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1. An Examination of Linear Combinations of Skew-Adjoint Type Algebraic Curvature Tensors

   Brundan, J ( March 2021 , PUMP journal of undergraduate research)

   Dunn, C ; Raianu, S (Ed.)

   In this paper a new potential invariant of algebraic curvature tensors, the sig- nature, will be examined. Furthermore, linear combinations of skew-adjoint type algebraic curvature tensors will be thoroughly examined so as to provide some insight into the possible forms of algebraic curvature tensors. 

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2. Concise tensors of minimal border rank

   https://doi.org/10.1007/s00208-023-02569-y  

   Jelisiejew, Joachim ; Landsberg, J. M. ; Pal, Arpan ( April 2023 , Mathematische Annalen)

   We determine defining equations for the set of concise tensors of minimal border rank in Cm⊗Cm⊗Cm when m = 5 and the set of concise minimal border rank 1∗-generic tensors when m = 5, 6. We solve the classical problem in algebraic complexity theory of classifying minimal border rank tensors in the special case m = 5. Our proofs utilize two recent developments: the 111-equations defined by Buczy´nska–Buczy´nski and results of Jelisiejew–Šivic on the variety of commuting matrices. We introduce a new algebraic invariant of a concise tensor, its 111-algebra, and exploit it to give a strengthening of Friedland’s normal form for 1-degenerate tensors satisfying Strassen’s equations. We use the 111-algebra to characterize wild minimal border rank tensors and classify them in C5⊗C5⊗C5. 

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3. Smooth connectivity in real algebraic varieties

   https://doi.org/10.1007/s11075-024-01952-3  

   Cummings, Joseph ; Hauenstein, Jonathan_D ; Hong, Hoon ; Smyth, Clifford_D ( October 2024 , Numerical Algorithms)

   A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach.

    

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4. The Octatope Abstract Domain for Verification of Neural Networks.

   https://doi.org/10.1007/978-3-031-27481-7_26  

   Bak, S. ; Dohmen, T. ; Subramani, K. ; Trivedi, A. ; Velasquez, A. ; Wojciechowski, P. ( March 2023 , Formal Methods. FM 2023.)

   Chechik, M. ; Katoen, JP. ; Leucker, M. (Ed.)

   Efficient verification algorithms for neural networks often depend on various abstract domains such as intervals, zonotopes, and linear star sets. The choice of the abstract domain presents an expressiveness vs. scalability trade-off: simpler domains are less precise but yield faster algorithms. This paper investigates the octatope abstract domain in the context of neural net verification. Octatopes are affine transformations of n-dimensional octagons—sets of unit-two-variable-per-inequality (UTVPI) constraints. Octatopes generalize the idea of zonotopes which can be viewed as an affine transformation of a box. On the other hand, octatopes can be considered as a restriction of linear star set, which are affine transformations of arbitrary H-Polytopes. This distinction places octatopes firmly between zonotopes and star sets in their expressive power, but what about the efficiency of decision procedures? An important analysis problem for neural networks is the exact range computation problem that asks to compute the exact set of possible outputs given a set of possible inputs. For this, three computational procedures are needed: 1) optimization of a linear cost function; 2) affine mapping; and 3) over-approximating the intersection with a half-space. While zonotopes allow an efficient solution for these approaches, star sets solves these procedures via linear programming. We show that these operations are faster for octatopes than the more expressive linear star sets. For octatopes, we reduce these problems to min-cost flow problems, which can be solved in strongly polynomial time using the Out-of-Kilter algorithm. Evaluating exact range computation on several ACAS Xu neural network benchmarks, we find that octatopes show promise as a practical abstract domain for neural network verification. 

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5. Introductory topics in derived algebraic geometry

   Tony Pantev ; Gabriele Vezzosi ( October 2021 , Panoramas et synthèses)

   Cantat, S. ; Massart, P ; Dalibard, A.-L. ; Mezard, A ; Guillin, A. ; Rigot, S. ; Remy, B. (Ed.)

   We give a quick introduction to derived algebraic geometry (DAG) sampling basic constructions and techniques. We discuss affine derived schemes, derived algebraic stacks, and the Artin-Lurie representability theorem. Through the example of deformations of smooth and proper schemes, we explain how DAG sheds light on classical deformation theory. In the last two sections, we introduce differential forms on derived stacks, and then specialize to shifted symplectic forms, giving the main existence theorems proved in [17]. 

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