### 2021 – 2022 Academic Year

**Organized by:** Daniel Dix ( dix@math.sc.edu )

This is a traditional in-person seminar. No recordings are planned. Come and participate!

Organizational Meeting

- Friday, Feb 4
- 3pm
- COL 1015

**Daniel Dix**

- Friday, Feb 11
- 2:15pm
- COL 1015

**Abstract**: This will be an overview of how an interesting groupoid can be derived from a molecular
system of three identical nuclei plus some number of electrons. The structure of the
groupoid will be fully determined, and that will significantly constrain the electronic
energy eigenvalue intersection patterns for the molecule.

**Daniel Dix**

- Friday, Feb 18
- 2:15pm
- COL 1015

**Abstract**: We will show how a groupoid arises from the tangent mapping of a section of an associated
bundle to the \(C^2\) invariant subspace bundle (that we derived from a triatomic
molecular system in Part 1) at a triple eigenvalue intersection point that has maximal
\(S_3\) symmetry. By linearization and passage to the range of the tangent mapping
we arrive at a computable groupoid that gives information about the eigenvalue intersections
of the molecular system.

**Daniel Dix**

- Friday, Feb 25
- 2:15pm
- COL 1015

**Abstract**: If \(f\colon \mathbb R^n\to M\) is a \(C^2\) mapping, where \(M\) is an \(m\)-dimensional
manifold, equipped with an atlas of homeomorphisms \(\phi_\mu\colon U_\mu\to V_\mu\),
where \(U_\mu\subset\mathbb R^m\) and \(V_\mu\subset M\) are open sets, with \(C^2\)
overlap mappings, and \(\mathbf l_0\in\mathbb R^n\), then there is a natural groupoid
defined as follows. The objects are pairs \((\mu,A)\), where \(f(\mathbf l_0)\in V_\mu\)
and \(A=D(\phi_\mu^{-1}\circ f)(\mathbf l_0)\). An arrow between objects \((\mu,A)\)
and \((\nu,B)\) is determined by a triple \((\mu,G_{\nu,\mu},\nu)\), where \(G_{\nu,\mu}\)
is a linear isomorphism so that \(B=G_{\nu,\mu}A\), i.e. \(G_{\nu,\mu}=D(\phi_\nu^{-1}\circ\phi_\mu)(\phi_\mu^{-1}(f(\mathbf
l_0)))\). This groupoid is another way of presenting the tangent mapping (differential)
of \(f\) at \(\mathbf l_0\). We apply this construction where \(n=3\) and \(M=\mathfrak
B\) and \(f(\mathbf l) =(\mathbf l,\Pi\breve{\mathcal H}(\mathbf l))\), where \(\Pi\breve{\mathcal
H}\) is the trace-free projection of the molecular electronic Hamiltonian restricted
to a 3-dimensional invariant subspace \(\mathcal F(\mathbf l)\), and where \(\mathbf
l_0\) is an equilateral triangle configuration at which the three lowest eigenvalues
of \(\breve{\mathcal H}(\mathbf l_0)\) coincide. This construction, combined with
certain functorial (groupoid homomorphism) images, leads to a groupoid we can completely
compute.

** **

**Ralph Howard**

- Friday, Mar 18
- 2:15pm
- COL 1015

**Abstract**: For curves in the plane which have linearly independent velocity and acceleration
vectors there a notion of affine arclength and affine curvature which is invariant
under area preserving affine maps of the plane. In terms of the Euclidean arclength
\(s\) and curvature \(\kappa\) the affine arclength is

\(\int_a^b \kappa^{1/3} ds\)

We will outline the basic theory of the differential geometryof affine curves and give some new results which estimate the area bounded by the curve and the segment between the endpoints of the curve in terms of the affine arclength of the curve and its affine curvature.

Most of the proofs do not involve any mathematics not in in Math 241 and 242 (or Math 550 and 520).

**Ralph Howard**

- Friday, Mar 18
- 2:15pm
- COL 1015

**Abstract**: For curves in the plane which have linearly independent velocity and acceleration
vectors there a notion of affine arclength and affine curvature which is invariant
under area preserving affine maps of the plane. In terms of the Euclidean arclength
\(s\) and curvature \(\kappa\) the affine arclength is

\(\int_a^b \kappa^{1/3} ds\)

We will outline the basic theory of the differential geometryof affine curves and give some new results which estimate the area bounded by the curve and the segment between the endpoints of the curve in terms of the affine arclength of the curve and its affine curvature.

Most of the proofs do not involve any mathematics not in in Math 241 and 242 (or Math 550 and 520).

** **

**Stephen Fenner**

- Friday, Apr 8
- 2:15pm
- COL 1015

**Abstract**: The quantum fanout gate has been used to speed up quantum algorithms such as the
quantum Fourier transform used in Shor's quantum algorithm for factoring. Fanout
can be implemented by evolving a system of qubits via a simple Hamiltonian involving
pairwise interqubit couplings of various strengths. We characterize exactly which
coupling strengths are sufficient for fanout: they are sufficient if and only if they
are odd multiples of some constant energy value J. We also investigate when these
couplings can arise assuming that strengths vary inversely proportional to the squares
of the distances between qubits.

This is joint work with Rabins Wosti.

**Rabins Wosti**, Computer Science and Engineering Department

- Friday, Apr 15
- 2:15pm
- COL 1015

**Abstract**: The quantum fanout gate has been used to speed up quantum algorithms such as the quantum
Fourier transform used in Shor's quantum algorithm for factoring. Fanout can be implemented
by evolving a system of qubits via a simple Hamiltonian involving pairwise interqubit
couplings of various strengths. We characterize exactly which coupling strengths
are sufficient for fanout: they are sufficient if and only if they are odd multiples
of some constant energy value J. We also investigate when these couplings can arise
assuming that strengths vary inversely proportional to the squares of the distances
between qubits.

This is joint work with Stephen Fenner.

**Margarite Laborde**

- Friday, Apr 22
- 2:15pm
- COL 1015

**Abstract**: