Calendar

In this talk, I will give an overview of recent developments in geometric constructions of field theory in string/M-theory and identifying higher form symmetries. The main focus will be on d>= 4 constructed from string/M-theory. I will also discuss realization in terms of holographic models in string theory. In the talk next week Lakshya Bhardwaj will speak about 1-form symmetries in class S, N=1 deformations thereof and the relation to confinement.

Zoom: https://harvard.zoom.us/j/977347126

In many problems in statistical learning, random matrix theory, and statistical physics, one needs to simulate dynamics on random matrix ensembles. A classical example is to use iterative methods to compute the extremal eigenvalues/eigenvectors of a (spiked) random matrix. Other examples include approximate message passing on dense random graphs, and gradient descent algorithms for solving learning and estimation problems with random initialization. We will show that all such dynamics can be simulated by an efficient matrix-free scheme, if the random matrix is drawn from an ensemble with translation-invariant properties. Examples of such ensembles include the i.i.d. Gaussian (i.e. the rectangular Ginibre) ensemble, the Haar-distributed random orthogonal ensemble, the Gaussian orthogonal ensemble, and their complex-valued counterparts.

A “direct” approach to the simulation, where one first generates a dense n × n matrix from the ensemble, requires at least O(n^2) resource in space and time. The new algorithm, named Householder Dice (HD), overcomes this O(n^2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(n T^2), respectively, with T being the number of iterations. When T ≪ n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint.

Finally, the HD algorithm is not just a computational trick. I will show how its construction can serve as a simple proof technique for several problems in high-dimensional estimation.

Zoom: https://harvard.zoom.us/j/99333938108

The geometric Satake equivalence due to Lusztig, Drinfeld, Ginzburg, Mirković and Vilonen is an indispensable tool in the Langlands program. Versions of this equivalence are known for different cohomology theories such as Betti cohomology or algebraic D-modules over characteristic zero fields and $\ell$-adic cohomology over arbitrary fields. In this talk, I explain how to apply the theory of motivic complexes as developed by Voevodsky, Ayoub, Cisinski-Déglise and many others to the construction of a motivic Satake equivalence. Under suitable cycle class maps, it recovers the classical equivalence. As dual group, one obtains a certain extension of the Langlands dual group by a one dimensional torus. A key step in the proof is the construction of intersection motives on affine Grassmannians. A direct consequence of their existence is an unconditional construction of IC-Chow groups of moduli stacks of shtukas. My hope is to obtain on the long run independence-of-$\ell$ results in the work of V. Lafforgue on the Langlands correspondence for function fields. This is ongoing joint work with Jakob Scholbach from Münster.

Zoom: https://harvard.zoom.us/j/99334398740

Password: The order of the permutation group on 9 elements.

This talk will discuss dependent type theory from the perspective of artificial intelligence and cognitive science.  From an artificial intelligence perspective it will be argued that type theory is central to defining the “game” of mathematics — an action space and reward structure for pure mathematics. From a cognitive science perspective type theory provides a model of the grammar of the colloquial (natural) language of mathematics.  Of particular interest is the notion of a signature-axiom structure class and the three fundamental notions of equality in mathematics — set-theoretic equality between structure elements, isomorphism between structures, and Birkoff and Rota’s notion of cryptomorphism between structure classes.  This talk will present a version of type theory based on set-theoretic semantics and the 1930’s notion of structure and isomorphism given by the Bourbaki group of mathematicians.  It will be argued that this “Bourbaki type theory” (BTT) is more natural and accessible to classically trained mathematicians than Martin-Löf type theory (MLTT). BTT avoids the Curry-Howard isomorphism and axiom J of MLTT.  The talk will also discuss BTT as a model of MLTT.  The BTT model is similar to the groupoid model in that propositional equality is interpreted as isomorphism but different in various details.  The talk will also briefly mention initial thoughts in defining an action space and reward structure for a game of mathematics.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

I will talk, pretty casually, about random walks, which I first learned about as part of my Harvard undergrad thesis in finite group theory, and which turn out to be at the heart of the mathematical analysis of gerrymandering; along the way I will talk about the project of doing mathematics in a way that engages with the world outside the math department walls.

Please go to the College Calendar to register.

Zoom: https://calendar.college.harvard.edu/event/open_neighborhood_seminar

Website: https://math.harvard.edu/ons