I will define and review the basics of exponential networks associated to CY 3-folds described by conic bundles. I will focus mostly on the mathematical aspects and general ideas behind this construction as well as its conjectural connection with generalized Donaldson–Thomas invariants. This is based on joint work with S. Banerjee and P. Longhi.

In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.

In this talk, we briefly review the distributional symmetries for *-random variables, which are defined by coactions of corepresentations of quantum groups. We classify all de Finetti type theorems for classical independence and free independence by studying vanishing conditions on the classical and free cumulants.  Examples for our de Finetti type theorems and approximation results in the spirit of Diaconis and Freedman are also provided.

In this talk, we will discuss the construction of Kahler Ricci flow on complete Kahler manifolds with unbounded curvature. As a corollary, we will discuss the application related to Yau’s
uniformization problem and the regularity of Gromov-Hausdorff’s limit. This is joint work with L.F. Tam.

— Organized by Professor Shing-Tung Yau

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $\Gamma_0(N)$ in the case where $N$ is prime and equal to the conductor
of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancellation in this sum than would be predicted by the common phenomenon of “square root cancellation”, while in another case we find more cancellation.

In this talk, I will describe an approach to proving the space (as opposed to time) version of this conjecture via spectral graph theory.  Specifically, I will explain how randomized space-bounded algorithms are described by random walks on directed graphs, and techniques in algorithmic spectral graph theory (e.g. solving Laplacian systems) have yielded deterministic space-efficient algorithms for approximating the behavior of such random walks on undirected graphs and Eulerian directed graphs (where every vertex has the same in-degree as out-degree).  If these algorithms can be extended to general directed graphs, then the aforementioned conjecture about derandomizing space-efficient algorithms will be resolved.

Joint works with Jack Murtagh, Omer Reingold, Aaron Sidford,  AmirMadhi Ahmadinejad, Jon Kelner, and John Peebles.

No additional detail for this event.

We will discuss the classical notion of J-convexity of subsets of complex manifolds, and the closely related notion of Stein manifolds. The theory is particularly subtle in complex dimension 2. Surprisingly, progress can be made using topological 4-manifold theory. Every tame CW 2-complex topologically embedded in a complex surface can be perturbed so that it becomes J-convex in the sense of being a nested intersection of Stein open subsets. These Stein neighborhoods are all topologically equivalent to each other, but can be very different when viewed in the smooth category. As applications, we obtain uncountable families of distinct smoothings of R^4 admitting convex or concave holomorphic structures. We can also generalize the notion of J-convex hypersurfaces to the topological category. The resulting topological embeddings behave like their smooth counterparts, but are much more common.

Future schedule is found here: https://scholar.harvard.edu/gerig/seminar

We will review the connection between Feynman integrals and Calabi–Yau periods. Then we explain some properties of the period geometry, in particular the Picard–Fuchs differential ideal and to which extend it allows to calculate the Feynman integral. The main result is an explicit analytic calculation of the 3-Loop Banana graph with all mass parameters using relative periods on a special family of K3 surfaces.

For any polarized hyperkahler manifold of K3 type whose dimension is divisible by 8, we produce a Lagrangian submanifold which is Fano arising as a connected component of the fixed locus of an involution on the hyperkahler manifold. This is an ongoing joint work with E. Macrì, K. O’Grady, and G. Saccà.

Multivariate public key cryptosystems (MPKC) are one of the four main families of post-quantum public key cryptosystems. In a MPKC, the public key is given by a set of quadratic polynomials and its security is based on the hardness of solving a set of multivariate polynomials. In this talk, we will give an introduction to the multivariate public key cryptosystems including the main designs, the main attack tools and the mathematical theory behind in particular algebraic geometry. We will also present state of the art research in the area.

Variational quantum algorithms such as VQE or QAOA aim to simulate low-energy properties of quantum many-body systems or find approximate solutions of combinatorial optimization problems. Such algorithms employ variational states generated by low-depth quantum circuits to minimize the expected value of a quantum or classical Hamiltonian. In this talk I will explain how to use general structural properties of variational states such as locality and symmetry to derive upper bounds on their computational power and, in certain cases, rule out potential quantum speedups. To overcome some of these limitations, we introduce the correlation rounding method and a recursive Quantum Approximate Optimization Algorithm.

Based on arXiv:1909.11485 and arXiv:1910.08980

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

We explain a new height gap result on holonomic power series
with rational coefficients, and prove the Schinzel-Zassenhaus conjecture
as its consequence: a monic irreducible non-cyclotomic integer polynomial
of degree $n > 1$ has at least one complex root of modulus exceeding
$2^{1/(4n)}$. The method, an arithmetic algebraization argument with input
from two fixed places of the global field $\mathbb{Q}$, takes a
combination of a $p$-adic overconvergence for a certain algebraic
function, at a fixed prime $p$ on the one hand (this is the place where
the cyclotomic examples get recognized and excluded); and, on the other
hand, of the automatic Archimedean analytic continuation for solutions of
linear ODE. The input at the Archimedean place has a potential-theoretic
flavor and comes out of Dubinin’s solution of an extremal problem of
Gonchar about harmonic measure. A possible connection to the $p$-curvature
conjecture is indicated, as an analogous height gap hypothesis for
G-operators with infinite monodromy.

Repeating the same pattern, we shall follow up by a sharp arithmetic
criterion on a formal power series to satisfy a linear differential
equation with polynomial coefficients. It upgrades the classical
Polya-Bertrandias rationality criterion, and we shall conclude by
explaining how this new criterion serves to amplify Calegari’s p-adic
counterpart of Apery’s theorem, yielding thus a proof of irrationality of
the Kubota-Leopoldt 2-adic zeta value $\zeta_2(5)$. The latter is joint
work with Frank Calegari and Yunqing Tang.

No additional detail for this event.

The works by J.S. Bach discussed in this talk will be performed in a free recital by Prof. Forger at Harvard’s Memorial Church at 7:30pm that evening (March 12th)

I will describe a collaborative project with the University of Michigan Organ Department to perfectly digitize many performances of difficult organ works (the Trio Sonatas by J.S. Bach) by students and faculty at many skill levels. We use these digitizations, and direct representations of the score to ask how music should encoded in the mind. Our results challenge the modern mathematical theory of music encoding, e.g., based on orbifolds, and reveal surprising new mathematical patterns in Bach’s music. We also discover ways in which biophysical limits of neuronal computation may limit performance.

Daniel Forger is the Robert W. and Lynn H. Browne Professor of Science, Professor of Mathematics and Research Professor of Computational Medicine and Bioinformatics at the University of Michigan. He is also a visiting scholar at Harvard’s NSF-Simons Center and an Associate of the American Guild of Organists.

Given a slice knot K and a prime power n, the n-th cyclic branched cover \Sigma_n(K) bounds a rational homology ball (Casson-Gordon). Even if one restricts to n=2, this gives a powerful sliceness obstruction, which for example sufficed determine the smoothly slice 2-bridge (Lisca) and odd 3-strand pretzel knots (Greene-Jabuka). It is natural to ask whether the property that all prime power cyclic branched covers bound rational homology ball characterizes slice knots. In this talk I will discuss recent joint work with P. Aceto, J. Meier, M. Miller, J. Park, and A. Stipsicz proving it does not.

Future schedule is found here: https://scholar.harvard.edu/gerig/seminar

No additional detail for this event.

Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne–Mumford moduli spaces of stable real curves (this is very different from Solomon’s approach).

via Zoom Video Conferencing:  https://harvard.zoom.us/j/738333299

The AdS/CFT correspondence stipulates that gravitational evolution in a bulk spacetime is dual to a boundary description that has no gravity. In the AdS/CFT picture the bulk spacetime evolves gravitationally against an anti-de Sitter space background, and the boundary dual theory is a conformal gauge theory in a spacetime of one dimension less. Recent insights by Ryu and Takayanagi have conjectured that quantum entangled boundary states quantitatively give rise to geometry in the bulk. They do so by explicitly referring to “minimal surfaces” in the bulk, connecting them to the entropy of a related area in the boundary. I will present a conceptually and technically powerful complementary holographic entanglement picture, reformulating Ryu–Takayanagi to no longer refer to minimal surfaces, and suggesting a new way to think about the holographic principle and the connection between spacetime gravitation and information. I will introduce the idea of bit threads, and show how they can be used for fun and profit.

via Zoom Video Conferencing: https://harvard.zoom.us/j/881015355

via Zoom Video Conferencing: TBA

Recent high-field low-temperature data shed new light on the mysterious pseudogap phase in cuprates. I will introduce a simple way
to synthesize the low-temperature data, the charge density wave, and the previous ARPES and optical data. In the meantime, I will discuss
the general problem of how fluctuating superconducting order changes the fermion spectrum and other response functions.

via Zoom Video Conferencing: https://harvard.zoom.us/j/136830668

The Sato-Tate group of an abelian variety A of dimension g defined over a number field is a compact real Lie subgroup of the unitary
simplectic group of degree 2g that conjecturally governs the limiting distribution of the normalized Frobenius elements acting on the Tate module
of A. In previous joint work with Kedlaya, Rotger and Sutherland, it was shown that there are 52 such subgroups (up to conjugacy) that occur as
Sato-Tate groups of abelian surfaces over number fields. In this talk I will present several aspects of the classification of the 410 subgroups (up
to conjugacy) of the unitary symplectic group of degree 6 that occur as the Sato-Tate groups of abelian threefolds over number fields. This is a joint
work with Kiran Kedlaya and Andrew Sutherland.

Seminar will meet via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373

via Zoom Video Conferencing: https://harvard.zoom.us/j/635180669

*note special time*

via Zoom Video Conferencing: https://harvard.zoom.us/j/362172385

via Zoom Video Conferencing:  https://harvard.zoom.us/j/779283357

A liquid crystal is a phase of matter in which order and disorder coexist: for some degrees of freedom, there is order, whereas for others, disorder. Such materials were discovered in the late XIXth century, but it took over a century to understand, from microscopic models, how such phases form. In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of such a liquid crystal phase. In this setting, this amounts to showing that dimers spontaneously align, but do not fully crystallize: there is no translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. In this talk, I will discuss a recent proof of this conjecture, that is, a proof of the emergence of a liquid crystal phase in the Heilmann-Lieb model. This is joint work with E.H. Lieb.