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January  January  January  January  1  CMSA EVENT: CMSA Algebraic Geometry in String Theory Seminar: Algebraic billiards and dynamical degrees
Speaker: Max Weinreich – Harvard 10:30 AM11:30 AM February 1, 2024 20 Garden Street, Cambridge, MA 02138 Billiards is one of the moststudied dynamical systems, modeling the behavior of a point particle bouncing around some space. If the space is a plane region bounded by an algebraic curve, then we may use techniques from algebraic geometry to study its billiards map. We explain how to view billiards as a complex algebraic correspondence, and we prove upper and lower bounds on the dynamical degree, the growth rate of the degrees of the iterates, in terms of the degree of the boundary curve. These degree growth rates are studied in mathematical physics, broadly speaking, as a way to identify integrable (exactly solvable) physical models. In our setting, this theory gives us an upper bound on the entropy, or chaos, of billiards in curves.  CMSA EVENT: CMSA Member Seminar: On complete CalabiYau metrics and MongeAmpere equations
Speaker: Freid Tong – Harvard 12:00 PM1:00 PM February 1, 2024February 2, 2024 CalabiYau metrics are central objects in K\”ahler geometry and also string theory. The existence of CalabiYau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture, but the situation in the noncompact setting is much more delicate, and many questions related to the existence and uniqueness of noncompact CalabiYau metrics remain unanswered. I will give an introduction to this subject and discuss some ongoing joint work with T. Collins and S.T. Yau, on a new relationship between complete CalabiYau metrics and a new MongeAmpere equation. Friday, Feb. 2nd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363  THURSDAY SEMINAR SEMINAR: Thursday Seminar: Ravenel’s Telescope Conjecture: Ambidexterity and chromatic cyclotomic extensions
Speaker: Michael Hopkins – Harvard 3:30 PM5:30 PM February 1, 2024 1 Oxford Street, Cambridge, MA 02138 USA This semester we will go through the work of Burklund, Hahn, Levy and Schlank on the construction of counterexamples to the telescope conjecture.
 2  CMSA EVENT: CMSA Member Seminar: On complete CalabiYau metrics and MongeAmpere equations
Speaker: Freid Tong – Harvard 12:00 PM1:00 PM February 2, 2024February 2, 2024 CalabiYau metrics are central objects in K\”ahler geometry and also string theory. The existence of CalabiYau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture, but the situation in the noncompact setting is much more delicate, and many questions related to the existence and uniqueness of noncompact CalabiYau metrics remain unanswered. I will give an introduction to this subject and discuss some ongoing joint work with T. Collins and S.T. Yau, on a new relationship between complete CalabiYau metrics and a new MongeAmpere equation. Friday, Feb. 2nd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363
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4  5  6  CMSA EVENT: CMSA General Relativity Seminar: Noncompact ndimensional Einstein spaces as attractors for the Einstein flow
Speaker: Jinhua Wang – Xiamen University 10:00 AM11:00 AM February 6, 2024 We prove that along with the Einstein flow, any small perturbations of an $n$($n\geq4$)dimensional, noncompact negative Einstein space with some “nonpositive Weyl tensor” lead to a unique and global solution, and the solution will be attracted to a noncompact Einstein space that is close to the background one. The $n=3$ case has been addressed by WangYuan, while in dimension $n\geq 4$, as we know, negative Einstein metrics in general have nontrivial moduli spaces. This fact is reflected on the structure of Einstein equations, which further indicates no decay for the spatial Weyl tensor. Furthermore, it is suggested in the proof that the mechanic preventing the metric from flowing back to the original Einstein metric lies in the nondecaying character of spatial Weyl tensor. In contrary to the compact case considered in AnderssonMoncrief, our proof is independent of the theory of infinitesimal Einstein deformations. Instead, we take advantage of the inherent geometric structures of Einstein equations and develop an approach of energy estimates for a hyperbolic system of Maxwell type. Please note: This seminar will take place on Zoom from 10:00 am to 11:00 am ET
Zoom: https://harvard.zoom.us/j/7855806609 Password: cmsa  SEMINARS: Probability Seminar: Fractal Geometry of Stochastic Partial Differential Equations
Speaker: Promit Ghosal – Brandeis 1:30 PM2:30 PM February 6, 2024 Stochastic partial differential equations (PDEs) find extensive applications across diverse domains such as physics, finance, biology, and engineering, serving as effective tools for modeling systems influenced by random factors. The analysis of the patterns in the peaks and valleys of stochastic PDEs is crucial for gaining deeper insights into the underlying physical phenomena. One notable example is the KPZ equation, a fundamental stochastic PDE associated with significant models like random growth processes, Burgers turbulence, interacting particle systems, and random polymers. The study of the fractal structures inherent in the KPZ equation provides a quantitative characterization of the intermittent nature of its peaks, as well as those of the stochastic heat equation—a subject that has been extensively explored over the past few decades. Conversely, the Parabolic Anderson model (PAM) serves as a prototypical framework for simulating the conduction of electrons in crystals containing defects. Investigating the intermittency of peaks in the PAM has been a prominent area of research, closely tied to the phenomenon of Anderson localization. In this presentation, we delve into the fractal geometry of both the KPZ equation and the PAM, unveiling their multifractal nature. Specifically, we demonstrate that the spatial and spatiotemporal peaks of these equations exhibit infinitely many distinct values. Furthermore, we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor) associated with these peaks. The key findings presented here stem from a series of works that employ a diverse array of tools, ranging from random matrix theory and the Gibbs property of random curves to the utilization of regularity structures and paracontrolled calculus.  HARVARDMIT ALGEBRAIC GEOMETRY SEMINAR: HarvardMIT Algebraic Geometry Seminar: Enumerativity of fixeddomain GromovWitten invariants
Speaker: Carl Lian – Tufts University 3:00 PM4:00 PM February 6, 2024 It is wellunderstood that GromovWitten (GW) invariants often fail to be enumerative. For example, when r is at least 3, the highergenus GW invariants of P^r fail to count smooth curves in projective space in any transparent sense. The situation seems to be better when one fixes the complex structure of the domain curve. It was originally speculated that if X is a Fano variety, then the “fixeddomain” GW count of curves of sufficiently large degree passing through the maximal number of general points is enumerative. I will discuss some positive and negative results in this direction, focusing on the case of hypersurfaces. The most recent results are joint with Roya Beheshti, Brian Lehmann, Eric Riedl, Jason Starr, and Sho Tanimoto, and build on earlier work with Rahul Pandharipande and Alessio Cela. For more information, please see https://researchseminars.org/seminar/harvardmitagseminar
 7  CMSA EVENT: CMSA/Tsinghua MathScience Literature Lecture: Stretching and shrinking: 85 years of the Hopf argument for ergodicity
Speaker: Amie Wilkinson – University of Chicago 9:00 AM10:30 AM February 7, 2024 The early 20th century witnessed an explosion of activity, much of it centered at Harvard, on rigorizing the property of ergodicity first proposed by Boltzmann in his 1898 Ergodic Hypothesis for ideal gases. Earlier, in the 1880’s, Henri Poincaré and Felix Klein had also initiated a study of discrete groups of hyperbolic isometries. The geodesics in hyperbolic manifolds were discovered to carry a rich structure, first investigated from a topological perspective by Emil Artin and Marston Morse. The time was ripe to investigate geodesics in hyperbolic manifolds from an ergodic theoretic (i.e., statistical) perspective, and indeed Gustav Hedlund proved in 1934 that the geodesic flow for closed hyperbolic surfaces is ergodic. In 1939, Eberhard Hopf published a proof of the ergodicity of geodesic flows for negatively curved surfaces containing a novel method, now known as the Hopf argument. The Hopf argument, a “soft” argument for ergodicity of systems with some hyperbolicity (the “stretching and shrinking” in the title) has since seen wide application in geometry, representation theory and dynamics. I will discuss three results relying on the Hopf argument: Theorem (E. Hopf, 1939, D. Anosov, 1967): In a closed manifold of negative sectional curvatures, almost every geodesic is directionally equidistributed. Theorem (G. Mostow, 1968) Let M and N be closed hyperbolic manifolds of dimension at least 3, and let f:M>N be a homotopy equivalence. Then f is homotopic to a unique isometry. Theorem (R. Mañé, 1983, A. Avila S. Crovisier A.W., 2022) The C^1 generic symplectomorphism of a closed symplectic manifold with positive entropy is ergodic.  CMSA EVENT: CMSA New Technologies in Mathematics Seminar: Large language models, mathematical discovery, and search in the space of strategies: an anecdote
Speaker: Jordan Ellenberg – Univ. of Wisconsin Dept. of Mathematics 1:00 PM2:00 PM February 7, 2024 20 Garden Street, Cambridge, MA 02138 Please note special time I spent a portion of 2023 working with a team at DeepMind on the “cap set problem” – how large can a subset of (Z/3Z)^n be which contains no three terms which sum to zero? (I will explain, for those not familiar with this problem, something about the role it plays in combinatorics, its history, and why number theorists care about it a lot.) By now, there are many examples of machine learning mechanisms being used to help generate interesting mathematical knowledge, and especially interesting examples. This project used a novel protocol; instead of searching directly for large cap sets, we used LLMs trained on code to search the space of short programs for those which, when executed, output large capsets. One advantage is that a program is much more humanreadable than a large collection of vectors over Z/3Z, bringing us closer to the notverywelldefinedbutimportant goal of “interpretable machine learning.” I’ll talk about what succeeded in this project (more than I expected!) what didn’t, and what role I can imagine this approach to the mathML interface playing in nearfuture mathematical practice. The paper: https://www.nature.com/articles/s41586023069246 https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09 Password: cmsa  HARVARDMIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Cluster algebras and scattering amplitudes
Speaker: Marcus Spradlin – Brown University 4:15 PM5:15 PM February 7, 2024 In recent years fruitful connections between math and physics have emerged from the study of scattering amplitudes. I will review and put into context some key concepts from this exchange of ideas, involving cluster algebras, positive geometries, and the amplituhedron. I will highlight further physicsmotivated conjectures that may provide fruitful avenues for continued exchange in the years to come. =============================== For more info, see https://math.mit.edu/combin/  OPEN NEIGHBORHOOD SEMINAR: Open Neighborhood Seminar: Symmetry in Deep Neural Networks
Speaker: Robin Walters – Northeastern 4:30 PM5:30 PM February 7, 2024 1 Oxford Street, Cambridge, MA 02138 USA
Deep learning has had transformative impacts in many fields including computer vision, computational biology, and dynamics by allowing us to learn functions directly from data. However, there remain many domains in which learning is difficult due to poor model generalization or limited training data. We’ll explore two applications of representation theory to neural networks which help address these issues. Firstly, consider the case in which the data represent a group equivariant function. In this case, we can consider spaces of equivariant neural networks which may more easily be fit to the data using gradient descent. Secondly, we can consider symmetries of the parameter space as well. Exploiting these symmetries can lead to models with fewer free parameters, faster convergence, and more stable optimization. =============================== https://people.math.harvard.edu/~gammage/ons/
 8  CMSA EVENT: CMSA Algebraic Geometry in String Theory Seminar: On (semi)stable reduction and KSBA moduli in positive characteristic
Speaker: Iacopo Brivio – Harvard 10:30 AM11:30 AM February 8, 2024 20 Garden Street, Cambridge, MA 02138 The moduli space M_g of genus g stable curves is perhaps the most studied of all algebraic varieties. Its higherdimensional generalization is the moduli functor M_{n,v} of ndimension stable varieties of volume v. It was proven only recently, and thanks to the joint effort of many over many years, that such functors are represented by projective algebraic spaces when working over the complex numbers. In this talk I will give some examples showing that the same moduli functors in positive characteristic are not even proper and, more in general, that the MMP fails to be functorial even in very nice families. In the second part I am going to explore some possible generalizations of the notion of stable variety that could be used as a replacement in positive characteristic. Zoom: https://harvard.zoom.us/j/93338480366?pwd=NEROWElhWStQVjVLRVZFSm1tV1ZCdz09  THURSDAY SEMINAR SEMINAR: Thursday Seminar: Ravenel’s Telescope Conjecture: Chromatically localized algebraic Ktheory
Speaker: Ishan Levy – Harvard 3:30 PM5:30 PM February 8, 2024 1 Oxford Street, Cambridge, MA 02138 USA This semester we will go through the work of Burklund, Hahn, Levy and Schlank on the construction of counterexamples to the telescope conjecture.
 9  CMSA EVENT: CMSA Member Seminar: The spectrum of some nonlinear random matrices
Speaker: Benjamin McKenna – Harvard 12:00 PM1:00 PM February 9, 2024 Modern data science often requires one to consider “nonlinear random matrices,” a broad term for randommatrix models whose construction involves a nonlinear function applied entrywise. Such models are typically far from classical random matrix theory, and in principle entrywise nonlinearities can affect the eigenvalues in a complicated way. However, recent years have seen a number of results on nonlinear models whose spectrum is surprisingly simple. We give one such result, emphasizing general randommatrix techniques like free probability and orthogonal polynomials. Joint work with Sofiia Dubova, Yue M. Lu, and HorngTzer Yau. Friday, Feb. 2nd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363  HARVARDMIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Reinforcement learning and pattern finding in combinatorics
Speaker: Adam Wagner – WPI 3:00 PM4:00 PM February 9, 2024 We will look at two ways we can use tools from machine learning to help us with research in combinatorics. First we discuss reinforcement learning, a method that gives us a way to check conjectures for counterexamples efficiently. While it usually does not perform as well as other simpler methods, there have been several examples of projects in the past few years where RL was crucial for success. In the second half of the talk we will consider the following question of Ellenberg: at most how many points can we pick in the N by N grid, without creating an isosceles triangle? The best known constructions, found by computer searches for small values of N, clearly follow a pattern which we do not yet understand. We will discuss how one can train transformers to understand this pattern, and use this trained transformer to help us find a bit better constructions for various N. This is joint work with Jordan Ellenberg, Marijn Heule, and Geordie Williamson =============================== For more info, see https://math.mit.edu/combin/  GAUGETOPOLOGYSYMPLECTIC SEMINAR: Gauge Theory and Topology Seminar: Twisted Knots and the Perturbed Alexander Invariant
Speaker: Joe Boninger – Boston College 3:30 PM4:30 PM February 9, 2024 1 Oxford Street, Cambridge, MA 02138 USA
The perturbed Alexander invariant, defined by BarNatan and van der Veen, is an infinite family of polynomial invariants of knots in the threesphere. The first polynomial, rho_1, is quick to calculate and may be better at distinguishing knots than practically any other computable invariant; it also has deep connections to both classical and quantum topology. We will discuss the perturbed Alexander invariant and rho_1 in particular, and give results on the behavior of rho_1 and the classical Alexander polynomial under the operation of applying full twists to a knot. Our arguments use a model of random walks on a knot diagram.
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11  12  CMSA EVENT: CMSA Colloquium: Machine learning and scientific computing: there is plenty of room in the middle
Speaker: Petros Koumoutsakos – Harvard SEAS 4:30 PM5:30 PM February 12, 2024 20 Garden Street, Cambridge, MA 02138 Over the last last thirty years we have experienced more than a billionfold increase in hardware capabilities and a dizzying pace of acquiring and transmitting massive amounts of data. Scientific Computing and, more lately, Artificial Intelligence (AI) has been key beneficiaries of these advances. In this talk I would outline the need for bridging the decades long advances in Scientific Computing with those of AI. I will use examples from fluid mechanics to argue for forming alloys of AI and simulations for their prediction and control. I will present novel algorithms for learning the Effective Dynamics (LED) of complex systems and a fusion of multi agent reinforcement learning and scientific computing (SciMARL) for modeling and control of turbulent flows. I will also show our recent work on Optimizing a Discrete Loss (ODIL) that outperforms popular techniques such as PINNs by several orders of magnitude. I will juxtapose successes and failures and argue that the proper fusion of scientific computing and AI expertise are essential to advance scientific frontiers.
 13  CMSA EVENT: CMSA General Relativity Seminar: Characteristic Initial Value Problem for the 3D Compressible Euler Equations
Speaker: Sifan Yu – NUS 11:00 AM12:00 PM February 13, 2024 20 Garden Street, Cambridge, MA 02138 We present the first result for the characteristic initial value problem of the compressible Euler equations in three space dimensions without any symmetry assumption. We allow presence of vorticity and consider any equation of state. Compared to the standard Cauchy problem, where initial data can be freely prescribed on a constanttime hypersurface, we formulate the problem by distinguishing between the “freecomponent” and the “constrainedcomponent” of the initial data. The latter is to be solved by the “freecomponent” utilizing the properties of the compressible Euler equations on the initial null hypersurfaces. Then, we establish a priori estimates, followed by a local wellposedness and a continuation criterion argument. Moreover, we prove a regularity theory in Sobolev norms. Our analysis critically relies on the vectorfield method due to the nature of the problem. This is a joint work with Jared Speck.
Zoom: https://harvard.zoom.us/j/7855806609 Password: cmsa  SEMINARS: Probability Seminar: POSTPONED
Speaker: Matthew Nicoletti – MIT 1:30 PM2:30 PM February 13, 2024 This seminar has been POSTPONED. Apologies for any inconveniences. Rescheduled date TBD. Recently, there has been much progress in understanding stationary measures for colored (also called multispecies or multitype) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (mASEP); (2) the qdeformed Totally Asymmetric Zero Range Process (TAZRP) also known as the qBoson particle system; (3) the qdeformed Pushing Totally Asymmetric Simple Exclusion Process (qPushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new “queue vertex models” on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. This is joint work with A. Aggarwal and L. Petrov.  HARVARDMIT ALGEBRAIC GEOMETRY SEMINAR: HarvardMIT Algebraic Geometry Seminar: POSTPONED
Speaker: Maksym Fedorchuk – Boston College 3:00 PM4:00 PM February 13, 2024 Seminar POSTPONED due to weather. Apologies for any inconveniences. Rescheduled date: February 27th, 3:004:00. See website for more details. A recent achievement in Kstability of Fano varieties is an algebrogeometric construction of a projective moduli space of Kpolystable Fanos. The ample line bundle on this moduli space is the CM line bundle of Tian. One of the consequences of the general theory is that given a family of Kstable Fanos over a punctured curve, the polystable filling is the one that minimizes the degree of the CM line bundle after every finite base change. A natural question is to ask what are the CMminimizers without base change. In answering this question, we arrive at a theory of Kollár stability for fibrations over onedimensional bases, and standard models of Fano fibrations. After explaining the general theory, I will sketch work in progress on standard models of quartic threefold hypersurfaces. This talk is based on joint work with Hamid Abban and Igor Krylov. For more information, please see https://researchseminars.org/seminar/harvardmitagseminar
 14  CMSA EVENT: CMSA New Technologies in Mathematics Seminar: What Algorithms can Transformers Learn? A Study in Length Generalization
Speaker: Preetum Nakkiran – Apple 2:00 PM February 14, 2024 Large language models exhibit many surprising “outofdistribution” generalization abilities, yet also struggle to solve certain simple tasks like decimal addition. To clarify the scope of Transformers’ outofdistribution generalization, we isolate this behavior in a specific controlled setting: lengthgeneralization on algorithmic tasks. Eg: Can a model trained on 10 digit addition generalize to 50 digit addition? For which tasks do we expect this to work? Our key tool is the recentlyintroduced RASP language (Weiss et al 2021), which is a programming language tailormade for the Transformer’s computational model. We conjecture, informally, that: Transformers tend to lengthgeneralize on a task if there exists a short RASP program that solves the task for all input lengths. This simple conjecture remarkably captures most known instances of length generalization on algorithmic tasks, and can also inform design of effective scratchpads. Finally, on the theoretical side, we give a simple separating example between our conjecture and the “mindegreeinterpolator” model of learning from Abbe et al. (2023). Joint work with Hattie Zhou, Arwen Bradley, Etai Littwin, Noam Razin, Omid Saremi, Josh Susskind, and Samy Bengio. To appear in ICLR 2024. https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09 Password: cmsa  HARVARDMIT COMBINATORICS SEMINAR: CANCELLED – Richard P. Stanley Seminar in Combinatorics: Colored Interacting Particle Systems on the Ring: Stationary Measures from Yang–Baxter Equation
Speaker: Matthew Nicoletti – MIT 3:00 PM4:00 PM February 14, 2024 Seminar CANCELLED due to speaker illness. Will be rescheduled at a different date Recently, there has been much progress in understanding stationary measures for colored (also called multispecies or multitype) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (mASEP); (2) the qdeformed Totally Asymmetric Zero Range Process (TAZRP) also known as the qBoson particle system; (3) the qdeformed Pushing Totally Asymmetric Simple Exclusion Process (qPushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new “queue vertex models” on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. This is joint work with A. Aggarwal and L. Petrov. =============================== For more info, see https://math.mit.edu/combin/  NUMBER THEORY SEMINAR: Number Theory Seminar: Compatibility of the canonical ladic local systems on exceptional Shimura varieties
Speaker: Stefan Patrikis – The Ohio State University 3:00 PM4:00 PM February 14, 2024 1 Oxford Street, Cambridge, MA 02138 USA Let G, X be a Shimura datum, and let K be a compact open subgroup of G(Af). One hopes that under mild assumptions on G and K, the points of the Shimura variety ShK(G, X) parametrize a family of motives; in abelian type this is wellunderstood, but in nonabelian type it is completely mysterious. I will discuss joint work with Christian Klevdal showing that for exceptional Shimura varieties the points (over number fields, say) at least yield compatible systems of ladic representations, which should be the ladic realizations of the conjectural motives. For more info, see https://ashvinswaminathan.github.io/home/NTSeminar.html  SEMINARS: Dynamics, Geometry and Moduli Spaces Seminar: The WeilPetersson gradient flow of renormalized volume uniformizes relatively acylindrical manifolds
Speaker: Martin Bridgeman – Boston College 4:00 PM5:00 PM February 14, 2024 We consider the WeilPetersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures of a (relatively) acylindrical manifold. Using a toy model for the flow, we show that the flow has a global attracting fixed point at the structure Mgeod the unique structure with totally geodesic convex core boundary. This is joint work with Kenneth Bromberg and Franco Vargas Pallete. See webpage for more details: https://people.math.harvard.edu/~ctm/sem/
 15  THURSDAY SEMINAR SEMINAR: Thursday Seminar: Cyclotomic redshift
Speaker: Andy Senger – Harvard 3:30 PM5:30 PM February 15, 2024 1 Oxford Street, Cambridge, MA 02138 USA I will discuss the interaction of telescopically localized algebraic K theory with the higher cyclotomic extensions introduced in Mike’s talk, and explain why this is a key step in the disproof of the telescope conjecture. Along the way, we will show that telescopically localized K theory commutes with (co)limits indexed by pi finite pspaces.
 16  CMSA EVENT: CMSA Member Seminar: Symmetries and algebraicity in the flux landscape
Speaker: Damian van de Heisteeg – Harvard 12:00 PM1:00 PM February 16, 2024 In this talk I consider potentials coming from fluxes in string theory. The minima of these potentials trace out special loci in the moduli space of CalabiYau manifolds. I discuss the structure that underlies these minima from a Hodgetheoretic point of view. Friday, Feb. 16th at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363  HARVARDMIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Direct Product Testing on High Dimensional Expanders
Speaker: Mitali Bafna – MIT 3:00 PM4:00 PM February 16, 2024 The problem of testing direct product functions lies at the intersection of many areas within theoretical computer science, such as error correcting codes, probabilistically checkable proofs (PCPs), hardness amplification and property testing. We want to efficiently encode a function from [n] to {0,1} using local views in a way that admits local testability and the direct product encoding gives us the restriction of f on various subsets of [n]. A set system is said to support a direct product test when the following property holds: whenever a natural 2query test passes with noticeable probability, the encoding is correlated to a directproduct encoding. We study the question of what hypergraph expansion properties are required of the set systems that support a direct product test in the listdecoding regime. In contrast to the uniquedecoding regime, we show that spectral expansion is insufficient and the setsystem must additionally possess a form of topological expansion called coboundary expansion. This also turns out to be sufficient, thus giving us a characterization of such set systems. Building set systems with these expansion properties would thus lead to the ultimate form of efficient direct product testers and perhaps also allow for efficient hardness amplification. Based on joint work with Dor Minzer (https://eccc.weizmann.ac.il/report/2023/120/) =============================== For more info, see https://math.mit.edu/combin/  GAUGETOPOLOGYSYMPLECTIC SEMINAR: Gauge Theory and Topology Seminar: The SeibergWitten Equations and Einstein Metrics on Finite Volume 4Manifolds
Speaker: Alex Xu – Columbia University 3:30 PM4:30 PM February 16, 2024 1 Oxford Street, Cambridge, MA 02138 USA
Irreducible solutions to the SeibergWitten equations give a priori estimates for the total scalar curvature of the underlying 4manifold. This was used by LeBrun in the late 90s to construct the first examples of closed 4manifolds that satisfy the strict HitchinThorpe inequality yet do not admit any Einstein metrics. In this talk, I will describe an extension of this story to the finite volume setting, where we consider complete metrics with asymptotically hyperbolic cusps. As an application we will construct an infinite family of finite volume 4manifolds with $T^3$ ends that do not admit any asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the HitchinThorpe inequality due to DaiWei.
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18  19  20  SEMINARS: Probability Seminar: Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices
Speaker: Patrick Lopatto – Brown University 1:30 PM2:30 PM February 20, 2024 We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of FyodorovHiaryKeating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.  HARVARDMIT ALGEBRAIC GEOMETRY SEMINAR: HarvardMIT Algebraic Geometry Seminar: BrillNoether loci
Speaker: Montserrat Teixidor – Tufts University 3:00 PM4:00 PM February 20, 2024 1 Oxford Street, Cambridge, MA 02138 USA BrillNoether loci are defined as the set of curves of genus g that have an unexpected linear series of degree d and dimension r. Pflueger showed that these loci are nonempty when the expected codimension is at most g3. By studying linear series on chains of elliptic curves, we give a new proof of a slightly refined version of this result. We can also look at the behavior of the generic curve in the locus. An interesting conjecture of Auel and Haburcak states that these loci are distinct and not contained in each other, unless they come from adding or removing fixed points. Their proof made use of curves contained in K3 surfaces and was sufficient to prove the result in small genus. Using chains of elliptic curves, we can obtain additional information. For more information, please see https://researchseminars.org/seminar/harvardmitagseminar  CMSA EVENT: Math Science Lectures in Honor of Raoul Bott: Maggie Miller: Fibered ribbon knots vs. major 4D conjectures, Lecture 1
Speaker: Maggie Miller – University of Texas at Austin 4:00 PM5:30 PM February 20, 2024 1 Oxford Street, Cambridge, MA 02138 View from the CMSA Events Page Fibered ribbon knots vs. major 4D conjectures Location: Harvard University Science Center Hall A & via Zoom webinar Dates: Feb 20 & 22, 2024 Time: 4:005:30 pm Directions and Recommended Lodging Registration is required. Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow. This will be the fourth annual Math Science Lecture Series held in Honor of Raoul Bott. Fibered ribbon knots vs. major 4D conjecturesFeb. 20, 2024 Title: Fibered ribbon knots and the Poincaré conjecture Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of CassonGordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for nonfibered examples. Feb. 22, 2024 Title: Fibered knots and the sliceribbon conjecture Abstract: The sliceribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of CassonGordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a nonribbon disk. In 2022, DaiKongMallickParkStoffregen showed that unfortunately (?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz. Talk Chair: Cliff Taubes (Harvard Mathematics) Moderator: Freid Tong (Harvard CMSA)
Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem. For more info, please see the article “Remembering Raoul Bott” from the American Mathematical Society.
 21  CMSA EVENT: CMSA New Technologies in Mathematics Seminar: Computers and mathematics in partial differential equations: new developments and challenges
Speaker: Javier GomezSerrano – Brown University Dept. of Mathematics 2:00 PM3:00 PM February 21, 2024 20 Garden Street, Cambridge, MA 02138 In this talk I will address the interaction between traditional and more modern mathematics and how computers have helped over the last decade providing rigorous (computerassisted) proofs in the context of partial differential equations. I will also describe new exciting future directions in the field. No background is assumed. https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09 Password: cmsa  NUMBER THEORY SEMINAR: Number Theory Seminar: The Average Size of 2Selmer Groups of Elliptic Curves over Function Fields
Speaker: Niven Achenjang – MIT 3:00 PM4:00 PM February 21, 2024 1 Oxford Street, Cambridge, MA 02138 USA Given an elliptic curve E over a global field K the abelian group E(K) is finitely generated, and so much effort has been put into trying to understand the behavior of E(K), as E varies. Of note, it is a folklore conjecture that, when all elliptic curves E/K are ordered by a suitably defined height, the average value of their ranks is exactly 1/2. One fruitful avenue for understanding the distribution of E(K) has been to first understand the distribution of the sizes of Selmer groups of elliptic curves. In this direction, various authors (including BhargavaShankar, PoonenRains, and BhargavaKaneLenstraPoonenRains) have made conjectures which predict, for example, that the average size of the nSelmer group of E/K is equal to the sum of the divisors of n. In this talk, I will report on some recent work verifying this average size prediction, “up to small error term,” whenever n=2 and K is any global *function* field. Results along these lines were previously known whenever K was a number field or function field of characteristic >5, so the novelty of my work is that it applies even in “bad” characteristic. For more info, see https://ashvinswaminathan.github.io/home/NTSeminar.html  HARVARDMIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: A subdivision of the permutahedron for every Coxeter element
Speaker: Melissa ShermanBennett – MIT 3:30 PM5:15 PM February 21, 2024 Please note the special time. I will discuss some regular subdivisions of the permutahedron in R^n, one for each Coxeter element in the symmetric group S_n. These subdivisions are “Bruhat interval” subdivisions, meaning that each face is the convex hull of the permutations in a Bruhat interval (regarded as vectors in R^n). Bruhat interval subdivisions in general correspond to cones in the positive tropical flag variety by a combination of results of JoswigLohoLuberOlarte and Boretsky; the subdivisions indexed by Coxeter elements are finest subdivisions and so correspond to a subset of the maximal cones. For a particular choice of Coxeter element, we recover a cubical subdivision of the permutahedron due to HaradaHoriguchiMasudaPark. Applications of these subdivisions include new formulas for the class of the permutahedral variety as a sum of Richardson classes in the cohomology ring of the flag variety. This is joint workinprogress with Mario Sanchez. =============================== For more info, see https://math.mit.edu/combin/
 22  CMSA EVENT: CMSA Algebraic Geometry in String Theory Seminar: Geometric origins of values of the Riemann Zeta functions at positive integers.
Speaker: Yan Zhou – Northeastern 10:30 AM11:30 AM February 22, 2024 20 Garden Street, Cambridge, MA 02138 Given a Fano manifold, Iritani proposed that the asymptotic behavior of solutions to the quantum differential equation of the Fano should be given by the socalled ‘Gamma class’ in its cohomology ring. Later, AbouzaidGanatraIritaniSheridan reformulated the ‘Gamma conjecture’ for CalabiYau manifolds via the tropical SYZ mirror symmetry and proposed that values of the Riemann Zeta function at positive integers have geometric origins in the tropical periods and singularities of the SYZ geometry. In this talk, we will first review the content of the Gamma conjecture. Then, we will discuss the first step of generalizing AGIS’ approach to Gamma conjecture for the GrossSiebert mirror families of a Fano manifold in dimension 2 cases, based on joint work with Bohan Fang and Junxiao Wang. Zoom: https://harvard.zoom.us/j/93338480366?pwd=NEROWElhWStQVjVLRVZFSm1tV1ZCdz09  THURSDAY SEMINAR SEMINAR: Thursday Seminar: Topological cyclic homology
Speaker: Maxime Ramzi – Copenhagen University & Harvard 3:30 PM5:30 PM February 22, 2024 1 Oxford Street, Cambridge, MA 02138 USA Maxime Ramzi speaks on Topological cyclic homology  CMSA EVENT: Math Science Lectures in Honor of Raoul Bott: Maggie Miller: Fibered ribbon knots vs. major 4D conjectures, Lecture 2
Speaker: Maggie Miller – University of Texas at Austin 4:00 PM5:30 PM February 22, 2024 1 Oxford Street, Cambridge, MA 02138 View from the CMSA Events Page Fibered ribbon knots vs. major 4D conjectures Location: Harvard University Science Center Hall A & via Zoom webinar Dates: Feb 20 & 22, 2024 Time: 4:005:30 pm Directions and Recommended Lodging Registration is required. Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow. This will be the fourth annual Math Science Lecture Series held in Honor of Raoul Bott. Fibered ribbon knots vs. major 4D conjecturesFeb. 20, 2024 Title: Fibered ribbon knots and the Poincaré conjecture Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of CassonGordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for nonfibered examples. Feb. 22, 2024 Title: Fibered knots and the sliceribbon conjecture Abstract: The sliceribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of CassonGordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a nonribbon disk. In 2022, DaiKongMallickParkStoffregen showed that unfortunately (?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz. Talk Chair: Cliff Taubes (Harvard Mathematics) Moderator: Freid Tong (Harvard CMSA)
Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem. For more info, please see the article “Remembering Raoul Bott” from the American Mathematical Society.
 23  CMSA EVENT: CMSA Member Seminar: Integrability and Hidden Symmetries in Black Hole Dynamics
Speaker: Uri Kol – Harvard 12:00 PM1:00 PM February 23, 2024 The last decade has produced a number of remarkable discoveries, such as the first direct observation of gravitational waves by the LIGO/Virgo collaboration and the first black hole image taken by the Event Horizon Telescope. These discoveries mark the beginning of a new precision era in black hole physics, which is expected to develop further by future experiments such as LISA, the Einstein Telescope and Cosmic Explorer. In the era of precision black hole measurements, there is a need for precision theoretical methods and accurate predictions. In this talk I will describe an integrable sector of the gravitational scattering problem – analogous to the hydrogen atom in quantum mechanics – in which exact predictions can be made, and the implications for astrophysical black holes and binary mergers. Friday, Feb. 23rd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363  HARVARDMIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Asymptotic separation index as a tool in descriptive combinatorics
Speaker: Anton Bernshteyn – Georgia Tech 3:00 PM4:00 PM February 23, 2024 A common theme throughout mathematics is the search for “constructive” solutions to problems as opposed to mere existence results. For problems over R and other wellbehaved spaces, this idea is nicely captured by the concept of a Borel construction. In particular, one can investigate Borel solutions to classical combinatorial problems such as graph colorings, perfect matchings, etc. The area studying these questions is called descriptive combinatorics. As I will explain in the talk, many facts in graph theory that we know and love—for example, Brooks’ theorem—turn out to be inherently “nonconstructive” in this sense. The main result of this talk is that Borel versions of various classical combinatorial theorems nevertheless hold on graphs that can, in some sense, be easily decomposed into subgraphs with finite components. No prior familiarity with Borel combinatorics or descriptive set theory will be assumed. Based on joint work with Felix Weilacher. =============================== For more info, see https://math.mit.edu/combin/  GAUGETOPOLOGYSYMPLECTIC SEMINAR: Gauge Theory and Topology Seminar: Spectral flow and reducible solutions to the massive VafaWitten equations
Speaker: Cliff Taubes – Harvard University 3:30 PM4:30 PM February 23, 2024 1 Oxford Street, Cambridge, MA 02138 USA
The VafaWitten equations (with or without a mass term) constitute a nonlinear, first order system of differential equations on a given oriented, compact, Riemannian 4manifold. Because these are the variational equations of a functional, the linearized equations at any given solution can be used to define an elliptic, first order, selfadjoint differential operator. This talk will describe bounds (upper and lower) for the spectral flow between respective versions of this operator that are defined by the elements in diverging sequences of reducible solutions. (The spectral flow is formally the difference between the respective Morse indices of the solutions when they are viewed as critical points of the functional.) In some cases, the absolute value of the spectral flow is bounded along the sequence, whereas in others it diverges. This is a curious state of affairs.
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25  26  SEMINARS: Arithmetic Statistics Seminar: Singularity probability of adjacency matrices of random regular digraphs
Speaker: Hoi Nguyen – Ohio State 3:00 PM4:00 PM February 26, 2024 1 Oxford Street, Cambridge, MA 02138 USA We will discuss an algebraic method by J. Huang to bound the singularity probability of the adjacency matrix of random dregular digraphs. Although we can slightly improve the bounds, the current estimates are still far from best possible, especially toward the problem of bounding the least singular values.  CMSA EVENT: CMSA Colloquium: Factorization algebras in quite a lot of generality
Speaker: Clark Barwick – University of Edinburgh 4:30 PM5:30 PM February 26, 2024 20 Garden Street, Cambridge, MA 02138 The objects of arithmetic geometry are not manifolds. Some concepts from differential geometry admit analogues in arithmetic, but they are not straightforward. How then can we hope to make precise sense of quantum field theories on these objects? I will propose the beginnings of a mathematical framework via a general theory of factorization algebras. A new feature is a subtle piece of additional structure on our objects – what I call a worldstructure – that is ordinarily left implicit.
 27  SEMINARS: Probability Seminar: Antonio Auffinger, Northwestern
Speaker: Antonio Auffinger – Northwestern 1:30 PM2:30 PM February 27, 2024  SEMINARS: Probability Seminar: Dimension Reduction Methods for Data Visualization
Speaker: Antonio Auffinger – Northwestern University 1:30 PM2:30 PM February 27, 2024 The purpose of dimension reduction methods for data visualization is to project high dimensional data to 2 or 3 dimensions so that humans can understand some of its structure. In this talk, we will give an overview of some of the most popular and powerful methods in this active area. We will then focus on two algorithms: Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). Here, we will present new rigorous results that establish an equilibrium distribution for these methods when the number of data points diverge in the presence of pure noise or with a planted signal. Based on joint work with Daniel Fletcher.  HARVARDMIT ALGEBRAIC GEOMETRY SEMINAR: HarvardMIT Algebraic Geometry Seminar: CMminimizers and standard models of Fano fibrations over curves
Speaker: Maksym Fedorchuk – Boston College 3:00 PM4:00 PM February 27, 2024 A recent achievement in Kstability of Fano varieties is an algebrogeometric construction of a projective moduli space of Kpolystable Fanos. The ample line bundle on this moduli space is the CM line bundle of Tian. One of the consequences of the general theory is that given a family of Kstable Fanos over a punctured curve, the polystable filling is the one that minimizes the degree of the CM line bundle after every finite base change. A natural question is to ask what are the CMminimizers without base change. In answering this question, we arrive at a theory of Kollár stability for fibrations over onedimensional bases, and standard models of Fano fibrations. After explaining the general theory, I will sketch work in progress on standard models of quartic threefold hypersurfaces. This talk is based on joint work with Hamid Abban and Igor Krylov. For more information, please see https://researchseminars.org/seminar/harvardmitagseminar
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