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These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones.…
This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that…
Recent studies have noted an intriguing phenomenon termed Neural Collapse, that is, when the neural networks establish the right correlation between feature spaces and the training targets, their last-layer features, together with the…
In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often…
Catastrophic forgetting is a major problem in continual learning, and lots of approaches arise to reduce it. However, most of them are evaluated through task accuracy, which ignores the internal model structure. Recent research suggests…
We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong…
This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a…
Graph convolutions have gained popularity due to their ability to efficiently operate on data with an irregular geometric structure. However, graph convolutions cause over-smoothing, which refers to representations becoming more similar…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
Developing a better understanding of surprising or counterintuitive phenomena has constituted a significant portion of deep learning research in recent years. These include double descent, grokking, and the lottery ticket hypothesis --…
The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is…
We prove that P = NP implies #P = FP by exploiting the topological structure of 3SAT solution spaces. The argument proceeds via a dichotomy: any polynomial-time algorithm for 3SAT either operates without global knowledge of the…
We study higher dimensional gravitational collapse to topological black holes in two steps. Firstly, we construct some (n+2)-dimensional collapsing space-times, which include generalised Lemaitre-Tolman-Bondi-like solutions, and we prove…
Lifted inference algorithms exploit symmetries in probabilistic models to speed up inference. They show impressive performance when calculating unconditional probabilities in relational models, but often resort to non-lifted inference when…
This article is an exposition of recent results on self-similar sets, asserting that if the dimension is smaller than the trivial upper bound then there are almost overlaps between cylinders. We give a heuristic derivation of the theorem…
We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author's pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples…
This paper explores the connection between two recently identified phenomena in deep learning: plasticity loss and neural collapse. We analyze their correlation in different scenarios, revealing a significant association during the initial…
In recent years, the expander decomposition method was used to develop many graph algorithms, resulting in major improvements to longstanding complexity barriers. This powerful hammer has led the community to (1) believe that most problems…
In this work, we introduce a novel methodology for divisive hierarchical clustering. Our divisive (``top-down'') approach is motivated by the fact that agglomerative hierarchical clustering (``bottom-up''), which is commonly used for…
In this paper the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style are studied when the input polynomial set to decompose has a chordal associated graph. In particular, we prove that the…