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BEGIN:VEVENT
SUMMARY:Tom Coates (Imperial)
DTSTART:20230824T100000Z
DTEND:20230824T110000Z
DTSTAMP:20260422T212858Z
UID:Danger2023/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/1
 /">Machine Learning Detects Terminal Singularities</a>\nby Tom Coates (Imp
 erial) as part of DANGER3: Data\, Numbers\, and Geometry\n\n\nAbstract\nI 
 will describe an example of AI-assisted mathematical discovery\, which is 
 joint work with Al Kasprzyk and Sara Veneziale. We consider the problem of
  determining whether a toric variety is a $\\mathbb{Q}$-Fano variety. $\\m
 athbb{Q}$-Fano varieties are Fano varieties that have mild singularities c
 alled terminal singularities\; they play a key role in the Minimal Model P
 rogramme. Except for the special case of weighted projective spaces\, no e
 fficient global algorithm for checking terminality of toric varieties 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 t
 o the neural network is the weights of the toric variety $X$\; this is a m
 atrix of integers that determines $X$. We use the neural network to give t
 he 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 learnin
 g can be a powerful tool in developing mathematical conjectures and accele
 rating theoretical discovery.\n
LOCATION:https://researchseminars.org/talk/Danger2023/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matija Kazalicki (Zagreb)
DTSTART:20230824T120000Z
DTEND:20230824T130000Z
DTSTAMP:20260422T212858Z
UID:Danger2023/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/2
 /">Ranks of elliptic curves and deep neural networks</a>\nby Matija Kazali
 cki (Zagreb) as part of DANGER3: Data\, Numbers\, and Geometry\n\n\nAbstra
 ct\nDetermining the rank of an elliptic curve $E/\\mathbb{Q}$ is a difficu
 lt problem. In applications such as the search for curves of high rank\, o
 ne often relies on heuristics to estimate the analytic rank (which is equa
 l to the rank under the Birch and Swinnerton-Dyer conjecture). \n\nIn this
  talk\, we discuss a novel rank classification method based on deep convol
 utional neural networks (CNNs). The method takes as input the conductor 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 the 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 used heu
 ristics for estimating the rank of elliptic curves.\n\nWe evaluate our met
 hod on two datasets: the LMFDB and a custom dataset consisting of elliptic
  curves with trivial torsion\, conductor up to $10^{30}$\, and rank up to 
 $10$. Our experiments demonstrate that the CNNs outperform the Mestre-Naga
 o sums on the LMFDB dataset. On the custom dataset\, the performance of th
 e CNNs and the Mestre-Nagao sums is comparable. This is joint work with Do
 magoj Vlah.\n
LOCATION:https://researchseminars.org/talk/Danger2023/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ed Hirst (Queen Mary)
DTSTART:20230824T131500Z
DTEND:20230824T141500Z
DTSTAMP:20260422T212858Z
UID:Danger2023/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/3
 /">Machine Learning Sasakian and G2 topology on contact Calabi-Yau 7-manif
 olds</a>\nby Ed Hirst (Queen Mary) as part of DANGER3: Data\, Numbers\, an
 d Geometry\n\n\nAbstract\nCalabi-Yau links are constructed for all 7555 we
 ighted projective spaces with Calabi-Yau 3-fold hypersurfaces. Topological
  properties such as the Crowley-Nordström invariants and Sasakian Hodge n
 umbers are computed\, leading to new invariant values and some conjectures
  on their construction. Machine learning methods are implemented to predic
 t these invariants\, as well as to optimise their computation via Gröbner
  bases.\n
LOCATION:https://researchseminars.org/talk/Danger2023/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kathlén Kohn (KTH)
DTSTART:20230824T143000Z
DTEND:20230824T153000Z
DTSTAMP:20260422T212858Z
UID:Danger2023/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/4
 /">Understanding Linear Convolutional Neural Networks via Sparse Factoriza
 tions of Real Polynomials</a>\nby Kathlén Kohn (KTH) as part of DANGER3: 
 Data\, Numbers\, and Geometry\n\n\nAbstract\nThis talk will explain that C
 onvolutional Neural Networks without activation parametrize semialgebraic 
 sets of real homogeneous polynomials that admit a certain sparse factoriza
 tion. We will investigate how the geometry of these semialgebraic sets (e.
 g.\, their singularities and relative boundary) changes with the network a
 rchitecture. Moreover\, we will explore how these geometric properties aff
 ect the optimization of a loss function for given training data. This talk
  is based on joint work with Guido Montúfar\, Vahid Shahverdi\, and Matth
 ew Trager.\n
LOCATION:https://researchseminars.org/talk/Danger2023/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Challenger Mishra (Cambridge)
DTSTART:20230825T100000Z
DTEND:20230825T110000Z
DTSTAMP:20260422T212858Z
UID:Danger2023/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/5
 /">Mathematical Conjecture Generation and Machine Intelligence</a>\nby Cha
 llenger Mishra (Cambridge) as part of DANGER3: Data\, Numbers\, and Geomet
 ry\n\n\nAbstract\nConjectures hold a special status in mathematics. Good c
 onjectures epitomise milestones in mathematical discovery\, and have histo
 rically inspired new mathematics and shaped progress in theoretical physic
 s. Hilbert’s list of 23 problems and André Weil’s conjectures oversaw
  major developments in mathematics for decades. Crafting conjectures can o
 ften be understood as a problem in pattern recognition\, for which Machine
  Learning (ML) is tailor-made. In this talk\, I will propose a framework t
 hat allows a principled study of a space of mathematical conjectures. Usin
 g this framework and exploiting domain knowledge and machine learning\, we
  generate a number of conjectures in number theory and group theory. I wil
 l present evidence in support of some of the resulting conjectures and pre
 sent a new theorem. I will lay out a vision for this endeavour\, and concl
 ude by posing some general questions about the pipeline.\n
LOCATION:https://researchseminars.org/talk/Danger2023/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Malik Amir (SolutionAI and Montréal)
DTSTART:20230825T120000Z
DTEND:20230825T130000Z
DTSTAMP:20260422T212858Z
UID:Danger2023/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/6
 /">Data-Driven Insights into the Rank of Elliptic Curves of Prime Conducto
 rs</a>\nby Malik Amir (SolutionAI and Montréal) as part of DANGER3: Data\
 , Numbers\, and Geometry\n\n\nAbstract\nIn this presentation\, we explore 
 the intersection of data science and elliptic curves of prime conductor. W
 e will begin with a quick introduction to elliptic curves before introduci
 ng the celebrated Birch and Swinnerton-Dyer conjecture. We will discuss th
 e original insight of Birch and Swinnerton-Dyer concerning the traces of F
 robenius and what they know about certain mathematical data attached to el
 liptic curves. We will be especially interested in the rank of elliptic cu
 rves of prime conductor. All along this talk\, we will present experiments
  performed on the largest known dataset of such elliptic  curves : the Ben
 nett-Gherga-Retchnizer dataset\, and will explicitly formulate open questi
 ons based on these observations. We will discuss some tension between data
  and the minimalist conjecture which stipulates that the average rank shou
 ld be $\\frac{1}{2}$. Among the various data scientific experiments that w
 ere performed\, we will describe an interesting bias that exists between t
 he distribution of the 2-torsion coefficients and the distribution of the 
 rank. Finally we will discuss the importance of simple machine learning mo
 dels for predicting the rank based on the traces of Frobenius.\n
LOCATION:https://researchseminars.org/talk/Danger2023/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elli Heyes (LIMS)
DTSTART:20230825T131500Z
DTEND:20230825T141500Z
DTSTAMP:20260422T212858Z
UID:Danger2023/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/7
 /">New Calabi-Yau Manifolds from Genetic Algorithms</a>\nby Elli Heyes (LI
 MS) as part of DANGER3: Data\, Numbers\, and Geometry\n\n\nAbstract\nCalab
 i-Yau manifolds can be obtained as hypersurfaces in toric varieties built 
 from reflexive polytopes. We generate reflexive polytopes in various dimen
 sions using a genetic algorithm. As a proof of principle\, we demonstrate 
 that our algorithm reproduces the full set of reflexive polytopes in two a
 nd three dimensions\, and in four dimensions with a small number of vertic
 es and points. Motivated by this result\, we construct five-dimensional re
 flexive polytopes with the lowest number of vertices and points. By calcul
 ating the normal form of the polytopes\, we establish that many of these a
 re not in existing datasets and therefore give rise to new Calabi-Yau four
 -folds.\n
LOCATION:https://researchseminars.org/talk/Danger2023/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katia Matcheva (Florida)
DTSTART:20230825T143000Z
DTEND:20230825T153000Z
DTSTAMP:20260422T212858Z
UID:Danger2023/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Danger2023/8
 /">Deep Learning Symmetries in Physics from First Principles</a>\nby Katia
  Matcheva (Florida) as part of DANGER3: Data\, Numbers\, and Geometry\n\n\
 nAbstract\nSymmetries are the cornerstones of modern theoretical physics\,
  as they imply fundamental conservation laws. The recent boom in AI algori
 thms and their successful application to high-dimensional large datasets f
 rom all aspects of life motivates us to approach the problem of discovery 
 and identification of symmetries in physics as a machine-learning task. In
  a series of papers\, we have developed and tested a deep-learning algorit
 hm for the discovery and identification of the continuous group of symmetr
 ies present in a labeled dataset. We use fully connected neural network ar
 chitectures to model the symmetry transformations and the corresponding ge
 nerators. Our proposed loss functions ensure that the applied transformati
 ons are symmetries and that the corresponding set of generators is orthono
 rmal and forms a closed algebra. One variant of our method is designed to 
 discover symmetries in a reduced-dimensionality latent space\, while anoth
 er variant is capable of obtaining the generators in the canonical sparse 
 representation. Our procedure is completely agnostic and has been validate
 d with several examples illustrating the discovery of the symmetries behin
 d the orthogonal\, unitary\, Lorentz\, and exceptional Lie groups.\n
LOCATION:https://researchseminars.org/talk/Danger2023/8/
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