# Department of Mathematics

## Analysis Seminar

We invite speakers to present original research in analysis.

### 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:

Symmetry laws showcase the elegant relationship between mathematics and physical systems. Noether’s theorem, which relates symmetries in a Hamiltonian with conserved physical quantities, is one of the most impactful theorems throughout physics. As such, describing this property in a Hamiltonian is of the utmost importance in many applications–from determining state transition laws to expressing resource theories. In this talk, I give algorithms to determine if a Hamiltonian is symmetric with respect to a discrete, finite group $$G$$ and its associated unitary representation $$\{U(g)\}_{g\in G}$$. Furthermore, I directly relate the acceptance probability of these algorithms with the typical commutation relationship for symmetry in quantum mechanics. I show that one of the algorithms can efficiently compute the normalized commutator of the group representation and Hamiltonian.

Joint work with Mark M. Wilde and available as arXiv:2203.10017