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BEGIN:VEVENT
SUMMARY:Tom Coates (Imperial)
DTSTART:20230824T100000Z
DTEND:20230824T110000Z
DTSTAMP:20260404T094802Z
UID:Danger2023/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/1/">Machine Learning Detects Terminal Singularities</a>\nby Tom Coat
 es (Imperial) as part of DANGER3: Data\, Numbers\, and Geometry\n\n\nAbstr
 act\nI will describe an example of AI-assisted mathematical discovery\, wh
 ich is joint work with Al Kasprzyk and Sara Veneziale. We consider the pro
 blem of determining whether a toric variety is a $\\mathbb{Q}$-Fano variet
 y. $\\mathbb{Q}$-Fano varieties are Fano varieties that have mild singular
 ities called terminal singularities\; they play a key role in the Minimal 
 Model Programme. Except for the special case of weighted projective spaces
 \, no efficient global algorithm for checking terminality of toric varieti
 es was known.\n\nWe show that\, for eight-dimensional Fano toric varieties
  $X$ of Picard rank two\, a simple feedforward neural network can predict 
 with 95% accuracy whether or not $X$ has terminal singularities. The input
  data to the neural network is the weights of the toric variety $X$\; this
  is a matrix of integers that determines $X$. We use the neural network to
  give the first sketch of the landscape of $\\mathbb{Q}$-Fano varieties in
  eight dimensions.\n\nInspired by the ML analysis\, we formulate and prove
  a new global\, combinatorial criterion for a toric variety of Picard rank
  two to have terminal singularities. This gives new evidence that machine 
 learning can be a powerful tool in developing mathematical conjectures and
  accelerating theoretical discovery.\n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matija Kazalicki (Zagreb)
DTSTART:20230824T120000Z
DTEND:20230824T130000Z
DTSTAMP:20260404T094802Z
UID:Danger2023/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/2/">Ranks of elliptic curves and deep neural networks</a>\nby Matija
  Kazalicki (Zagreb) as part of DANGER3: Data\, Numbers\, and Geometry\n\n\
 nAbstract\nDetermining the rank of an elliptic curve $E/\\mathbb{Q}$ is a 
 difficult problem. In applications such as the search for curves of high r
 ank\, one often relies on heuristics to estimate the analytic rank (which 
 is equal to the rank under the Birch and Swinnerton-Dyer conjecture). \n\n
 In this talk\, we discuss a novel rank classification method based on deep
  convolutional neural networks (CNNs). The method takes as input the condu
 ctor of $E$ and a sequence of normalized Frobenius traces $a_p$ for primes
  $p$ in a certain range ($p<10^k$ for $k=3\,4\,5$)\, and aims to predict t
 he rank or detect curves of "high" rank. We compare our method with eight 
 simple neural network models of the Mestre-Nagao sums\, which are widely u
 sed heuristics for estimating the rank of elliptic curves.\n\nWe evaluate 
 our method on two datasets: the LMFDB and a custom dataset consisting of e
 lliptic curves with trivial torsion\, conductor up to $10^{30}$\, and rank
  up to $10$. Our experiments demonstrate that the CNNs outperform the Mest
 re-Nagao sums on the LMFDB dataset. On the custom dataset\, the performanc
 e of the CNNs and the Mestre-Nagao sums is comparable. This is joint work 
 with Domagoj Vlah.\n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ed Hirst (Queen Mary)
DTSTART:20230824T131500Z
DTEND:20230824T141500Z
DTSTAMP:20260404T094802Z
UID:Danger2023/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/3/">Machine Learning Sasakian and G2 topology on contact Calabi-Yau 
 7-manifolds</a>\nby Ed Hirst (Queen Mary) as part of DANGER3: Data\, Numbe
 rs\, and Geometry\n\n\nAbstract\nCalabi-Yau links are constructed for all 
 7555 weighted projective spaces with Calabi-Yau 3-fold hypersurfaces. Topo
 logical properties such as the Crowley-Nordström invariants and Sasakian 
 Hodge numbers are computed\, leading to new invariant values and some conj
 ectures on their construction. Machine learning methods are implemented to
  predict these invariants\, as well as to optimise their computation via G
 röbner bases.\n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kathlén Kohn (KTH)
DTSTART:20230824T143000Z
DTEND:20230824T153000Z
DTSTAMP:20260404T094802Z
UID:Danger2023/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/4/">Understanding Linear Convolutional Neural Networks via Sparse Fa
 ctorizations of Real Polynomials</a>\nby Kathlén Kohn (KTH) as part of DA
 NGER3: Data\, Numbers\, and Geometry\n\n\nAbstract\nThis talk will explain
  that Convolutional Neural Networks without activation parametrize semialg
 ebraic sets of real homogeneous polynomials that admit a certain sparse fa
 ctorization. We will investigate how the geometry of these semialgebraic s
 ets (e.g.\, their singularities and relative boundary) changes with the ne
 twork architecture. Moreover\, we will explore how these geometric propert
 ies affect the optimization of a loss function for given training data. Th
 is talk is based on joint work with Guido Montúfar\, Vahid Shahverdi\, an
 d Matthew Trager.\n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Challenger Mishra (Cambridge)
DTSTART:20230825T100000Z
DTEND:20230825T110000Z
DTSTAMP:20260404T094802Z
UID:Danger2023/5
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/5/">Mathematical Conjecture Generation and Machine Intelligence</a>\
 nby Challenger Mishra (Cambridge) as part of DANGER3: Data\, Numbers\, and
  Geometry\n\n\nAbstract\nConjectures hold a special status in mathematics.
  Good conjectures epitomise milestones in mathematical discovery\, and hav
 e historically inspired new mathematics and shaped progress in theoretical
  physics. Hilbert’s list of 23 problems and André Weil’s conjectures 
 oversaw major developments in mathematics for decades. Crafting conjecture
 s can often be understood as a problem in pattern recognition\, for which 
 Machine Learning (ML) is tailor-made. In this talk\, I will propose a fram
 ework that allows a principled study of a space of mathematical conjecture
 s. Using this framework and exploiting domain knowledge and machine learni
 ng\, we generate a number of conjectures in number theory and group theory
 . I will present evidence in support of some of the resulting conjectures 
 and present a new theorem. I will lay out a vision for this endeavour\, an
 d conclude by posing some general questions about the pipeline.\n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Malik Amir (SolutionAI and Montréal)
DTSTART:20230825T120000Z
DTEND:20230825T130000Z
DTSTAMP:20260404T094802Z
UID:Danger2023/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/6/">Data-Driven Insights into the Rank of Elliptic Curves of Prime C
 onductors</a>\nby Malik Amir (SolutionAI and Montréal) as part of DANGER3
 : Data\, Numbers\, and Geometry\n\n\nAbstract\nIn this presentation\, we e
 xplore the intersection of data science and elliptic curves of prime condu
 ctor. We will begin with a quick introduction to elliptic curves before in
 troducing the celebrated Birch and Swinnerton-Dyer conjecture. We will dis
 cuss the original insight of Birch and Swinnerton-Dyer concerning the trac
 es of Frobenius and what they know about certain mathematical data attache
 d to elliptic curves. We will be especially interested in the rank of elli
 ptic curves of prime conductor. All along this talk\, we will present expe
 riments performed on the largest known dataset of such elliptic  curves : 
 the Bennett-Gherga-Retchnizer dataset\, and will explicitly formulate open
  questions based on these observations. We will discuss some tension betwe
 en data and the minimalist conjecture which stipulates that the average ra
 nk should be $\\frac{1}{2}$. Among the various data scientific experiments
  that were performed\, we will describe an interesting bias that exists be
 tween the distribution of the 2-torsion coefficients and the distribution 
 of the rank. Finally we will discuss the importance of simple machine lear
 ning models for predicting the rank based on the traces of Frobenius.\n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elli Heyes (LIMS)
DTSTART:20230825T131500Z
DTEND:20230825T141500Z
DTSTAMP:20260404T094802Z
UID:Danger2023/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/7/">New Calabi-Yau Manifolds from Genetic Algorithms</a>\nby Elli He
 yes (LIMS) as part of DANGER3: Data\, Numbers\, and Geometry\n\n\nAbstract
 \nCalabi-Yau manifolds can be obtained as hypersurfaces in toric varieties
  built from reflexive polytopes. We generate reflexive polytopes in variou
 s dimensions using a genetic algorithm. As a proof of principle\, we demon
 strate that our algorithm reproduces the full set of reflexive polytopes i
 n two and three dimensions\, and in four dimensions with a small number of
  vertices and points. Motivated by this result\, we construct five-dimensi
 onal reflexive polytopes with the lowest number of vertices and points. By
  calculating the normal form of the polytopes\, we establish that many of 
 these are not in existing datasets and therefore give rise to new Calabi-Y
 au four-folds.\n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katia Matcheva (Florida)
DTSTART:20230825T143000Z
DTEND:20230825T153000Z
DTSTAMP:20260404T094802Z
UID:Danger2023/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/Dange
 r2023/8/">Deep Learning Symmetries in Physics from First Principles</a>\nb
 y Katia Matcheva (Florida) as part of DANGER3: Data\, Numbers\, and Geomet
 ry\n\n\nAbstract\nSymmetries are the cornerstones of modern theoretical ph
 ysics\, as they imply fundamental conservation laws. The recent boom in AI
  algorithms and their successful application to high-dimensional large dat
 asets from all aspects of life motivates us to approach the problem of dis
 covery and identification of symmetries in physics as a machine-learning t
 ask. In a series of papers\, we have developed and tested a deep-learning 
 algorithm for the discovery and identification of the continuous group of 
 symmetries present in a labeled dataset. We use fully connected neural net
 work architectures to model the symmetry transformations and the correspon
 ding generators. Our proposed loss functions ensure that the applied trans
 formations are symmetries and that the corresponding set of generators is 
 orthonormal and forms a closed algebra. One variant of our method is desig
 ned to discover symmetries in a reduced-dimensionality latent space\, whil
 e another variant is capable of obtaining the generators in the canonical 
 sparse representation. Our procedure is completely agnostic and has been v
 alidated with several examples illustrating the discovery of the symmetrie
 s behind the orthogonal\, unitary\, Lorentz\, and exceptional Lie groups.\
 n
LOCATION:https://stable.researchseminars.org/talk/Danger2023/8/
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