebraic Reasoning (AR), operates. The DD manages formal geometric deductions by applying predefined theorems and axioms, while AR complements this by resolving complex algebraic expressions inherent in geometric proofs. Furthermore, I will explain the Neural Language Model utilized in AlphaGeometry, highlighting how it differs from known large language models such as GPT-4. This comparison will shed light on the potential of general-purpose language models in solving mathematical problems, offering insights into the broader capabilities of large language models in Math. |
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4:00pm to 5:30pm - RH 440 R - Logic Set Theory Yeonwook Jung - (UC Irvine) Topological Erdős Similarity Conjecture and Strong Measure Zero Sets It is well-known that a finite set is universal, that is, each Lebesgue measurable set with positive measure contains an affine copy of a finite set. The Erdős similarity conjecture, which remains open, states that there is no infinite universal set. In 2022, Gallagher, Lai, and Weber considered a topological version of this conjecture, defining a set to be topologically universal if each dense G-delta set contains an affine copy of the set. They conjectured that there are no such uncountable sets. In this thesis, we give a full classification of topologically universal sets as a special subfamily of measure zero sets. As a corollary, we prove that the topological Erdős similarity conjecture is independent of ZFC. We generalize this result to arbitrary locally compact Polish groups, and use the measure-category duality to pose and investigate the full-measure Erdős similarity conjecture. |
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4:00pm to 5:00pm - RH 306 - Differential Geometry Ilka Agricola - (University of Marburg) G_2 and SU(3) manifolds via spinors. Abstract: We present a uniform description of SU(3) structures in dimension 6 as well as G_2 structures in dimension 7 in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach sheds new light on connections with torsion and their invariants. |
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2:00pm to 3:00pm - RH340P - Special Colloquium Ilka Agricola - (University of Marburg) The challenges of predatory journals and paper mills in mathematics: A glimpse at the parallel universe of fake science. In November 2023, Clarivate Plc announced that it had excluded the entire field of mathematics from the most recent edition of its influential list of authors of highly cited papers because of massive citation manipulation, which in return influences the so-called “Shanghai ranking” of top universities (or those claiming to be top). While most mathematicians would probably not care, the exclusion is in fact the tip of the iceberg of a parallel universe of predatory and mega-journals whose main purpose is to offer publishing opportunities for whoever is willing to pay the right price. I will explain how the system works, why we should care, and what measures we can all take against. In preparation, I invite you to think about the following questions: How often have you been contacted in the past months to attend a conference not in your field / submit a paper to or edit a special issue in a journal you don’t know / review an article within 10 days or so? What do you know about the following journals: “Mathematics”, “Axioms” (published by MDPI), “Chaos, solitons, fractals” (Elsevier), “Journal of Difference Equations” (Springer)? Do you know the following mathematicians: Abdon Atangana, Dumitru Baleanu, Hari M. Srivastava? This talk is related to my work as Chair of the Committee on Electronic Information and Communication (= publishing) of the International Mathematical Union. |
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3:00pm to 4:00pm - RH 510R - Combinatorics and Probability Pedro Abdalla - (UCI) Introduction to Robust Statistics II Robust Statistics is a classical topic that dates back to the seminal work of Huber in the 1980s. In essence, the main goal of the field is to account for the effect of outliers when performing estimation tasks, such as mean estimation. A recent line of research, inspired by the seminal work of Catoni, has revisited some classical problems in robust statistics from a non-asymptotic perspective. The goal of this short seminar series is to introduce the key ideas related to robust estimation and discuss various notions of robustness, including heavy-tailed distributions and adversarial contamination. The primary example will be the mean estimation problem, and if time permits, I will also cover covariance estimation |
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9:00am to 9:50am - Zoom - Inverse Problems Yavar Kian - (University of Rouen Normandy) Determination of quasilinear terms from restricted data |
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3:00pm to 4:00pm - RH 306 - Number Theory Sehun Jeong - (Claremont Graduate University) Primitive elements in number fields and Diophantine avoidance The famous primitive element theorem states that every number field K is of the form Q(a) for some element a in K, called a primitive element. In fact, it is clear from the proof of this theorem that not only there are infinitely many such primitive elements in K, but in fact most elements in K are primitive. This observation raises the question about finding a primitive element of small “size”, where the standard way of measuring size is with the use of a height function. We discuss some conjectures and known results in this direction, as well as some of our recent work on a variation of this problem which includes some additional avoidance conditions. Joint work with Lenny Fukshansky at Claremont McKenna College. |
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4:00pm to 5:00pm - RH 306 - Colloquium Ilka Agricola - (University of Marburg) Holonomy - a success concept of modern differential geometry Holonomy is a prime example of mathematical intuition and creativity - it generalises our school knowledge about the sum of angles in a triangle and led to `Berger’s holonomy theorem’ from 1954 which turned out to be a most successful research programme for differential geometry for over 50 years. We are going to tell the story of this development, how holonomy relates to curvature and advanced symmetry concepts, including a small detour to theoretical physics and what Calabi-Yau manifolds have to do with it. We conclude by a small outlook to recent results. |