Calendar

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