PROGRAM

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

15:00-19:30

#### REGISTRATION

Hotel Fortin Plaza

18:00-20:00

#### WELCOME RECEPTION

Presidencia Municipal

09:00-10:00

#### INAUGURAL CEREMONY

Macedonio Alcala Theater
10:00-10:15

10:15-11:15

#### Chenyang Xu 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
11:15-12:15

#### Jaime Urrutia FucugauchiPublic Lecture

Macedonio Alcala Theater
12:15-15:00

15:00-19:30

### Special Sessions Group 1

Instituto Tecnológico de Oaxaca
Ing. Sistemas Computacionales Building 1

Coffee Break From 17:00 to 17:30

19:30-20:00
09:00-10:00

#### David Brydges 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
10:00-11:00

#### Geoff Whittle 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.
11:00-12:00

#### Manuel del Pino 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.
12:00-15:00

15:00-19:30

### Special Sessions Group 1

Instituto Tecnológico de Oaxaca
Ing. Sistemas Computacionales Building 1

Coffee Break From 17:00 to 17:30

19:30-20:00
20:00-21:00

#### Mariel Vazquez 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.
09:00-10:00

#### Hee Oh 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.
10:00-11:00

#### William Duke 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.
11:00-12:00

#### Fernando Codá Marques 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.
12:00-15:00

15:00-20:00
FREE AFTERNOON
20:00-

#### Banquet

Hotel Quinta Real
09:00-10:00

#### Ana Amador 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.
10:00-11:00

#### Hiraku Nakajima 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.
11:00-12:00

#### Lexing Ying 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.
12:00-15:00

15:00-19:30

### Special Sessions Group 2

Instituto Tecnológico de Oaxaca
Ing. Sistemas Computacionales Building 1

Coffee Break from 17:00 to 17:30

CANCELED

20:00-21:00

09:00-10:00

#### Mariel Vazquez 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.
10:00-11:00

#### Alexey Bondal 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.
11:00-12:00

#### Ian Agol 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.
12:00-15:00

15:00-19:30

### Special Sessions Group 2

Instituto Tecnológico de Oaxaca
Ing. Sistemas Computacionales Building 1

Coffee Break from 17:00 to 17:30

CANCELED

Sunday 13th

#### Welcome Reception

18:00-20:00

Palacio Municipal de Oaxaca

Wednesday 16th

#### Banquet

20:00

Quinta Real Hotel

Matroid Theory
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

Bergman Spaces and Toeplitz operators
Salvador Perez Esteva (IM-Cuer) Kehe Zhu (St. Univ New York Albay)
spestevaim.unam.mx

Dynamical Systems and Geometry
Carlos Cabrera-Aubin Arroyo (IM-Cuer), Tomoki Kawahira (Tokyo I. Of Technology), Federico Rodriuez (Penn State), Andres Navas (U. De santiago)
aubinarroyoim.unam.mx

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

Current Trends in Symplectic Geometry and Topology.
Cheol-Hyun Cho (Seoul National. Univ.), Eduardo Gonzalez (UMASS, Boston), Kaoru Ono (RImS, Kyoto), Andres Pedroza (Colima), Yasha Savelyev (Colima)
andres_pedrozaucol.mx

Symmetric Structures in Discrete Geometry
Asia Ivic Weiss ( York Univ.), Daniel Pellicer (CCM), Egon Schulte (NE University, Boston)
pellicermatmor.unam.mx

Algebraic and Enumerative Combinatorics
Ernesto Vallejo (CCM) Marcelo Aguiar (Cornell) Nantel Bergeron (York, Canada)
ernevallejogmail.com

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

Mathematical Ecology and Epidemiology - CANCELED

Advances in Geometric Methods in Mechanics with Applications
Antonio Hernández Garduño (UAM - Iztapalapa), Vakhtang Putkaradza (Alberta), Dmitry Zenkov (NCSU)

Nonlinear Elliptic PDE and Systems
Nils Ackerman (IM) Jaeyoung Byeon (South Korea) Yihong Du (Australia) Monica Musso (Chile)
nilsackermath.info

Matrix Analysis and its Applications
Qing-Wen Wamng (Shangai Univ.), Fuhzen Zhang (Miami), Yang Zhang (Mantoba)
zhangnova.edu

Instability of natural convection in cylindrical containers
Eduardo Ramos (IER, UNAM)
ermier.unam.mx

Algebraic Geometry
Pedro Luis del Angel (CIMAT), Keinichiro Kimura (Tsukuba)
luiscimat.mx

Groups and orbifolds in geometry and topology
luperciomath.cinvestav.mx

Special Functions, Orthogonal Polynomials and Applications
Manuel Dominguez (IM, UNAM), Luis E. Garza Gaona (Colima)
mdi29im.unam.mx

Singularities of Spaces and Mappings
Jawad Snousi-Jose Seade (IM-Cuer) David Massey (USA) Masaharu Ishikawa (Japan) Nguyen Viet Dung (Vietnam)
jsnoussiim.unam.mx

Poisson Geometry and Its Applications
Misael Avendaño (UNISON), Matias del Hoyo (UFF), Rui Loja Fernandes (U. Illinois-Urbana), Yuri Vorobiev (UNISON)
misaelavemat.uson.mx

Geometry and Topology
Daniel Juan (CCM), Chris Connell (Indiana), Fuquan Fang (Capital Normal Univ. China)
danielmatmor.unam.mx

Inverse problems and PDE Control Theory
Luz de Teresa (IM-UNAM), Héctor Morales Bárcenas (UAM-Mexico), Abdon Choque (IFM)
abdon.ifmgmail.com

Commutative Algebra: Homological and Combinatorial Methods
Louiza Fouli (New Mexico State Univ.), Luis Nuñez Betancourt (CIMA), Rafael Heraclio Villareal (Cinvestav), Yuji Yoshino (Okayama Univ.)
rhvillarrealgmail.com

Stochastic Processes and Applications
Victor Rivero (CIMAT) Joaquin Fontbona (DIM Chile) Kouji Yano (Kyoto)
jcpardocimat.mx