Welcome to the Third Pacific Rim Mathematical Association.

Plenary Lecture PL1 & Public Lecture **Monday 14**: Macedonio Alcala Theater

Plenary Lectures PL2 to PL13 **Tuesday to Friday**: Hotel Fortin Plaza

All Special Sessions **Monday to Friday**: Instituto Tecnológico de Oaxaca

Ing. Sistemas Computacionales Building 1

Plenary Lecture PL 1

The space parametrising objects is called moduli space. In algebraic geometry,
to construct moduli spaces for appropriate objects and to study their geometry
is a central topic. In this talk, we will discuss the recent progress on the
construction of moduli spaces for higher dimensional varieties, and its relations
to topics like minimal model program, Kahler-Einstein metric etc.

Macedonio Alcala Theater

Public Lecture

Macedonio Alcala Theater

Instituto Tecnológico de Oaxaca

Ing. Sistemas Computacionales Building 1

Coffee Break From 17:00 to 17:30

Plenary Lecture PL 2

The 1982 Nobel Prize in Physics was awarded to Ken Wilson for \emph{
his theory for critical phenomena in connection with phase
transitions}. His work stimulated many authors to search for a
rigorous systematic realisation of his calculus (The Renormalisation
Group). With the same objective we will give an introduction for a
general audience to recent work of Bauerschmidt, Brydges and Slade,
and other authors, which starts with the $\varphi^{4}$ model on the
lattice $Z^{d}$ at weak coupling. This model is a variation on the
Ising model: instead of the spin $\varphi_{x}$, for every lattice site
$x \in Z^{d}$, taking values in $\{-1,1 \}$, $\varphi_{x}$ is real
valued, but concentrated near $\pm 1$ by a weight that includes
factors $\exp \big[-g (\varphi_{x}^{2}- 1)^{2} \big]$. Weak coupling
means $g$ is positive and small. More generally we study $O (n)$
models that are defined by replacing $\varphi_{x}^{2}$ by
$\|\varphi_{x}\|^{2}$ with $\varphi \in R^{n}$. There is also a
precise mathematical definition of the $n=0$ component model and it
turns out to be a natural model for self-repelling walk. Our results
include a proof for $d=4$ that the susceptibility diverges with a
logarithmic correction to the mean-field behaviour with exponent $(n +
2)/(n + 8)$.
Recently G. Slade has extended these methods to other dimensions
$d$. He studies $O(n)$ models, including $n=0$, with a long range
ferromagnetic spin-spin interaction that decays with distance $r$ as
$r^{-(d+\alpha)}$, for $\alpha \in (0,2)$. These models have upper
critical dimension $d_c=2\alpha$. For dimensions $d=1,2,3$ and small
$\epsilon>0$, he sets $\alpha = \frac 12 (d+\epsilon)$, so that
$d=d_c-\epsilon$ is below the upper critical dimension $d_{c}$. For
small $\epsilon$ and weak coupling he proves that the susceptibility
decays with exponent $\gamma = 1 + \frac{n+2}{n+8} \frac
\epsilon\alpha + O(\epsilon^2)$. Expansion in $\epsilon$ for such
long-range models was first carried out in the physics literature in

Plenary Lecture PL 3

Recently Jim Geelen, Bert Gerards and I announced that we had a proof of
Rota’s Conjecture, which concerns matroids representable over finite fields.
In this talk, rather than discuss the proof, I will attempt to give a feel
for the milieu in which the conjecture arose and to communicate some of the
reasons why, from the time I was a graduate student, I became fascinated with it.

Plenary Lecture PL 4

A fundamental question in nonlinear evolution equations is the analysis of
solutions which develop singularities (blow-up) in finite time or as time
goes to infinity. We review recent results on the construction of solutions
to certain notable nonlinear parabolic PDE which exhibit this kind of
behavior in the form of "bubbling". This means solutions that at main order
look like asymptotically singular time-dependent scalings of a fixed finite
energy entire steady state. We carry out this analysis for the classical
two-dimensional harmonic map flow into the sphere and the energy-critical
semilinear heat equation.

Instituto Tecnológico de Oaxaca

Ing. Sistemas Computacionales Building 1

Coffee Break From 17:00 to 17:30

Public Lecture

Panhuelito Garden (Pañuelito)

¿Qué tienen en común los aros de humo y las molecular circulares de ADN?
Los dos están sujetos a procesos de reconección local, que son communes tanto
en biología como en física. En biología, rearreglos cromosómicos surgen a
raíz de errores en la replicación del ADN, como daños ocasionados por
radiación, y caracterizan a ciertas celulas cancerígenas. En esta plática
mostramos como se aplican herramientas de topología, simulaciones y
visualización computacional para analizar datos de reconección.

Plenary Lecture PL 5

The prime number theorem states that the number of primes of size at most
T grows like T/log T. Geometric analogues of this profound fact have been of
great interest over the years. In my lecture, I will discuss refined versions
of the prime number theorem for hyperbolic 3-manifolds and for hyperbolic rational maps.

Plenary Lecture PL 6

My aim is to give a general and accessible description of joint work with
Ozlem Imamoglu and Arpad Toth on some new geometric invariants associated
to real quadratic fields. These are hyperbolic surfaces bounded by closed
geodesics whose geometric properties are highly arithmetic.
Key to their study are automorphic forms and $L$-functions. Several areas
of math enter, including algebraic and analytic number theory, geometry and
spectral theory.

Plenary Lecture PL 7

In 1911, Hermann Weyl proved a universal formula that describes
the asymptotic behavior of the eigenvalues of the Laplacian. I will discuss
a proof (joint work with Liokumovich and Neves) of a Weyl law for the
volume spectrum, as conjectured by Gromov. The eigenvalues of the Laplacian
are replaced by the areas of minimal hypersurfaces constructed by minimax
methods.

FREE AFTERNOON

Plenary Lecture PL 8

Birdsong is a complex motor activity that emerges from the interaction between
the peripheral system (PS), the central nervous system (CNS) and the environment.
The similarities to human speech, both in production and learning, have positioned
songbirds as unique animal models for studying this learned motor skill. In this
talk I will present a low dimensional dynamical system as a model of the avian
vocal apparatus. Inputs can be related to physiological variables, being the
output a synthetic song (SYN) that is a copy of the recorded birdsong (BOS).
To go beyond sound comparison, we measured neural activity highly tuned to BOS
and found that the patterns of neural response to BOS and SYN were remarkable
similar. This work allowed to relate motor gestures and neural activity, making
specific predictions on the timing of the neural activity. To study the dynamical
emergence of these features, we developed a neural model in which the variables
were the average activities of different neural populations within the nuclei of
the song system. This model can reproduce the measured respiratory patterns and the
timing of the neural activity. In this talk, I will present experimental data in
accordance with the dynamical model. This interdisciplinary work shows how low
dimensional models for the PS and CNS can be a valuable tool for studying the
neuroscience of generation and control of complex motor tasks.

Plenary Lecture PL 9

Quiver gauge theories give two types of algebraic symplectic
varieties, which are called quiver varieties and Coulomb branches
respectively. The first ones were introduced by the speaker in 1994, and
their homology groups are representations of Kac-Moody Lie algebras. The
second ones were introduced by the speaker and Braverman, Finkelberg in
2016. Two types of varieties are very different (e.g., dimensions are
different), but expected to be related in rather mysterious ways. As an
example of mysterious links, I would like to explain a conjectural
realization of Kac-Moody Lie algebras representations on homology groups of
Coulomb branches. It nicely matches with geometric Satake correspondence
for usual finite dimensional complex simple groups and loop groups.

Plenary Lecture PL 10

High frequency wave propagation has been a longstanding challenge in
scientific computing. For the time-harmonic problems, the linear systems
resulting from PDE and/or integral formulations are difficult to solve for
standard iterative methods since they are highly indefinite. In this talk,
we consider several such cases that arise from applications. For each one,
we construct a sparsifying preconditioner that results in small numbers of
iterations when combined with a standard iterative solver.

Instituto Tecnológico de Oaxaca

Ing. Sistemas Computacionales Building 1

Coffee Break from 17:00 to 17:30

Plenary Lecture PL 11

Reconnection processes appear at widely different scales, from microscopic
DNA recombination to large-scale reconnection of vortices in fluid
turbulence. Motivated by biological data, in this talk I illustrate how our
group uses tools from knot theory and low-dimensional topology, combined
with computer simulations and visualization methods to characterize the
process of topology simplification by local reconnection. Replication of
circular DNA yields 2-component links of type T(2,2n). Unlinking the DNA
circles is essential to cell survival. We provide mathematical proof that
there is a unique minimal pathway of DNA unlinking by local reconnection
assuming that at every step the topological complexity goes down. We also
investigate, both analytically and numerically, whether there are other
minimal pathways of unlinking replication links by local reconnection when
we relax the complexity assumption. We introduce a Monte Carlo method to
simulate local reconnection, provide a quantitative measure to distinguish
among pathways and conclude that the unique unlinking pathway found under
the strict assumption remains the most probable after the assumption is
lifted. These results point to a universal property relevant to any local
reconnection event between two sites along one or two circles, such as the
reconnection of knotted fluid vortices.

Plenary Lecture PL 12

The lecture is devoted to categorical aspects of Algebraic Geometry. This is about description of the derived categories of coherent sheaves on algebraic varieties and their behavior under various geometric operations, especially those appearing in the Minimal Model Program of Birational Geometry.

A description of general (enhanced) triangulated categories via generators will be given with emphasizing the role of exceptional collections and tilting generators. Then a description of the derived categories of toric varieties via perverse (topological) sheaves on stratified spaces will be outlined. A motivation from Mirror Symmetry will be presented.

The role of the relative canonical class in constructing tilting relative generators will be illustrated on the class of birational morphisms between smooth varieties with dimension of fibers bounded by 1.

A homotopical viewpoint on the algebra of functors via categorification of perverse sheaves on stratified spaces will be discussed. Current results and conjectures on its relevance to birational transformations, such as flops, will be presented.

A description of general (enhanced) triangulated categories via generators will be given with emphasizing the role of exceptional collections and tilting generators. Then a description of the derived categories of toric varieties via perverse (topological) sheaves on stratified spaces will be outlined. A motivation from Mirror Symmetry will be presented.

The role of the relative canonical class in constructing tilting relative generators will be illustrated on the class of birational morphisms between smooth varieties with dimension of fibers bounded by 1.

A homotopical viewpoint on the algebra of functors via categorification of perverse sheaves on stratified spaces will be discussed. Current results and conjectures on its relevance to birational transformations, such as flops, will be presented.

Plenary Lecture PL 13

We'll discuss Thurston's conjecture
that hyperbolic manifolds admit a finite-sheeted
cover which fibers over the circle. We'll then
discuss some of the tools of geometric group
theory used to resolve this conjecture, combining
results of Kahn and Markovic and Wise and his
collaborators. We'll mention some related results
as well.

Instituto Tecnológico de Oaxaca

Ing. Sistemas Computacionales Building 1

Coffee Break from 17:00 to 17:30

Carolyn Chun, US Naval Academy, 121 Blake Rd, Annapolis, MD 21402, United States. E-mail: chchchungmail.com.

Criel Merino, UNAM, León 2, altos, centro 68000, Oaxaca, Oax., México. Email: merinomatem.unam.mx, main contact organizer.

Peter Nelson, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON N2L3G1, Canada. E-mail: apnelsonuwaterloo.ca

Salvador Perez Esteva (IM-Cuer) Kehe Zhu (St. Univ New York Albay)

spestevaim.unam.mx

Carlos Cabrera-Aubin Arroyo (IM-Cuer), Tomoki Kawahira (Tokyo I. Of Technology), Federico Rodriuez (Penn State), Andres Navas (U. De santiago)

aubinarroyoim.unam.mx

Kazuo Akutagawa Tokyo I. of Tech.), Akito Futaki (U. of Tokyo), Jimy Petean (CIMAT)

jimmycimat.mx

Cheol-Hyun Cho (Seoul National. Univ.), Eduardo Gonzalez (UMASS, Boston), Kaoru Ono (RImS, Kyoto), Andres Pedroza (Colima), Yasha Savelyev (Colima)

andres_pedrozaucol.mx

Asia Ivic Weiss ( York Univ.), Daniel Pellicer (CCM), Egon Schulte (NE University, Boston)

pellicermatmor.unam.mx

Ernesto Vallejo (CCM) Marcelo Aguiar (Cornell) Nantel Bergeron (York, Canada)

ernevallejogmail.com

Natalia García Colín (Infotec), Luis Goddyn (Simon Fraser Univ.), Amanda Montejano (UNAM)

garciacolin.nataliagooglemail.com

Antonio Hernández Garduño (UAM - Iztapalapa), Vakhtang Putkaradza (Alberta), Dmitry Zenkov (NCSU)

putkaradualberta.ca

Nils Ackerman (IM) Jaeyoung Byeon (South Korea) Yihong Du (Australia) Monica Musso (Chile)

nilsackermath.info

Qing-Wen Wamng (Shangai Univ.), Fuhzen Zhang (Miami), Yang Zhang (Mantoba)

zhangnova.edu

Eduardo Ramos (IER, UNAM)

ermier.unam.mx

Ernesto Lupercio (CINVESTAV) Alejandro Adem (UBC) Jianzhang Pan (Academia Sinica, China)

luperciomath.cinvestav.mx

Manuel Dominguez (IM, UNAM), Luis E. Garza Gaona (Colima)

mdi29im.unam.mx

Jawad Snousi-Jose Seade (IM-Cuer) David Massey (USA) Masaharu Ishikawa (Japan) Nguyen Viet Dung (Vietnam)

jsnoussiim.unam.mx

Misael Avendaño (UNISON), Matias del Hoyo (UFF), Rui Loja Fernandes (U. Illinois-Urbana), Yuri Vorobiev (UNISON)

misaelavemat.uson.mx

Daniel Juan (CCM), Chris Connell (Indiana), Fuquan Fang (Capital Normal Univ. China)

danielmatmor.unam.mx

Luz de Teresa (IM-UNAM), Héctor Morales Bárcenas (UAM-Mexico), Abdon Choque (IFM)

abdon.ifmgmail.com

Louiza Fouli (New Mexico State Univ.), Luis Nuñez Betancourt (CIMA), Rafael Heraclio Villareal (Cinvestav), Yuji Yoshino (Okayama Univ.)

rhvillarrealgmail.com

Victor Rivero (CIMAT) Joaquin Fontbona (DIM Chile) Kouji Yano (Kyoto)

jcpardocimat.mx

Sunday 13th## Welcome Reception

18:00-20:00Palacio Municipal de Oaxaca

Wednesday 16th## Banquet

20:00Quinta Real Hotel