Calendar

On Feb 27-March 1, 2023 the CMSA will host a Conference on Geometry and Statistics.

For more information, please see: https://cmsa.fas.harvard.edu/event/geometry-and-statistics/

On Feb 27-March 1, 2023 the CMSA will host a Conference on Geometry and Statistics.

For more information, please see: https://cmsa.fas.harvard.edu/event/geometry-and-statistics/

Brill-Noether theory answers the question of whether a general curve of genus $g$ admits $g^r_d$, a linear system of rank $r$ and degree $d$. A refined Brill-Noether theory hopes to answer the question of whether a “general curve with a $g^r_d$” admits a $g^{r’}_{d’}$. In other words, we want to know about the relative position between Brill-Noether loci in the moduli space of curves of genus $g$. I’ll explain a strategy for distinguishing Brill-Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill-Noether loci with respect to containment. Via an analysis of the stability of Lazarsfeld-Mukai bundles, we obtain new lifting results for linear systems of rank 3 which suffice to prove the maximal Brill-Noether loci conjecture in genus 9-19, 22, and 23. This is joint work with Richard Haburcak.

Brill-Noether theory answers the question of whether a general curve of genus $g$ admits $g^r_d$, a linear system of rank $r$ and degree $d$. A refined Brill-Noether theory hopes to answer the question of whether a “general curve with a $g^r_d$” admits a $g^{r’}_{d’}$. In other words, we want to know about the relative position between Brill-Noether loci in the moduli space of curves of genus $g$. I’ll explain a strategy for distinguishing Brill-Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill-Noether loci with respect to containment. Via an analysis of the stability of Lazarsfeld-Mukai bundles, we obtain new lifting results for linear systems of rank 3 which suffice to prove the maximal Brill-Noether loci conjecture in genus 9-19, 22, and 23. This is joint work with Richard Haburcak.

Feedforward neural networks with ReLU activation are a class of parameterized functions that have proven remarkably successful in supervised learning tasks. They do so even in regimes where classical notions of complexity like the parametric dimension indicate that they ought to be overfitting the training data.In this talk, we argue that–contrary to conventional intuition–parametric dimension is a highly inadequate complexity measure for the class of ReLU neural network functions. We introduce the notion of the local functional dimension of a ReLU network parameter, discuss its relationship to the geometry of the underlying decomposition of the domain into linear regions, and present some preliminary experimental results suggesting that functional dimension is highly inhomogeneous for many architectures. Moreover, this inhomogeneity should have significant implications for the dynamics of training ReLU networks via gradient descent. Some of this work is joint with Kathryn Lindsey, Rob Meyerhoff, and Chenxi Wu, and some is joint with Kathryn Lindsey and David Rolnick.

On Feb 27-March 1, 2023 the CMSA will host a Conference on Geometry and Statistics.

For more information, please see: https://cmsa.fas.harvard.edu/event/geometry-and-statistics/

Title: TBA

Abstract: TBA

One of the first deep things we learned in elementary school is the act of taking negatives of numbers. I’ll explain the geometry that underlies this activity, which leads to algebraic K-theory and other gadgets. I also want to explain the joy of doing math with your friends.

For more information, please see: https://people.math.harvard.edu/~ana/ons/

This seminar will take place in SC 507 at 3:30pm.

String stars, or Horowitz-Polchinski solutions, are string theory saddles with normalizable condensates of thermal-winding strings. In the past, string stars were offered as a possible description of stringy (Euclidean) black holes in asymptotically flat spacetime, close to the Hagedorn temperature. I will discuss the thermodynamic properties of string stars in asymptotically (thermal) anti-de Sitter background (including AdS3 with NS-NS flux), their possible connection to small black holes in AdS, and their implications for holography. I will also present new “winding-string gas” saddles for confining holographic backgrounds such as the Witten model, and their relation to the deconfined phase of 3+1 pure Yang-Mills.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

Khovanov homology can be upgraded to an invariant of pairs (K,V) where K is a framed knot and V is an object of the annular Bar-Natan category (ABN). In this context, the pair (K,V) is called a colored knot, and its Khovanov invariant is called colored Khovanov homology.  In my talk I will discuss recent joint work with David Rose and Paul Wedrich, in which we construct an object in ABN (more accurately, an ind-object therein), called a Kirby color, whose associated colored Khovanov invariant satisfies the important handle-slide relation.  Our work gives an annular perspective on the Manolescu-Neithalath 2-handle formula for sl(2) skein lasagna modules.