Abstract TBA

The Math Picture Language seminar will be held at 9:30 a.m. Boston time.
Click the link for a Zoom Link for Tuesday Math Picture Language Seminars.
Recorded seminars can be viewed on the Mathematical Picture Language YouTube channel

In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent.

 Generative models have attracted intense interests recently. In this talk, I will discuss one class of generative models, Generative Adversarial Networks (GANs).  I will first provide a gentle review of the mathematical framework behind GANs. I will then proceed to discuss a few challenges in GANs training from an analytical perspective. I will finally report some recent progress for GANs training in terms of its stability and convergence analysis.

A great tool in the arsenal of combinatorialists is modeling problems as counting  the lattice points of some convex polytope. Let $P\subseteq\R^d$ be a rational  convex polytope, that is, a polytope with vertices in $\mathbb{Q}^d$, then the Ehrhart function of the polytope $i(P,n)$ counts the number of integer lattice of the dilation $nP$ (here $nP$ denotes the polytope obtained from dilating $P$ by a factor n). Ehrhart functions have a rich history and many wonderful properties (e.g., Ehrhart himself proved that when $P$ is a lattice polytope, then $i(P,n)$ is a polynomial of degree $dim(P)$. The connections to Hilbert series are legendary). This topic has appeared in algebraic combinatorics, representation theory, algebraic geometry and others areas. But what if we count the integer lattice points with *weights*? Say  $w: \R^d \to \R$   a  function, often called a  \emph{weight function}. We can consider the, \emph{weighted Ehrhart} function: \[ i(P,w,n)=\sum_ {x\in nP \cap \Z^d} w(x).  \] (Here $w(x) := w(x_1,\dots,x_d)$ runs over the set of integer points belonging to $P$)

In this lecture I review what we know about weighted Ehrhart functions.

Some basic things remain true, other classical results have delicate variations and extensions. I will discuss several new theorems:

1) We generalized R. Stanley’s theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes.  We show that these results continue to hold for \emph{weighted} Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope.

2) We also investigated  nonnegativity of the $h^\ast$-polynomial as a real-valued function for  a larger family of weights. In fact, discuss the case of  counting lattice points of a polytope that are weighted not by a simple polynomial, but by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials. We obtain new identities in representation theory and semigroup theory similar to RSK.

This work comprises 3 papers joint work with subsets of the following wonderful people: Esme Bajo, Rob Davis, Laura Escobar, Alexey Garber, Katharina Jochemko, Nathan Kaplan, Sofia Garzon-Mora, Josephine Yu, Rafael Villarreal, and Chengyang Wang.

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For information about the Combinatorics Seminar, please visit:

http://math.mit.edu/seminars/combin/

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One of the most remarkable possibilities of General Relativity concerns gravitational collapse to black holes, leaving behind a geometry with light rings, ergoregions and horizons. These peculiarities are responsible for uniqueness properties and energy extraction mechanisms that turn black holes into ideal laboratories of strong gravity, of particle physics (yes!) and of possible quantum-gravity effects. I will discuss some of the latest progress in tests of General Relativity with black holes.

Zoom: https://harvard.zoom.us/j/7855806609

For more information, please see:  Algebraic Dynamics Seminar at Harvard

This is joint work, the forward direction with Slaven Jabuka and the partial reverse direction with Brendan Owens.

*Rescheduled from April 11*

Stabilizer states and Clifford unitaries have played important roles in quantum information and computation, such as quantum error correcting code and measurement-based quantum computation. In this work, I will introduce a convolutional framework to study stabilizer states and channels based on qudits. Moreover, we establish a quantum central limit theorem, based on iterating the convolution of a quantum state, and show this converges to a stabilizer state. This talk is based on the joint work with Weichen Gu, and Arthur Jaffe (arXiv:2302.07841, 2302.08423).

The Math Picture Language seminar will be held at 9:30 a.m. Boston time.
Click the link for a Zoom Link for Tuesday Math Picture Language Seminars.
Recorded seminars can be viewed on the Mathematical Picture Language YouTube channel.

Let Z be the germ of a complex hypersurface isolated singularity of equation f. We consider the family of analytic D-modules generated by the powers of 1/f and relate it to the pole order filtration on the de Rham cohomology of the complement of \{f=0\}. This work builds on Vilonen’s characterization of the intersection homology D-module.

Modern Hopfield Networks or Dense Associative Memories are recurrent neural networks with fixed point attractor states that are described by an energy function. In contrast to conventional Hopfield Networks, which were popular in the 1980s, their modern versions have a very large memory storage capacity, which makes them appealing tools for many problems in machine learning and cognitive and neuro-sciences. In this talk I will introduce an intuition and a mathematical formulation of this class of models, and will give examples of problems in AI that can be tackled using these new ideas. Particularly, I will introduce an architecture called Energy Transformer, which replaces the conventional attention mechanism with a recurrent Dense Associative Memory model. I will explain the theoretical principles behind this architectural choice and show promising empirical results on challenging computer vision and graph network tasks.

This seminar will be on Zoom:

https://harvard.zoom.us/j/93216906981?pwd=NEJJM1g5QUFQZ1RQNURjSVRmU2I3UT09

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian Wachs q-analogue (q-CSF) have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013).

In the first part of the talk, I will discuss a linear relation of the q-CSF for abelian paths in terms of the Garsia–Remmel q-rook and q-hit numbers originally due to Guay-Paquet and its relation to the
e-positivity conjecture of Stanley–Stembridge and Shareshian–Wachs. This is joint work with Colmenarejo and Panova. In the second part of the talk, I will discuss the Newton polytope of CSFs of Dyck paths,
whether it is saturated, and a conjectured Lorenztian property for these CSFs that is true for the abelian case. This is joint work with Matherne and Selover.

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For more info, see https://math.mit.edu/combin/

In this talk, I will discuss two recent projects from my lab that involve lineage trees of cells (the branching diagram that represents the ancestry and division history of individual cells). In the first project, we reconstructed the lineage trees of individual cancer cells from the patterns of randomly occurring mutations in these cells. We then inferred the age at which the cancer mutation first occurred and the rate of expansion of the population of cancer cells within each patient. To our surprise, we discovered that the cancer mutation occurs decades before diagnosis. For the second project, we developed microfluidic ‘mother machines’ that allow us to observe mammalian cells dividing across tens of generations. Using our observations, we calculated the correlation between the duration of cell cycle phases in pairs of cells, as a function of their lineage distance. These correlations revealed many surprises that we are trying to understand using hidden Markov models on trees. For both projects, I will discuss the mathematical challenges that we have faced and open problems related to inference from lineage trees.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/active-matter-seminar/

In this talk, I will discuss two recent projects from my lab that involve lineage trees of cells (the branching diagram that represents the ancestry and division history of individual cells). In the first project, we reconstructed the lineage trees of individual cancer cells from the patterns of randomly occurring mutations in these cells. We then inferred the age at which the cancer mutation first occurred and the rate of expansion of the population of cancer cells within each patient. To our surprise, we discovered that the cancer mutation occurs decades before diagnosis. For the second project, we developed microfluidic ‘mother machines’ that allow us to observe mammalian cells dividing across tens of generations. Using our observations, we calculated the correlation between the duration of cell cycle phases in pairs of cells, as a function of their lineage distance. These correlations revealed many surprises that we are trying to understand using hidden Markov models on trees. For both projects, I will discuss the mathematical challenges that we have faced and open problems related to inference from lineage trees.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/active-matter-seminar/

Zoom: https://harvard.zoom.us/j/7855806609

On May 16 – May 19, 2023 the CMSA will be hosting a four-day workshop on GRAMSIA: Graphical Models, Statistical Inference, and Algorithms. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by David Gamarnik (MIT), Kavita Ramanan (Brown), and Prasad Tetali  (Carnegie Mellon).

For more information, please see: https://cmsa.fas.harvard.edu/event/gramsia-graphical-models-statistical-inference-and-algorithms/

On May 16 – May 19, 2023 the CMSA will be hosting a four-day workshop on GRAMSIA: Graphical Models, Statistical Inference, and Algorithms. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by David Gamarnik (MIT), Kavita Ramanan (Brown), and Prasad Tetali  (Carnegie Mellon).

For more information, please see: https://cmsa.fas.harvard.edu/event/gramsia-graphical-models-statistical-inference-and-algorithms/

Let S be a subset of the Boolean cube that is both an antichain and a distance-r code. How large can S be? I will discuss the solution to this problem and its connections with combinatorial proofs of anticoncentration theorems.

Based on joint work with Xiaoyu He, Bhargav Narayanan, and Sam Spiro.

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For more info, see https://math.mit.edu/combin/

Let S be a subset of the Boolean cube that is both an antichain and a distance-r code. How large can S be? I will discuss the solution to this problem and its connections with combinatorial proofs of anticoncentration theorems.

Based on joint work with Xiaoyu He, Bhargav Narayanan, and Sam Spiro.

===============================

For more info, see https://math.mit.edu/combin/

On May 16 – May 19, 2023 the CMSA will be hosting a four-day workshop on GRAMSIA: Graphical Models, Statistical Inference, and Algorithms. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by David Gamarnik (MIT), Kavita Ramanan (Brown), and Prasad Tetali  (Carnegie Mellon).

For more information, please see: https://cmsa.fas.harvard.edu/event/gramsia-graphical-models-statistical-inference-and-algorithms/

On May 16 – May 19, 2023 the CMSA will be hosting a four-day workshop on GRAMSIA: Graphical Models, Statistical Inference, and Algorithms. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop is organized by David Gamarnik (MIT), Kavita Ramanan (Brown), and Prasad Tetali  (Carnegie Mellon).

For more information, please see: https://cmsa.fas.harvard.edu/event/gramsia-graphical-models-statistical-inference-and-algorithms/

Through the work of Simpson, complex variations of hodge structures (C-VHS) play a central role in the study of the moduli space of local systems over a complex algebraic variety. In this talk I will consider one of the simplest examples, namely C-VHS of rank 2 over curves. These objects are known to hyperbolic geometers as « branched hyperbolic surfaces ».

I will review what is known about their monodromy, and discuss in particular a joint result with Bertrand Deroin: every PSL(2,R)-local system of Euler class 2g-3 over a curve of genus g admits an isomonodromic deformation that supports a C-VHS.