**Ciprian Manolescu,** Professor of Mathematics at
UCLA, presented the Myhill Lecture Series 2014-15. This 3-part
lecture series was held in April 2015, and included a reception in
honor of Professor Manolescu after the first lecture. Manolescu
earned his BA from Harvard in 2001. In 2002 he won the Morgan Prize
for undergraduate research. He continued at Harvard, earning his
PhD in 2004 under the supervision of Peter Kronheimer.

From 2004-2008 he was a Clay Research Fellow while holding postdoctoral appointments at Princeton & IAS and then Columbia University. Since 2008 he has been at UCLA, where he is Professor of Mathematics. He has held visiting research positions at MSRI, the Simons Center for Geometry and Physics, and the National Center for Scientific Research in Paris.

Manolescu is the only person in history to earn a perfect
score three times at the International Math Olympiad, representing
Romania. In 2012, he was awarded a prize of the European Congress
of Mathematics, held every four years. EMS prizes are awarded to
researchers not older than 35 years, of European nationality or
working in Europe, in recognition of excellent contributions in
mathematics. The citation for Manolescu reads *“An
EMS-prize goes to him for his deep and highly influential work on
Floer theory, successfully combining techniques from gauge theory,
symplectic geometry, algebraic topology, dynamical systems and
algebraic geometry to study low-dimensional manifolds, and in
particular for his key role in the development of combinatorial
Floer theory.”*

**Lecture 1: Manifolds and triangulations**

I will introduce the triangulation problem for manifolds: When is a
topological manifold (a space that looks locally like Euclidean
space) homeomorphic to a simplicial complex? I will describe the
history of this problem, and in particular its reduction to a
question in low dimensional topology. This reduction is work of
Galewski-Stern and Matumoto from the 1970s.

**Lecture 2: Seiberg-Witten theory**

The Seiberg-Witten equations are a set of elliptic partial
differential equations that can be written on 4-manifolds or
3-manifolds. An important property is that the moduli space of
solutions to these equations is compact. As a consequence, one can
use them to define topological invariants of 3- and
4-manifolds.

Together, these invariants give rise to a variant of a TQFT (topological quantum field theory). In turn, this leads to several topological applications.

**Lecture 3: The triangulation conjecture**

The triangulation conjecture claimed that any manifold can be
triangulated. I will sketch the disproof of the conjecture in
dimensions at least 5. This uses the TQFT properties of the
Seiberg-Witten equations, as well as their Pin(2) symmetry.