Home » Articles posted by Julien Dubedat Added on April 15, 2024 by Julien Dubedat
Abstract:
A matrix is called rigid if one must change many of its entries before it becomes a low-rank matrix. Leslie Valiant introduced the notion in 1977 as a tool to prove lower bounds on the number of arithmetic operations needed to compute linear transformations like the discrete Fourier transform. Since then, many connections have been demonstrated between matrix rigidity and other topics in the theory of computation.
Unfortunately, proving that matrices of interest are rigid has shown to be a major challenge. In fact, it remains an open problem to prove that any explicit family of matrices is rigid. By contrast, it is known that a random matrix is rigid with high probability.
In this talk, I’ll give a brief overview of matrix rigidity and its uses in computer science. I’ll then discuss some recent results, including proofs that matrices like Hadamard and Fourier matrices, which were previously conjectured to be rigid, are in fact not rigid. The talk will not assume the audience has a background in computer science.
Time and location: tbd, 4:30-5:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
 Print this page Added on April 11, 2024 by Julien Dubedat
Title: Symmetries of manifolds
Abstract: Whenever one studies a mathematical object one ought also to study its symmetries. Manifolds are the central objects of study in topology and geometry, and their groups of symmetries come in many flavours (isometries, diffeomorphisms, homeomorphisms, …). I will discuss some classical and recent results about the spaces of all symmetries of certain simple manifolds, and report on an emerging conjectural picture.
Time and location: Wed. Apr. 17, 4:30-5:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
Print this pageAdded on February 22, 2024 by Julien DubedatSpecial time, location: Tue. Feb. 27, 4:10-5:25pm, 407 Math
Title:
Universally counting curves in Calabi–Yau threefolds
Abstract:
Statements such as “there is a unique line between any pair of distinct points in the plane” and “there are 27 lines on any cubic surface” have given rise to the modern theory of enumerative geometry.
To define such “curve counts” in a general setting usually involves choosing a particularly nice compactification of the space of smooth embedded curves (one which admits a natural “virtual fundamental class”). I will propose a new perspective on enumerative invariants which is based instead on a certain “Grothendieck group of 1-cycles” and the “universal” curve enumeration invariant taking values in this group. It turns out that if we restrict to complex threefolds with nef anticanonical bundle, this group has a very simple structure: it is generated by “local curves”. This generation result implies some new cases of the MNOP conjecture relating Gromov–Witten and Donaldson–Pandharipande–Thomas invariants of complex threefolds.
Print this pageAdded on February 16, 2024 by Julien Dubedat
Title: Curvature blow up at big bang singularities
Abstract: Singularities have been accepted as a natural feature in general relativity since the appearance of the singularity theorems of Hawking and Penrose. But these theorems do not say much concerning the nature of singularities. Do the gravitational fields become unbounded? Can the spacetime be extended through the singularity? Recently, many results demonstrating the stability of spatially homogeneous solutions with big bang singularities have appeared. As a consequence, there is an open set of initial data yielding big bang singularities with curvature blow up. However, the purpose of the talk is to illustrate that it is possible to go beyond stability results. In fact, I will present a new result (joint work with Hans Oude Groeniger and Oliver Petersen) in which we identify a general condition on initial data ensuring big bang formation. The solutions need, in this case, not be close to symmetric background solutions. Moreover, the result reproduces previous results in the Einstein-scalar field and Einstein-vacuum settings. Finally, the result is in the Einstein-non-linear scalar field setting, and therefore yields future and past global non-linear stability of large classes of spatially locally homogeneous solutions.
Time and location: Wed. Feb. 28, 4:30-5:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
Print this pageAdded on January 12, 2024 by Julien Dubedat
Title: On High Girth Steiner-Triple Systems and Subspace Designs
Abstract: We discuss the recent resolutions of the 1973 conjecture of Erdős on the existence of high girth Steiner triple systems and the existence of subspace designs. The talk will focus on placing these results within the context of classical design theory and within recent advances in the absorption method in combinatorics.
Based on joint works w. Peter Keevash, Matthew Kwan, Ashwin Sah, and Michael Simkin
Time and location: Wed. Jan. 17, 4:30-5:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
Print this pageAdded on February 27, 2020 by Julien DubedatTitle: A dynamical approach to universality in probability theory with applications in random matrices read more »
Print this pageAdded on September 30, 2019 by Julien DubedatTitle: The Local Behavior of Random Lozenge Tilings read more »
Print this pageAdded on September 20, 2019 by Julien DubedatTitle: Sphere packing, Fourier interpolation, and ground states in 8 and 24 dimensions read more »
Print this pageAdded on October 23, 2018 by Julien DubedatTitle: Combinatorial Applications of Computational Topology and Algebraic Geometry read more »
Print this pageAdded on October 05, 2018 by Julien DubedatTitle: Microlocal methods in chaotic dynamics read more »
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