In this talk, I will discuss another possibility that arises from the interplay of integrability and symmetry; in integrable one dimensional quantum magnets with complex symmetries, spin transport is neither ballistic nor diffusive, but rather superdiffusive. Using a novel method for the simulation of quantum dynamics (termed Density Matrix Truncation), I will present a detailed analysis of spin transport in a variety of integrable quantum magnets with various symmetries. Crucially, our analysis is not restricted to capturing the dynamical exponent of the transport dynamics and enables us to fully characterize its universality class: for all superdiffusive models, we find that transport falls under the celebrated Kardar-Parisi-Zhang (KPZ) universality class.

Finally, I will discuss how modern atomic, molecular and optical platforms provide an important bridge to connect the microscopic interactions to the resulting hydrodynamical transport dynamics. To this end, I will present recent experimental results, where this KPZ universal behavior was observed using atoms confined to an optical lattice.

[1] Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems
B Ye†, FM*, J Kemp*, RB Hutson, NY Yao
(PRL in press) – arXiv:2205.02853

[2] Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion
D Wei, A Rubio-Abadal, B Ye, FM, J Kemp, K Srakaew, S Hollerith, J Rui, S Gopalakrishnan, NY Yao, I Bloch, J Zeiher
Science (2022) — arXiv:2107.00038

Electromagnetic fields in a magneto-electric medium behave in close analogy to photons coupled to the hypothetical elementary particle, the axion. This emergent axion electrodynamics is expected to provide novel ways to detect and control material properties with electromagnetic fields. Despite having been studied intensively for over a decade, its theoretical understanding remains mostly confined to the static limit. Formulating axion electrodynamics at general optical frequencies requires resolving the difficulty of calculating optical magneto-electric coupling in periodic systems and demands a proper generalization of the axion field. In this talk, I will introduce a theory of optical axion electrodynamics that allows for a simple quantitative analysis. Then, I will move on to discuss the issue of the Kerr effect in axion antiferromagnets, refuting the conventional wisdom that the Kerr effect is a measure of the net magnetic moment. Finally, I will apply our theory to a topological antiferromagnet MnBi2Te4.

Semiconductor bilayer heterostructures provide a remarkable platform for simulating Hubbard models on an emergent lattice defined by moire potential minima. As a hallmark of Hubbard model physics, the Mott insulator state with local magnetic moments has been observed at half filling of moire band. In this talk, I will describe new phases of matter that grow out of the canonical 120-degree antiferromagnetic Mott insulator on the triangular lattice. First, in an intermediate range of magnetic fields, doping this Mott insulator gives rise to a dilute gas of spin polarons, which form a pseudogap metal. Second, the application of an electric field between the two layers can invert the many-body gap of a charge-transfer Mott insulator, resulting in a continuous phase transition to a quantum anomalous Hall insulator with a chiral spin structure. Experimental results will be discussed and compared with theoretical predictions.

Relative trace formulas are central tools in the study of relative functoriality. In many cases of interest, basic stability problems have not previously been addressed. In this talk, I discuss a theory of endoscopy in the context of symmetric varieties with the global goal of stabilizing the associated relative trace formula. I outline how, using the dual group of the symmetric variety, one can give a good notion of endoscopic symmetric variety and conjecture a matching of relative orbital integrals in order to stabilize the relative trace formula, which can be proved in some cases. Time permitting, I will explain my proof of these conjectures in the case of unitary Friedberg–Jacquet periods.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 2nd.

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

The wonderful variety of a realizable matroid and its Chow ring have played key roles in solving many long-standing open questions in combinatorics and algebraic geometry. Yet, its $K$-rings are underexplored until recently. I will be sharing with you some discoveries on the $K$-rings of the wonderful variety associated with a realizable matroid: an exceptional isomorphism between the $K$-ring and the Chow ring, with integral coefficients, and a Hirzebruch–Riemann–Roch-type formula. These generalize a recent construction of Berget–Eur–Spink–Tseng on the permutohedral variety. We also compute the Euler characteristic of every line bundle on wonderful varieties, and give a purely combinatorial formula. This in turn gives a new valuative invariant of an arbitrary matroid. As an application, we present the $K$-rings and compute the Euler characteristic of arbitrary line bundles of the Deligne–Mumford–Knudsen moduli spaces of rational stable curves with distinct marked points. Joint with Matt Larson, Sam Payne and Nick Proudfoot.

For more information on the speaker, please see: http://www.shiyue.li

Collective cell migration drives numerous physiological processes such as tissue morphogenesis, wound healing, tumor progression and cancer invasion. However, how the interplay of mechanical interactions and the modes of collective self-organization among cells informs such processes is yet to be established. In this talk, I will focus on the role of three-dimensional force transmission, from a theoretical and computational perspective, on two phenomena: (1) cell extrusion from a cellular monolayer and (2) density-independent solid-like to fluid-like transition of active cell layers. For the first topic, I will focus on how increasing cell-cell adhesion relative to cell-substrate adhesion enables cells to collectively exploit distinct mechanical pathways – leveraging defects in nematic and hexatic phases associated with cellular arrangement – to eliminate an unwanted cell. For the second topic, I will show how solid-like to fluid-like transition in active cell layers is linked to the percolation of isotropic stresses. This is achieved via two distinct and independent paths to model this transition by increasing (a) cell-cell adhesion and (b) active traction forces. Additionally, using finite-size scaling analyses, the phase transition associated with each path is mapped onto the 2D site percolation universality class. Our results highlight the importance of force transmission in informing the collective behavior of living cells and opens the door to new sets of questions for those interested in connecting the physics of cellular self-organization to the dynamics of biological systems.

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/active-matter-seminar

The pentagram map was introduced by Schwartz as a dynamical system on convex polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that it can be viewed as a translation map on a family of real tori. We will explain how the real, complex, and finite field dynamics of the pentagram map are all related by the following generalization: the pentagram map’s first or second iterate is birational to a translation on a family of Jacobian varieties (except possibly in characteristic 2). The second hour will get into the details of the proof, especially the definition of the Lax representation and the spectral curve.

Bordered Floer homology is an extension of Heegaard Floer homology to manifolds with parametrized boundary, and in the case of manifolds with torus boundary knot Floer homology gives another such extension. In earlier joint work with J. Rasmussen and L. Watson, it was shown that in this setting the bordered Floer invariant, which is equivalent to the UV=0 truncation of the knot Floer complex, can be encoded geometrically as a collection of immersed curves in the punctured torus and a pairing theorem recovers HF-hat (the simplest version of Heegaard Floer homology) of a glued manifold via Floer homology of immersed curves. In this talk, we will survey some applications of this result and then discuss a generalization that encodes the full knot Floer complex of a knot as a collection of decorated immersed curves in the torus. When two manifolds with torus boundary are glued, a pairing theorem computes HF^- of the resulting manifold as the Floer homology of certain immersed curves associated with each side. We remark that the curves we describe are invariants of knots, but we expect they are in fact invariants of the knot complements; if this is true, they may be viewed as defining a minus type bordered Floer invariant for manifolds with torus boundary.

Recently there has been lots of activity surrounding generalized notions of symmetry in quantum field theory, including “categorical symmetries”, “higher symmetries”, “noninvertible symmetries”, etc. Inspired by definitions of abstract (finite) groups and algebras and their linear actions, we introduce a framework for these symmetries in field theory and a calculus of topological defects based on techniques in topological field theory.  This is joint work with Constantin Teleman and Greg Moore.

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

I’ll discuss a number of interactions between algebraic geometry, low-dimensional topology, and arithmetic, arising from the study of local systems on moduli spaces of curves. I’ll explain how, in joint work with Aaron Landesman, we exploit these connections to resolve open questions of Prill, Esnault-Kerz, and others.

CMSA/Tsinghua Math-Science Literature Lecture

Prof. Hugh Woodin will present a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.

Date: Wednesday, November 9, 2022

Time: 9:30 – 11:00 am ET

Location: Via Zoom Webinar and Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

Registration is required.

• Register here to attend virtually:  Zoom Webinar Registration
• Register here to attend in-person: In-Person Registration

Title: Large cardinals and small sets: The AD⁺ Duality Program

Abstract: The determinacy axiom, AD, was introduced by Mycielski and Steinhaus over 60 years ago as an alternative to the Axiom of Choice for the study of arbitrary sets of real numbers.  The modern view is that determinacy axioms concern generalizations of the borel sets, and deep connections with large cardinal axioms have emerged.

The study of determinacy axioms has led to a specific technical refinement of AD, this is the axiom AD⁺. The further connections with large axioms have in turn implicitly led to a duality program, this is the AD⁺ Duality Program.

The main open problems here are intertwined with those of the Inner Model Program, which is the central program in the study of large cardinal axioms.

This has now all been distilled into a series of specific conjectures.

Talk chair: Horng-Tzer Yau (Harvard Mathematics & CMSA)

Moderator: Alejandro Poveda Ruzafa (Harvard CMSA)

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

An automorphic form can appear in multiple degrees of the cohomology of arithmetic manifolds, and this happens mostly when the arithmetic manifolds are not algebraic. This phenomenon is a part of the “derived” structures of the Langlands program, suggested by Venkatesh. However, even over algebraic arithmetic manifolds, certain automorphic forms like weight-one elliptic modular forms possess a derived structure. In this talk, we discuss this idea over Shimura varieties. A part of the story is the construction of archimedean/p-adic “derived” operators on the cohomology of Shimura varieties, using complex/p-adic Hodge theory.

Based on a joint work with Paul Bourgade.

Based on a joint work with Paul Bourgade.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 9th.

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

A hyperplane arrangement of braid type is a collection of hyperplanes in R^n of the form {x_i-x_j=s}, where i,j are indices in [n] and s is an integer. Classical families include the Catalan, Shi, Semi-order and Linial arrangements. In this talk we will discuss some bijections between the regions of braid type arrangements and some labeled plane trees. This bijective framework applies to the braid type arrangements which satisfy a particular property that we call “transitivity” (the above classical families are all transitive). Time permitting we will then discuss some recent progress in extending this bijective framework in two directions: (a) extension of the bijections to lower dimensional faces, and (b) extension to arrangements of other Coxeter types (which include hyperplanes of the form {x_i+x_j=s}). Part of this work is joint with Te Cao.

Title: TBA

Abstract: TBA

We present large classes of non-extremal solitons in gravity that are asymptotic to four-dimensional Minkowski spacetime plus extra compact dimensions. They correspond to smooth horizonless geometries induced by topology in spacetime and supported by electromagnetic flux, which characterize coherent states of quantum gravity. We discuss a new approach to deal with Einstein-Maxwell equations in more than four dimensions, such that they decompose into a set of Ernst equations. We generate the solitons by applying different techniques associated with the Ernst formalism. We focus on solitons with zero net charge yet supported by flux, and compare them to Schwarzschild black holes. These are also ultra-compact geometries with very high redshift but differ in many aspects. At the end of the talk, we discuss the stability properties of the solitons and their gravitational signatures.

This seminar will be held over Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/general-relativity-2021-22/

A link of an isolated complex surface singularity (X, 0) is a 3-manifold Y which is the boundary of the intersection of X with a small ball centered at 0. Smoothings of the singularity give non-singular 4-manifolds, the Milnor fibers, with the same boundary Y. The Milnor fibers carry symplectic (even Stein) structures, and thus provide fillings of the canonical contact structure on Y; another Stein filling comes from the minimal resolution of (X, 0). An important question is whether all Stein fillings of the link come from this algebraic construction: this is true in some simple cases such as lens spaces. However, even in the “next simplest” case, for many rational singularities, we are able to construct “unexpected” Stein fillings that do not arise from Milnor fibers. To this end, we encode Stein fillings via curve arrangements, motivated by  T.de Jong-D.van Straten’s description of smoothings  of certain rational surface singularities in terms of deformations of associated singular plane curves. We then use classical projective geometry  to construct unexpected line arrangements and unexpected fillings. This is a topological story, with minimal input from algebraic geometry.  Joint work with L. Starkston.

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

The Emergence Proposal claims that in Quantum Gravity the kinetic terms of the the fields in the IR emerge from integrating out (infinite) towers of particles up to the QG cutoff. After introducing this proposal in the context of the Swampland Program, I will explain why it is natural to identify this QG cutoff with the Species Scale, motivating it by direct computation in the presence of the relevant towers. Then, I will present evidence for this proposal by directly studying how it is realized in different string theory setups, where the kinetic terms of scalars, p-forms and even scalar potentials can be shown to emerge after integrating out such towers.

Given integers g, m, and n, the heavy/light moduli space Mbar_{g, m|n} is a compactification of the moduli space of smooth (m+n)-marked curves of genus g. These spaces are particular examples of Hassett’s moduli spaces of weighted stable curves. Their rational cohomology gives a rich family of representations of products of symmetric groups. I’ll discuss recent work on the structure of this family of representations, and how they relate to the S_n-representations determined by the cohomology of Deligne-Mumford compactifications. This talk is based on joint work with Stefano Serpente and Claudia Yun.

https://sites.google.com/view/harvardmitag

When a collection of electronic excitations are strongly coupled to a single mode cavity, mixed light-matter excitations called polaritons are created. The situation is especially interesting when the strength of the light-matter coupling Ω_R is such that the coupling energy becomes close to the one of the bare matter resonance ω₀. For this value of parameters, the system enters the so-called ultra-strong coupling regime, in which a number of very interesting physical effects were predicted caused by the counter-rotating and diamagnetic terms of the Hamiltonian.

In a microcavity, the strength of the electric field caused by the vacuum fluctuations, to which the strength of the light-matter coupling Ω_Ris proportional, scales inversely with the cavity volume. One very interesting feature of the circuit-based metamaterials is the fact that this volume can be scaled down to deep subwavelength values in all three dimension of space.¹ Using metamaterial coupled to two-dimensional electron gases under a strong applied magnetic field, we have now explored to which extend this volume can be scaled down and reached a regime where the stability of the polariton is limited by diffraction into a continuum of plasmon modes².

We have also used transport to probe the ultra-strong light-matter coupling³, and show now that the latter can induce a breakdown of the integer quantum Hall effect⁴. The phenomenon is explained in terms of cavity-assisted hopping, an anti-resonant process where an electron can scatter from one edge of the sample to the other by “borrowing” a photon from the cavity⁵. We are also evaluating a proposal suggesting that the value of the quantization voltage can be renormalized by the cavity⁶.

1. Scalari, G. et al. Ultrastrong Coupling of the Cyclotron Transition of a 2D Electron Gas to a THz Metamaterial. Science 335, 1323–1326 (2012).
2. Rajabali, S. et al. Polaritonic Nonlocality in Light Matter Interaction. Nat Photon 15, 690–695 (2021).
3. Paravicini-Bagliani, G. L. et al. Magneto-Transport Controlled by Landau Polariton States. Nat. Phys. 15, 186–190 (2019).
4. Appugliese, F. et al. Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect. Science 375, 1030–1034 (2022).
5. Ciuti, C. Cavity-mediated electron hopping in disordered quantum Hall systems. Phys. Rev. B 104, 155307 (2021).
6. Rokaj, V., Penz, M., Sentef, M. A., Ruggenthaler, M. & Rubio, A. Polaritonic Hofstadter butterfly and cavity control of the quantized Hall conductance. Phys. Rev. B 105, 205424 (2022).

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

In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle, but the role of symmetry breaking is not well understood. Here, we develop a Lagrangian formulation to study the geometry of learning dynamics in neural networks and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. Then, we generalize Noether’s theorem known in physics to describe a unique symmetry breaking mechanism in learning and derive the resulting motion of the Noether charge: Noether’s Learning Dynamics (NLD). Finally, we apply NLD to neural networks with normalization layers and discuss practical insights. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

To a connected reductive group G over a local field F we define a compact topological group π_1~(G) and an extension G(F)_∞ of G(F) by π_1~(G). From any character x of π_1~(G) of order n we obtain an n-fold cover G(F)_x of the topological group G(F). We also define an L-group for G(F)_x, which is a usually non-split extension of the Galois group by the dual group of G, and deduce from the linear case a refined local Langlands correspondence between genuine representations of G(F)_x and L-parameters valued in this L-group.

This construction is motivated by Langlands functoriality. We show that a subgroup of the L-group of G of a certain kind naturally leads to a smaller quasi-split group H and a double cover of H(F). Genuine representations of this double cover are expected to be in functorial relationship with representations of G(F). We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal L-parameters when p is sufficiently large, and one to the theory of endoscopy.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 16th.

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

I will discuss joint work with Christin Bibby, Nir Gadish, and Claudia Yun.  The historical antecedents are in earlier work of Billera-Holmes-Vogtmann around 2000 and others, who study an interesting space of metric trees on n labelled taxa. This is a space that is shellable and whose top homology was calculated, as an S_n-representation, by Robinson-Whitehouse.  These spaces have a reinterpretation as moduli of tropical curves of genus 0.  Other historical antecedents of our work are in the study of configuration spaces of n points on a graph, whose topological invariants are quite interesting. For example, Ko-Park proved that failure of planarity of a graph can be detected by torsion in H_1 of its unordered configuration space.

The work I will then describe concerns a genus g>0 analogue of the space of phylogenetic trees: the moduli space of tropical curves of genus g. Roughly speaking, this space parametrizes n-marked graphs of first Betti number g.  These spaces are no longer shellable for g>1, and their homology groups, as S_n-representations, are quite mysterious.  I will explain how making precise connections to compactified configuration spaces on graphs made it possible for us to make calculations in Sage when g=2 in a range beyond what was previously accessible.  These in turn produced new calculations and conjectures concerning the rational cohomology of M_{2,n}. No tropical geometry prerequisites are assumed in this talk.

Quasi-normal modes are complex exponential frequencies appearing in long time expansions of solutions to linear wave equations on black hole backgrounds. They appear in particular during the ringdown phase of a black hole merger when the dynamics is expected to be driven by linear effects. In this talk I give an overview of various results in pure mathematics which relate asymptotic behavior of quasi-normal modes at high frequency to the geometry of the set of trapped null geodesics, such as the photon sphere in Schwarzschild(-de Sitter). These trapped geodesics have two kinds of behavior: the geodesic flow is hyperbolic in directions normal to the trapped set (a feature stable under perturbations) and it is completely integrable on the trapped set. It turns out that normal hyperbolicity gives information about the rate of decay of quasi-normal modes, while complete integrability gives rise to a quantization condition.

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/general-relativity-2021-22/

My lab is developing biophysical methods to achieve multicolor and dynamic biological imaging at the molecular scale. Our approach to capturing the dynamics of cellular processes involves cryo-vitrifying samples after known time delays following stimulation using custom cryo- plunging and high-pressure freezing instruments. To achieve multicolor electron imaging, we are exploring the property of cathodoluminescence — optical emission induced by the electron beam. We are developing nanoprobes (“cathodophores”) that will be used as luminescent protein tags in electron microscopy. We are applying these new methods to study G-protein- coupled receptor signaling and to visualize the formation of biomolecular condensates.

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/active-matter-seminar

McMullen proved that a stable family of rational maps is either trivial or all its members are Lattès. His proof relies on Thurston’s theorem on postcritically finite maps, which uses Teichmüller theory. I will discuss a recent new proof of McMullen’s theorem due to Zhuchao Ji and Junyi Xie which instead uses Berkovich dynamics over the complex Levi-Civita field.

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

In string theory one finds that states become massless as one approaches boundaries in Calabi-Yau moduli spaces. In this talk we look in the opposite direction, that is, we search for points where the mass gap for these light states is maximized — the so-called desert. In explicit examples we identify these desert points, and find that they correspond to special points in the moduli space of the CY, such as orbifold points and rank two attractors.

Beyond the tight/overtwisted dichotomy, 3-dimensional contact topology would appear to be dominated by flexibility: a central result of Eliashberg and Mishachev says that the contactomorphism group of the standard contact 3-ball has the homotopy type of the diffeomorphism group. In contrast with this, I will discuss how monopole Floer homology imposes constraints on the behaviour of families of contact structures on 3-manifolds. Applications include detecting exotic contactomorphisms given by certain “Dehn twists” on embedded spheres.

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

The Bistritzer-MacDonald (BM) model predicted the existence of the narrow bands in the magic-angle twisted bilayer graphene (MATBG), and nowadays is a starting point for most theoretical works. In this talk, I will briefly review the BM model and then present a continuum field theory [1] for graphene bilayer system allowing any smooth lattice deformation including the small twist angle. With the gradient expansion to the second order, the continuum theory for MATBG [2] produces the spectrum that almost perfectly matches the spectrum of the microscopic model, suggesting the validity of this theory. In the presence of the lattice deformation, the inclusion of the pseudo-vector potential does not destroy but shift the flat band chiral limit to a smaller twist angle. Furthermore, the continuum theory contains another important interlayer tunneling term that was overlooked in all previous works. This term non-negligibly breaks the particle-hole symmetry of the narrow bands and may be related with the experimentally observed particle-hole asymmetry.

1. O. Vafek and JK, arXiv: 2208.05933.
2. JK and O. Vafek, arXiv: 2208.05953.

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

On November 28 – Dec 1, 2022, the CMSA will host a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry.

For more information, please see: https://cmsa.fas.harvard.edu/event/representation-theory-calabi-yau-manifolds-and-mirror-symmetry/

On November 28 – Dec 1, 2022, the CMSA will host a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry.

For more information, please see: https://cmsa.fas.harvard.edu/event/representation-theory-calabi-yau-manifolds-and-mirror-symmetry/

I will discuss recent work calculating the top weight cohomology of the moduli space A_g of principally polarized abelian varieties of dimension g for small values of g. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of A_g and the moduli space of tropical abelian varieties. This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

For more information, please see: https://sites.google.com/view/harvardmitag

On November 28 – Dec 1, 2022, the CMSA will host a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry.

For more information, please see: https://cmsa.fas.harvard.edu/event/representation-theory-calabi-yau-manifolds-and-mirror-symmetry/

Let S be a Shimura variety such that the connected components of the set of complex points S(C) are of the form D/Γ, where Γ is a torsion-free arithmetic group acting on the Hermitian symmetric domain D. Borel proved that any holomorphic map from any complex algebraic variety into S(C) is an algebraic map. In this talk I shall describe ongoing joint work with Ananth Shankar and Xinwen Zhu, where we prove a p-adic analogue of this result of Borel for compact Shimura varieties of abelian type.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 30th.

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

Braids on a given number of strands n can be concatenated and thereby form a group Bn. The latter possesses two different incarnations: it can be presented on a simple set of generators and relations due to E. Artin (1947), or it can be realized as the fundamental group of the space of configurations Xn of n points in the complex plane. I will explain how each of these incarnations leads to a class of representations of Bn. The topological representations arise from differential equations of Xn which are symmetric under the algebra gl_n of nxn matrices. The algebraic representations arise instead from a deformation of this algebra known as the quantum group U_q(gl_n). Finally, I will tie the knot by relating these two classes of representations.