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The Central Limit Theorem is an example of the ubiquitous yet still surprising phenomena in probability that many random inputs often combine to give an output insensitive to the input distributions. We will explore an example of how this plays out in the construction of random abelian groups from random integral matrices. As an example we will see the probability, as n goes to infinity, that a random linear map from Z^(n+1) to Z^n is surjective.

This talk includes joint work with Hoi Nguyen.

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

In this talk I will explore the thermodynamic response near extremality of charged black holes in four-dimensional Einstein-Maxwell theory with a positive cosmological constant. The latter exhibit three different extremal limits, dubbed cold, Nariai and ultracold configurations, with different near-horizon geometries. For each of these three cases I will analyze small deformations away from extremality, and construct the effective two-dimensional theory, obtained by dimensional reduction, that captures these features. The ultracold case in particular shows an interesting interplay between the entropy variation and charge variation, realizing a different symmetry breaking with respect to the other two near-extremal limits.

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

I will focus on the interaction between different active matter systems. In particular, I will describe recent experimental and modeling results that reveal how interaction forces between adhesive cells generate activity in the cell layer and lead to a potentially new mode of phase segregation. I will then discuss mechanics of how cells use finger-like protrusions, known as filopodia, to interact with their surrounding medium. First, I will present experimental and theoretical results of active mirror-symmetry breaking in subcellular skeleton of filopodia that allows for rotation, helicity, and buckling of these cellular fingers in a wide variety of cells ranging from epithelial, mesenchymal, cancerous and stem cells. I will then describe in-vivo experiments together with theoretical modeling showing how during embryo development specialized active cells probe and modify other cell layers and integrate within an active epithelium.

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/

A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.

Andrew Strominger will give the Third Annual Yip Lecture on February 2, 2023.

For more information, please see: https://cmsa.fas.harvard.edu/event/yip-2023/

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

We show that for any closed symplectic manifold, the number of 1-periodic orbits of any non-degenerate Hamiltonian is bounded from below by an integral version of total Betti number which takes account of torsions of all characteristics. The proof is based on a perturbation scheme (FOP perturbations) which produces pseudo-cycles from normally complex derived orbifolds, and a regularization procedure of moduli spaces of J-holomorphic curves (extending recent work of Abouzaid-McLean-Smith) which produces coherent smooth structures on the moduli spaces of Floer trajectories. I will outline the proof and indicate how these results may fit into other topics including Floer homotopy theory. This is based on joint work with Guangbo Xu.

In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a projective surface of non-negative Kodaira dimension to a projective line has at least three singular fibers. Furthermore, I will explain a proof of Popa’s conjecture on the superadditivity of the log Kodaira dimension over bases of dimension at most three. These theorems are applications of the main technical result, namely the logarithmic base change theorem.

Many decision and optimization problems over random structures exhibit an apparent gap between the existentially optimal values and algorithmically achievable values. Examples include the problem of finding a largest independent set in a random graph, the problem of finding a near ground state in a spin glass model, the problem of finding a satisfying assignment in a random constraint satisfaction problem, and many many more. Unfortunately,  at the same time no formal computational hardness results exist  which  explains this persistent algorithmic gap.

In the talk we will describe a new approach for establishing an algorithmic intractability for these problems called the overlap gap property. Originating in statistical physics theory of spin glasses, this is a simple to describe property which a) emerges in most models known to exhibit an apparent algorithmic hardness; b) is consistent with the hardness/tractability phase transition for many models analyzed to the day; and, importantly, c) allows to mathematically rigorously rule out a large class of algorithms as potential contenders, specifically the algorithms which exhibit a form of stability/noise insensitivity.

We will specifically show how to use this property to obtain stronger (stretched exponential) than the state of the art (quasi-polynomial) lower bounds on the size of constant depth Boolean circuits for solving the two of the aforementioned problems: the problem of finding a large independent set in a sparse random graph, and the problem of finding a near ground state of a p-spin model.

Joint work with Aukosh Jagannath and Alex Wein

Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), but one can also extend the idea further and ask whether E is ‘r-modular’ for r=2,3… . To define this generalization, the modular curve gets replaced with Drinfeld’s concept of a ‘shtuka space’. The r-modularity of E is predicted by Tate’s conjecture. In joint work with Adam Logan, we give some classes of elliptic curves E which are 2- and 3-modular.

Title: TBA

Abstract: TBA

I will discuss an approach to log-Sobolev inequalities that  combines the Bakry-Emery theory with renormalisation and present several  applications. These include log-Sobolev inequalities with polynomial  dependence for critical Ising models on Z^d when d>4 and singular SPDEs  with uniform dependence of the log-Sobolev constant on both the  regularisation and the volume. The talk is based on joint works with  Thierry Bodineau and Benoit Dagallier.

This seminar will be held on Zoom. They will also project the Zoom in the seminar room G-10 at CMSA, 20 Garden Street. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/probability-seminar/

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, February 8th.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

Quasinormal modes of gravitational waves and Ruelle resonances in hyperbolic classical dynamics share many general properties and can be considered “scattering resonances”: they appear in expansions of correlations, as poles of Green functions and are associated to trapping of trajectories (and are both notoriously hard to observe in nature, unlike, say, quantum resonances in chemistry or scattering poles in acoustical scattering). I will present a mathematical perspective that also includes zeros of the Riemann zeta function (scattering resonances for the Hamiltonian given by the Laplacian on the modular surface) and stresses the importance of different kinds of trapping phenomena, resulting, for instance, in fractal counting laws for resonances.

The CMSA will host a series of three 90-minute lectures on the subject of machine learning for protein folding.

Thursday, Feb 9, 2023:  3:30–5:00 pm ET

Thursday, Feb 16, 2023:  3:30–5:00 pm ET

Thursday, March 9, 2023: 3:30–5:00 pm ET

Further details TBA.

Quantum systems in 3+1-dimensions that are invariant under gauging a one-form symmetry enjoy novel non-invertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain infrared dynamics. We show that such non-invertible symmetries often forbid a symmetry-preserving vacuum state with a gapped spectrum, leaving only two possibilities for the infrared dynamics: a gapless state or spontaneous breaking of the non-invertible symmetries. These non-invertible symmetries are realized in lattice gauge theories, which serve to illustrate our results.

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

Heegaard Floer homology is an invariant of 3-manifolds, and knots and links within them, introduced by P. Oszváth and Z. Szabó in the early 2000s. Because of its relative computability by the standards of gauge and Floer theoretic invariants, it has enjoyed considerably popularity. However, it is not immediately obvious from the construction that Heegaard Floer homology is natural, that is, that it assigns to a basepointed 3-manifold a well-defined module over an appropriate base ring rather than an isomorphism class of modules, and well-defined cobordism maps to 4-manifolds with boundary. This situation was improved in the 2010s when A. Juhász, D. Thurston, and I. Zemke showed naturality of the various versions of Heegaard Floer homology. In this talk we consider involutive Heegaard Floer homology, a refinement of the theory introduced by C. Manolescu and I in 2015, whose definition relies on Juhász-Thurston-Zemke naturality but which is itself not obviously natural even given their results. We prove that involutive Heegaard Floer homology is a natural invariant of basepointed 3-manifolds together with a framing of the basepoint, and has well-defined maps associated to cobordisms, and discuss some consequences and implications. This is joint work with J. Hom, M. Stoffregen, and I. Zemke.

The swampland cobordism conjecture provides a convenient way to discuss conserved charges associated with the topology of spacetime. However, much of the power of the cobordism conjecture comes from a mathematical black box: the Adams spectral sequence. In this talk, I will give physical meaning to this black box through a concrete example: domain walls arising from the spontaneously breaking of parity symmetry, which arise in particle physics in Nelson-Barr models. I will argue that parity domain walls are exactly stable, and interpret this stability as the result of an unusual type of gauge symmetry that can only occur in gravitational theories.

Complete Calabi-Yau metrics are fundamental objects in Kahler geometry arising as singularity models or “bubbles” in degenerations of compact Calabi-Yau manifolds.  The existence of these metrics and their relationship with algebraic geometry are the subjects of several long standing conjectures due to Yau and Tian-Yau.  I will describe some recent progress towards the question of existence, and explain some future directions, highlighting connections with notions of algebro-geometric stability.

The purpose of this talk is to discuss Newton-Okounkov bodies,  a convex geometric construction associated to divisors on projective  varieties. We will touch on their relationship with local positivity,  multiplicative filtrations on section rings, and applications.

This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set $Y \subset \mathbb{R}^D$. When can we construct a smooth $d$-dimensional submanifold $\widehat{M} \subset \mathbb{R}^D$ to approximate $Y$, and how well can $\widehat{M}$ estimate $Y$ in terms of distance and smoothness? To address these problems, various mathematical approaches have been proposed (see Fefferman et al. (2016, 2018, 2021b)). However, many of these methods rely on restrictive assumptions, making extending them to efficient and workable algorithms challenging. As the manifold hypothesis (non-Euclidean structure exploration) continues to be a foundational element in statistics, the manifold fitting Problem, merits further exploration and discussion within the modern statistical community. The talk will be partially based on a recent work Yao and Xia (2019) along with some on-going progress. Relevant reference: https://arxiv.org/abs/1909.10228

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, February15th.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

In 1902, Painleve introduced a collection of 6 differential equations yielding all second order equations with no movable singularities. It turns out that algebraic solutions to Painleve’s sixth equation correspond to canonical triples of 2 by 2 complex matrices. We will describe the history of what is known about these canonical tuples of matrices. These canonical tuples can also be viewed as canonical representations of fundamental groups of Riemann surfaces, and also as local systems on certain covers of the moduli space of curves. This talk includes joint work with Daniel Litt.

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

Over the last two decades, major progress has been made in understanding the self-organization principles of active matter.  A wide variety of experimental model systems, from self-driven colloids to active elastic materials, has been established, and an extensive theoretical framework has been developed to explain many of the experimentally observed non-equilibrium pattern formation phenomena. Two key challenges for the coming years will be to translate this foundational knowledge into functional active materials, and to identify quantitative mathematical models that can inform and guide the design and production of such materials. Here, I will describe joint efforts with our experimental collaborators to realize self-growing bacterial materials [1], and to implement computational model inference schemes for active and living systems dynamics [2,3].

[1] Nature 608: 324, 2022
[2] PNAS 120: e2206994120, 2023
[3] eLife 10: e68679, 2021

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 explain how to describe quasinormal modes with large real frequencies using Penrose limits.To do so, I first recall relevant aspects of the Penrose limit, and its resulting plane wave spacetimes, as well as quasinormal modes to subsequently tie these together. Having established the main principle, I will illustrate the usefulness of this point of view with the geometric realization of the emergent symmetry algebra underlying the quasinormal modes in the large real frequency limit and present its application to the astrophysically important example of Kerr black holes. Based on arXiv:2301.06999.

This seminar will be held on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/general-relativity/

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

The CMSA will host a series of three 90-minute lectures on the subject of machine learning for protein folding.

Thursday, Feb 9, 2023:  3:30–5:00 pm ET

Thursday, Feb 16, 2023:  3:30–5:00 pm ET

Thursday, March 9, 2023: 3:30–5:00 pm ET

Further details TBA.

While many-body quantum systems can in principle host exotic quantum spin liquid (QSL) states, realizing them as ground states in experiments can be prohibitively difficult. In this talk, we show how non-equilibrium dynamics can provide a streamlined route toward creating QSLs. In particular, we show how a simple Hamiltonian parameter sweep can dynamically project out condensed anyons from a family of initial product states (e.g. dynamically  “un-Higgs”), yielding a QSL-like state. We christen such states “quantum spin lakes” which, while not thermodynamically large QSLs, enable their study in NISQ-era quantum simulators. Indeed, we show that this mechanism sheds light on recent experimental and numerical observations of the dynamical state preparation of the ruby lattice spin liquid in Rydberg atom arrays. Time permitting, we will discuss how our theory motivates a tree tensor network-based numerical tool—reliant on our theory—that quantitatively reproduces the experimental data two orders of magnitude faster than conventional brute-force simulation methods. Finally, we will highlight that even spin liquid states that are unstable in equilibrium—namely, 2 + 1D U(1) spin liquid states—can be robustly prepared by non-equilibrium dynamics.

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

While many-body quantum systems can in principle host exotic quantum spin liquid (QSL) states, realizing them as ground states in experiments can be prohibitively difficult. In this talk, we show how non-equilibrium dynamics can provide a streamlined route toward creating QSLs. In particular, we show how a simple Hamiltonian parameter sweep can dynamically project out condensed anyons from a family of initial product states (e.g. dynamically  “un-Higgs”), yielding a QSL-like state. We christen such states “quantum spin lakes” which, while not thermodynamically large QSLs, enable their study in NISQ-era quantum simulators. Indeed, we show that this mechanism sheds light on recent experimental and numerical observations of the dynamical state preparation of the ruby lattice spin liquid in Rydberg atom arrays. Time permitting, we will discuss how our theory motivates a tree tensor network-based numerical tool—reliant on our theory—that quantitatively reproduces the experimental data two orders of magnitude faster than conventional brute-force simulation methods. Finally, we will highlight that even spin liquid states that are unstable in equilibrium—namely, 2 + 1D U(1) spin liquid states—can be robustly prepared by non-equilibrium dynamics.

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

Given a spatial trivalent graph G, we will first review some of the objects and results from Kronheimer and Mrowka’s theory about the instanton invariant J#(G). Motivated by the Tutte relation and the 4-color theorem, we will then proceed to studying decompositions of graphs into two 4-ended tangle-graphs along a 4-punctured sphere. The resulting Lagrangian Floer theory happens to be on the pillowcase. Viewing J# invariant from this angle, we will propose several modifications to the construction of representation varieties: two different reductions at two basepoints, and a passage to the equivariant theory (which corresponds to the wrapped Floer theory on the pillowcase). The advantage is that the resulting equivariant invariants are as simple as possible, and can be used to recover the initial unreduced invariants via mapping cones. Based on this we will speculate on the existence of the corresponding instanton-theoretic curve invariants in the pillowcase, and indicate an open-ended strategy for how to study the Tutte relation for J#(G). This is joint work in progress with Fan Ye.

The period-index problem is a longstanding question about the complexity of Brauer classes over a field. I will discuss some Hodge-theoretic aspects of the problem for complex function fields, and give some applications to Brauer groups and the integral Hodge conjecture.

The black hole information paradox — whether information escapes an evaporating black hole or not —  remains one of the most longstanding mysteries of theoretical physics. The apparent conflict between validity of semiclassical gravity at low energies and unitarity of quantum mechanics has long been expected to find its resolution in a complete quantum theory of gravity. Recent developments in the holographic dictionary, and in particular its application to entanglement and complexity, however, have shown that a semiclassical analysis of gravitational physics can reproduce a hallmark feature of unitary evolution. I will describe this recent progress and discuss some promising indications of a full resolution of the information paradox.

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Abstract: TBA

We show that if each edge of the complete bipartite graph K_{n,n} is given a random list of C(\log n) colors from [n], then with high probability, there is a proper edge coloring where the color of each edge comes from the corresponding list. We also prove analogous results for Latin squares and Steiner triple systems. This resolves several related conjectures of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus. I will discuss some of the main ingredients which go into the proof: the Kahn-Kalai conjecture, absorption, and the Lovasz Local Lemma distribution. Based on joint work with Huy Tuan Pham.

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

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, February 22nd.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

The purpose of this talk is to give an overview of a semi-global existence result and a trapped surface formation results in the context of the Einstein-Yang-Mills system. Adopting a “signature for decay rates” approach first introduced by An, we develop a novel gauge (and scale) invariant hierarchy of non-linear estimates for the Yang-Mills curvature which, together with the estimates for the gravitational degrees of freedom, yield the desired semi-global existence result. Once semi-global existence has been established, we will explain how the formation of a trapped surface follows from a standard ODE argument. This is joint work with Puskar Mondal and Shing-Tung Yau.

This seminar will be held on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/general-relativity/

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

In this talk, I will show my recent work on general inverse $\sigma_k$ equations and the deformed Hermitian—Yang—Mills equation (hereinafter the dHYM equation). First, I will show my recent results. This result states that if a level set of a general inverse $\sigma_k$ equation (after translation if needed) is contained in the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the Monge—Ampère equation, the J-equation, the dHYM equation, the special Lagrangian equation, etc. Second, I will introduce some semialgebraic sets and a special class of univariate polynomials and give a Positivstellensatz type result. These give a numerical criterion to verify whether the level set will be contained in the positive orthant. Last, as an application, I will prove one of the conjectures by Collins—Jacob—Yau when the dimension equals four. This conjecture states that under the supercritical phase assumption, if there exists a C-subsolution to the dHYM equation, then the dHYM equation is solvable.

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/algebraic-geometry-in-string-theory/

Given a knot or link in the 3-sphere, its Murasugi signature is an integer-valued invariant which can easily be computed from a diagram. Work of Herald and Lin gives an alternative description of knot signatures, as signed counts of SU(2)-representations of the knot group which are traceless around meridians. There is a version of singular instanton homology for links which categorifies the Murasugi signature. We construct unoriented skein exact triangles for these Floer groups, categorifying the behavior of the Murasugi signature under unoriented skein relations. More generally, we construct skein exact triangles in the setting of equivariant singular instanton theory. This is joint work with Ali Daemi.

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.