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April | 1 | 2 - SEMINARS: Mathematical Picture Language Seminar: Hydrodynamics and Corrections to Random Matrix Universality in Quantum Chaos
Speaker: Brian Swingle – Brandeis University 9:30 AM-10:30 AM May 2, 2023 - HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR: Harvard–MIT Algebraic Geometry Seminar: Moduli spaces of cubic hypersurfaces
Speaker: Sebastian Casalaina-Martin – University of Colorado 3:00 PM-5:00 PM May 2, 2023 1 Oxford Street, Cambridge, MA 02138 USA
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.
| 3 - CMSA EVENT: CMSA Colloquium: Generative Adversarial Networks (GANs): An Analytical Perspective
Speaker: Xin Guo – UC Berkeley 12:30 PM-1:30 PM May 3, 2023 20 Garden Street, Cambridge, MA 02138
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.
- CMSA EVENT: CMSA Probability Seminar: Random Neural Networks
Speaker: Boris Hanin – Princeton University 3:30 PM-4:30 PM May 3, 2023 20 Garden Street, Cambridge, MA 02138
Fully connected neural networks are described two by structural parameters: a depth L and a width N. In this talk, I will present results and open questions about the asymptotic analysis of such networks with random weights and biases in the regime where N (and potentially L) are large. The first set of results are for deep linear networks, which are simply products of L random matrices of size N x N. I’ll explain how the setting where the ratio L / N is fixed with both N and L large reveals a number of phenomena not present when only one of them is large. I will then state several results about non-linear networks in which this depth-to-width ratio L / N again plays a crucial role and gives an effective notion of depth for a random neural network.
- CMSA EVENT: CMSA Probability Seminar: Random Neural Networks
Speaker: Boris Hanin – Princeton University 3:30 PM-4:30 PM May 3, 2023 20 Garden Street, Cambridge, MA 02138
Fully connected neural networks are described two by structural parameters: a depth L and a width N. In this talk, I will present results and open questions about the asymptotic analysis of such networks with random weights and biases in the regime where N (and potentially L) are large. The first set of results are for deep linear networks, which are simply products of L random matrices of size N x N. I’ll explain how the setting where the ratio L / N is fixed with both N and L large reveals a number of phenomena not present when only one of them is large. I will then state several results about non-linear networks in which this depth-to-width ratio L / N again plays a crucial role and gives an effective notion of depth for a random neural network.
- HARVARD-MIT COMBINATORICS SEMINAR: MIT-Harvard-MSR Combinatorics Seminar: Weighted Ehrhart Theory and why you should care!
Speaker: Jesús A. De Loera – UC Davis 4:15 PM-5:15 PM May 3, 2023
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. ======================================================= For information about the Combinatorics Seminar, please visit: http://math.mit.edu/seminars/combin/ =============================================
| 4 - CMSA EVENT: CMSA General Relativity Seminar: Testing GR with GWs
Speaker: Vitor Cardoso – IST, Lisbon and The Niels Bohr Institute, Copenhagen 9:30 AM-10:30 AM May 4, 2023
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 - SEMINARS: Algebraic Dynamics Seminar: The Zariski dense orbit conjecture
Speaker: Sina Saleh – Harvard University 4:00 PM-6:00 PM May 4, 2023
In this talk, I will present an adelic version of the Zariski dense orbit conjecture by Junyi Xie which is a strengthening of the original conjecture formulated independently by Medvedev-Scanlon, Amerik-Campana, and Zhang. I will introduce the adelic topology, which is a stronger topology than the Zariski topology, and its main properties. Using the adelic topology, one can give simpler proofs of the conjecture in the case of endomorphisms of abelian varieties and endomorphisms of A^n given by the coordinatewise action of one-variable polynomials. If time permits, we will discuss Xie’s proof of the conjecture in the case of the endomorphisms of P^2.
For more information, please see: Algebraic Dynamics Seminar at Harvard
| 5 - CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Detecting central charge in a superconducting quantum processor
Speaker: Sona Najafi – IBM Quantum 10:00 AM-11:30 AM May 5, 2023 20 Garden Street, Cambridge, MA 02138
Physical systems at the continuous phase transition point exhibit conformal symmetry rendering local scaling invariance. In two dimensions, the conformal group possesses infinite generators described by Virasoro algebra with an essential parameter known as a central charge. While the central charge manifests itself in a variety of quantities, its detection in experimental setup remains elusive. In this work, we utilize Shannon-Renyi entropy on a local basis of a one-dimensional quantum spin chain at a critical point. We first use a simulated variational quantum eigen solver to prepare the ground state of the critical transfer field Ising model and XXZ model with open and periodic boundary conditions and perform local Pauli X and Z basis measurements. Using error mitigation such as probabilistic error cancellation, we extract an estimation of the local Pauli observables needed to determine the Shannon-Renyi entropy with respect to subsystem size. Finally, we obtain the central charge in the sub-leading term of Shannon-Renyi entropy.
This seminar will be hybrid – in person and virtual. Password: cmsa For more information, please see:
- SEMINARS: Gauge Theory and Topology Seminar: Cube tilings and alternating links
Speaker: Joshua Greene – Boston College 3:30 PM-4:30 PM May 5, 2023 1 Oxford Street, Cambridge, MA 02138 USA
Consider a planar graph G, and form the lattice of integer-valued flows on G. Is it the case that this lattice embeds into the lattice of integer points in Euclidean space in such a way that each unit cube with integer vertices contains a point of the embedded sublattice? Consider instead an alternating link L, and form the double-cover of the three-sphere branched along L. Is it the case that this space bounds a smooth four-manifold with trivial rational homology groups? Under the correspondence that takes L to its Tait graph G, we conjecture that the answers to these two questions are the same. I will explain why a positive answer to the second implies a positive answer to the first using Floer homology. I will then explain why a positive answer to the first implies a positive answer to the second under the added hypothesis that each unit cube contains a *unique* point of the embedded sublattice. This is joint work, the forward direction with Slaven Jabuka and the partial reverse direction with Brendan Owens.
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7 | 8 | 9 | 10 - CMSA EVENT: CMSA New Technologies in Mathematics Seminar: Modern Hopfield Networks for Novel Transformer Architectures
Speaker: Dmitry Krotov – IBM Research – Cambridge 2:00 PM-3:00 PM May 10, 2023
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: - HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Linear relations and Lorentzian property of chromatic symmetric functions
Speaker: Alejandro Morales – UMass Amherst 4:15 PM-5:15 PM May 10, 2023
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. =============================== For more info, see https://math.mit.edu/combin/
| 11 - CMSA EVENT: CMSA Active Matter Seminar: Insights from single cell lineage tree
Speaker: Sahand Hormoz – Harvard Medical School, Dana-Farber Cancer Institute 1:00 PM-2:00 PM May 11, 2023
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/ - CMSA EVENT: CMSA Active Matter Seminar: Insights from single cell lineage tree
Speaker: Sahand Hormoz – Harvard Medical School, Dana-Farber Cancer Institute 1:00 PM-2:00 PM May 11, 2023
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/ - CMSA EVENT: CMSA General Relativity Seminar: Positivity of Static quasi-local Mass in general relativity
Speaker: Aghil Alaee – Clark University 1:30 PM-2:30 PM May 11, 2023
In this talk, we review results on the PMT of quasi-local masses and prove the positivity of static quasi-local masses with respect to the AdS and AdS Schwarzschild spacetimes.
Zoom: https://harvard.zoom.us/j/7855806609 - CMSA EVENT: CMSA Probability Seminar: How do the eigenvalues of a large non-Hermitian random matrix behave?
Speaker: Giorgio Cipolloni – Princeton University 1:30 PM-2:30 PM May 11, 2023 **note unusual day, time and location**
We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. Then, motivated by the long time behaviour of the ODE \dot{u}=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X.
| 12 - CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Anomalies of (1+1)D categorical symmetries
Speaker: Carolyn Zhang – University of Chicago 10:00 AM-11:30 AM May 12, 2023
We present a general approach for detecting when a fusion category symmetry is anomalous, based on the existence of a special kind of Lagrangian algebra of the corresponding Drinfeld center. The Drinfeld center of a fusion category $A$ describes a $(2+1)D$ topological order whose gapped boundaries enumerate all $(1+1)D$ gapped phases with the fusion category symmetry, which may be spontaneously broken. There always exists a gapped boundary, given by the \emph{electric} Lagrangian algebra, that describes a phase with $A$ fully spontaneously broken. The symmetry defects of this boundary can be identified with the objects in $A$. We observe that if there exists a different gapped boundary, given by a \emph{magnetic} Lagrangian algebra, then there exists a gapped phase where $A$ is not spontaneously broken at all, which means that $A$ is not anomalous. In certain cases, we show that requiring the existence of such a magnetic Lagrangian algebra leads to highly computable obstructions to $A$ being anomaly-free. As an application, we consider the Drinfeld centers of $\mathbb{Z}_N\times\mathbb{Z}_N$ Tambara-Yamagami fusion categories and recover known results from the study of fiber functors.
This seminar will be virtual. Password: cmsa For more information, please see:
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14 | 15 | 16 | 17 - CMSA EVENT: GRAMSIA: Graphical Models, Statistical Inference, and Algorithms
All day May 17, 2023-May 19, 2023 20 Garden Street, Cambridge, MA 02138 - HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Antichain Codes
Speaker: Benjamin Gunby – Rutgers University 3:00 PM-4:00 PM May 17, 2023
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/ - HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Antichain Codes
Speaker: Benjamin Gunby – Rutgers University 3:00 PM-4:00 PM May 17, 2023
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/
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21 | 22 | 23 - HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR: Harvard–MIT Algebraic Geometry Seminar: Complex variations of Hodge structures of rank 2 over curves
Speaker: Nicolas Tholozan – École Normale Supérieure 3:00 PM-4:00 PM May 23, 2023 1 Oxford Street, Cambridge, MA 02138 USA
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.
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28 | 29 | 30 | 31 | June | June | June |