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November  November  1  MATHEMATICAL PICTURE LANGUAGE SEMINAR
10:00 AM11:00 AM December 1, 2020  CMSA EVENT: CMSA Computer Science for Mathematicians: Some extensions on argumentation frameworks via hypergraphs
11:30 AM12:30 PM December 1, 2020 The Dung Abstract Argumentation Framework (AAF) is an effective formalism for modelling disputes between two or more agents. Generally, the Dung AF is extended to include some unique interactions between agents. This has further been explained with the Bipolar Argumentation Framework (BAF). In the academic space, the use of AAF is highly signified. We can use the AF as a means to resolve disagreements that allows for the determination of a winning argument. In general, there can be imperfect ontologies that affect how reasoning is defined. Typical logicbased AFs apply to the incoherent/uncertain ontologies. However, Dung demonstrated a stable extension of AF to support an “acceptable standard of behavior”. This talk will align with present endeavors on extending the Dung AAF to consider the notion of conflictfreeness in relation to persistence over a hypergraph. With a generic type of argumentation, there are some methods that can exploit certain complex decision procedures. Argument and attack relations within the Dung AAF, thus are further defined to obtain a graphical formula of Kripke groundedness. The incorporating of multiple levels of knowledge aligns with a computational linguistics aspect for the defining of a classification criteria for AAF. In the construction, I will provide some treatment of ‘good’ modeltheoretic properties that bridge AAF with Zarankiewicz’s problem to introduce how arguments are consistent with bipartite hypergraphs. The Zarankiewicz problem appears with the communication complexity on AF graphs. Zoom: https://harvard.zoom.us/j/98231541450  HARVARDMIT ALGEBRAIC GEOMETRY SEMINAR
3:00 PM4:00 PM December 1, 2020 Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties. We show that the “top open positroid variety” has mixed Hodge polynomial given by the q,trational Catalan numbers (up to a simple factor). Unlike the Grassmannian, the cohomology of open positroid varieties is not pure. The q,trational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties. Our work is also related to knot homology and to the cohomology of compactified Jacobians. This talk is based on joint work with Pavel Galashin. Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09
 2  CMSA EVENT: CMSA Math Science Literature Lecture Series
8:00 AM9:30 AM December 2, 2020 TITLE: Is relativity compatible with quantum theory? ABSTRACT: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13decimalpoint precision—the most accurate experiments ever performed. Talk chair: Zhengwei Liu Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.” For more information, please visit the event page.  CMSA EVENT: CMSA Strongly Correlated Quantum Materials and HighTemperature Superconductors Series: Interplay between superconductivity and nonFermi liquid at a quantum critical point in a metal
12:00 PM1:30 PM December 2, 2020 I discuss the interplay between nonFermi liquid behaviour and pairing near a quantumcritical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/Ω^γ (the γmodel). This model describes, in particular, the pairing at a 2D Isingnematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get rearranged in the presence of a small phase variation. I show that a new nonsuperconducting ground state emerges at γ >2. Zoom: https://harvard.zoom.us/j/977347126  RANDOM MATRIX SEMINAR
2:00 PM3:00 PM December 2, 2020 The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of nonoverlapping hypercubes of sidelengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309340). Zoom: https://harvard.zoom.us/j/98520388668  NUMBER THEORY SEMINAR
3:00 PM4:00 PM December 2, 2020 The Cohen–Lenstra–Martinet conjectures have been verified in only two cases. Davenport–Heilbronn compute the average size of the 3torsion subgroups in the class group of quadratic fields and Bhargava computes the average size of the 2torsion subgroups in the class groups of cubic fields. The values computed in the above two results are remarkably stable. In particular, work of Bhargava–Varma shows that they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions. However for certain “thin” families of cubic fields, namely, families of monogenic and nmonogenic cubic fields, the story is very different. In this talk, we will determine the average size of the 2torsion subgroups of the class groups of fields in these thin families. Surprisingly, these values differ from the Cohen–Lenstra–Martinet predictions! We will also provide an explanation for this difference in terms of the Tamagawa numbers of naturally arising reductive groups. This is joint work with Manjul Bhargava and Jon Hanke. Zoom: https://harvard.zoom.us/j/96767001802 Password: The order of the permutation group on 9 elements.  MATH TABLE
4:30 PM5:30 PM December 2, 2020 pAdic numbers have always been primarily associated with pure Mathematics, and have become especially relevant in algebra and modern number theory. But why did Computer Scientists become interested in them? In this talk we will introduce padic numbers and survey their main properties. We will then introduce Dixon’s algorithm, which is the first algorithm that used padic numbers to compute the exact rational solution to an integer linear system of equations. We will also explore the latest runtime improvements in padic linear algebra algorithms, and discuss whether we can solve linear equation systems faster than matrix multiplication. Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09
 3  CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Domain Wall Fermions and Chiral Gauge theories: Topological Insulators in Particle Physic
10:30 AM12:00 PM December 3, 2020 Ideas from the early 1990s for regulating chiral fermions in lattice gauge theory led to a number of developments which paralleled roughly concurrent and independent discoveries in condensed matter physics. I show how the Integer Quantum Hall Effect, Chern Insulators, Topological Insulators, and Majorana edge states all play a role in lattice gauge theories, and how one can also find relativistic versions of the Fractional Quantum Hall Effect, the Quantum Spin Hall Effect and related exotic forms of matter. How to construct a nonperturbative regulator for chiral gauge theories (like the Standard Model!) remains an open challenge, however, one that may require new insights from condensed matter physics into exotic states of matter. Zoom: https://harvard.zoom.us/j/977347126
 4  CMSA EVENT: CMSA Math Science Literature Lecture Series
8:00 AM9:30 AM December 4, 2020 TITLE: Michael Atiyah: Geometry and Physics ABSTRACT: In mid career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists. Talk chair: Peter Kronheimer Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.” For more information, please visit the event page.
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6  7  CMSA EVENT: CMSA Mathematical Physics Seminar: Moduli Space Holography and the Finiteness of Flux Vacua
10:30 AM11:30 AM December 7, 2020 In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, nearboundary regions in complex structure moduli spaces of CalabiYau manifolds using asymptotic Hodge theory. For real twodimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The bulk reconstruction from the boundary data is provided by the sl(2)orbit theorem of Schmid and Cattani, Kaplan, Schmid, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of CalabiYau fourfold compactifications and discuss how this allows us, in work with C. Schnell, to prove that the number of selfdual flux vacua is finite. Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09
 8  MATHEMATICAL PICTURE LANGUAGE SEMINAR
10:00 AM11:00 AM December 8, 2020 In this talk, I will present a new way to look at symmetry. We show that symmetry can be viewed as a noninvertible gravitational anomaly, and a noninvertible gravitational anomaly is classified by topological order in one higher dimension. This leads to a holographic view of symmetry: symmetry is a shadow of topological order in one higher dimension. This point of view allows us to see the duality (i.e. the equivalence) between symmetries that look very different. It also gives rise to a more general symmetry – algebraic higher symmetry, which is beyond higher group description. Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09  HARVARDMIT ALGEBRAIC GEOMETRY SEMINAR
3:00 PM4:00 PM December 8, 2020 Moduli spaces of curves admit finite covers by moduli spaces which parametrize curves together with socalled level structures. In my talk, I will discuss how the cohomology of these spaces at infinite level is related to a profinite property of the mapping class group. I will then explain why tools from padic geometry yield vanishing statements for these cohomologies in high degree. Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09
 9  CMSA EVENT: CMSA Strongly Correlated Quantum Materials and HighTemperature Superconductors Series: Signatures of anomalous symmetry breaking in the cuprates
10:30 AM12:00 PM December 9, 2020 The temperature versus doping phase diagram of the cuprate highT_{c }superconductors features an enigmatic pseudogap region whose microscopic origin remains a subject of intensive study. Experimentally resolving its symmetry properties is imperative for narrowing down the list of possible explanations. In this talk I will give an overview of how optical second harmonic generation (SHG) can be used as a sensitive probe of symmetry breaking, and recap the ways it has been used to solve outstanding problems in condensed matter physics. I will then describe how we have been applying SHG polarimetry and spectroscopy to interrogate the cuprate pseudogap. In particular, I will discuss our data on YBa_{2}Cu_{3}O_{y }[1], which show an order parameterlike increase in SHG intensity below the pseudogap temperature T* across a broad range of doping levels. I will then focus on our more recent results on a model parent cuprate Sr_{2}CuO_{2}Cl_{2 }[2], where evidence of anomalous broken symmetries surprisingly also exists. Possible connections between these observations will be speculated upon. [1] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh, “A global inversionsymmetrybroken phase inside the pseudogap region of YBa_{2}Cu_{3}O_{y},” Nature Phys. 13, 250 (2017). [2] A. de la Torre, K. L. Seyler, L. Zhao, S. Di Matteo, M. S. Scheurer, Y. Li, B. Yu, M. Greven, S. Sachdev, M. R. Norman and D. Hsieh. “Anomalous mirror symmetry breaking in a model insulating cuprate Sr_{2}CuO_{2}Cl_{2},” Preprint at https://arxiv.org/abs/2008.06516. Zoom: https://harvard.zoom.us/j/977347126  NUMBER THEORY SEMINAR
3:00 PM4:00 PM December 9, 2020 There are several ways to specify a number field. One can provide the minimal polynomial of a primitive element, the multiplication table of a $\bf Q$basis, the traces of a large enough family of elements, etc. From any way of specifying a number field one can hope to deduce a bound on the number $N_n(H)$ of number fields of given degree $n$ and discriminant bounded by $H$. Experimental data suggest that the number of isomorphism classes of number fields of degree $n$ and discriminant bounded by $H$ is equivalent to $c(n)H$ when $n\geqslant 2$ is fixed and $H$ tends to infinity. Such an estimate has been proved for $n=3$ by Davenport and Heilbronn and for $n=4$, $5$ by Bhargava. For an arbitrary $n$ Schmidt proved a bound of the form $c(n)H^{(n+2)/4}$ using Minkowski’s theorem. Ellenberg et Venkatesh have proved that the exponent of $H$ in $N_n(H)$ is less than subexponential in $\log (n)$. I will explain how Hermite interpolation (a theorem of Alexander and Hirschowitz) and geometry of numbers combine to produce short models for number fields and sharper bounds for $N_n(H)$. Zoom: https://harvard.zoom.us/j/96767001802 Password: The order of the permutation group on 9 elements.  CMSA EVENT: CMSA New Technologies in Mathematics: Machine learning and SU(3) structures on six manifolds
3:00 PM4:00 PM December 9, 2020 In this talk we will discuss the application of Machine Learning techniques to obtain numerical approximations to various metrics of SU(3) structure on six manifolds. More precisely, we will be interested in SU(3) structures whose torsion classes make them suitable backgrounds for various string compactifications. A variety of aspects of this topic will be covered. These will include learning moduli dependent RicciFlat metrics on CalabiYau threefolds and obtaining numerical approximations to torsional SU(3) structures. Zoom: https://harvard.zoom.us/j/96047767096?pwd=M2djQW5wck9pY25TYmZ1T1RSVk5MZz09
 10  CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: An analytic bootstrap approach for CFTs on RP^d and CFTs with boundaries
10:30 AM12:00 PM December 10, 2020 In this talk, I will introduce an analytic bootstrap approach for twopoint correlation functions in CFTs on real projective space, and CFTs with a conformal boundary. We will use holography as a kinematical tool to derive universal results. By examining the conformal block decomposition properties of exchange diagrams in AdS space, we identify a useful new basis for decomposing correlators. The dual basis gives rise to a basis of functionals, whose actions we can compute explicitly via holography. Applying these functionals to the crossing equations, we can systematically extract constraints on the CFT data in the form of sum rules. I will demonstrate this analytic method in the canonical example of \phi^4 theory in d=4\epsilon, fixing the CFT data to \epsilon^2. Zoom: https://harvard.zoom.us/j/977347126
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10:00 AM11:00 AM December 15, 2020 A twodimensional rotating shallowwater model describes a layer of water, in guise of oceans covering the Earth. It is formally analogue to a Schrödinger equation where the tools from topological insulators are relevant. Once regularized at small scale by an oddviscous term, such a model has a welldefined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulkedge correspondence. We study a continuous family of boundary conditions with a rich phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson’s theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, which is ultimately the reason for the violation. (Joint work with H. Jud and C. Tauber.) Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09  CMSA EVENT: CMSA Computer Science for Mathematicians: Lightning Network Economics: Costminimal channels and their implications for network structure
11:30 AM12:30 PM December 15, 2020 The Lightning Network is a secondlayer solution built above Bitcoin, aimed to solve Bitcoin’s scalability and immediacy problems. A channel in the Lightning Network allows two parties to secure bitcoin payments and escrow holdings between them. Designed to increase transaction immediacy and reduce blockchain congestion, this has the potential to solve many issues associated with Bitcoin. In this talk, we study the economics of the Lightning Network. We present conditions under which two parties optimally establish a channel and give explicit formulas for channels’ costs. Using these, we derive implications for the network’s structure under cooperation assumptions among small sets of users. We show both local implications, such as the wastefulness of certain structures, and global implications, such as a (low) upper bound on the Lightning Network’s average degree. Zoom: https://harvard.zoom.us/j/98231541450
 16  CMSA EVENT: CMSA Strongly Correlated Quantum Materials and HighTemperature Superconductors Series: Organizing Principle of Mottness and Complex Phenomenon in High Temperature Superconductors
10:30 AM12:00 PM December 16, 2020 The complex phenomenon in the highTc cuprate calls for a microscopic understanding based on general principles. In this Lecture, an exact organizing principle for a typical doped Mott insulator will be presented, in which the fermion sign structure is drastically reduced to a mutual statistics. Its nature as a longrange spincharge entanglement of manybody quantum mechanics will be exemplified by exact numerical calculations. The phase diagram of the cuprate may be unified in a “bottomup” fashion by a “parent” ground state ansatz with hidden orders constructed based on the organizing principle. Here the pairing mechanism will go beyond the “RVB” picture and the superconducting state is of nonBCS nature with modified London equation and novel elementary excitations. In particular, the Bogoliubov/Landau quasiparticle excitation are emerging with a twogap structure in the superconducting state and the Fermi arc in a pseudogap regime. A mathematic framework of fractionalization and duality transformation guided by the organizing principle will be introduced to describe the above emergent phenomenon. Zoom: https://harvard.zoom.us/j/977347126
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