Satellite Meeting: A Pan-Hemispheric Celebration of Mathematics in Miami
Complete Schedule  PDF
Plenary and Special Public Lectures
Special Sessions
Algebraic Geometry and Topology
Combinatorics
Differential Geometry and Geometric Analysis
Mathematical Biology
Abstracts  PDF



Complete Schedule of Talks
Thursday, July 26, 2018
8:00am - 9:00am Breakfast and Registration (Cox Science Veranda)
9:00am - 9:30am Welcome (Cox Science Bldg., Room 145)
9:30am - 10:45am Victor Reiner, Finite Reflection Groups and General Linear Groups, Part I
(Cox Science Bldg., Room 145)
10:45am - 11:15am Coffee Break (Cox Science Veranda)
11:15am - 12:30pm Victor Reiner, Finite Reflection Groups and General Linear Groups, Part II
(Cox Science Bldg., Room 145)
12:30pm - 2:00pm Lunch (Cox Science Veranda)
2:00pm - 2:45pm Special Session Talks, Part I
Algebraic Geometry: Ernesto Lupercio (Memorial Bldg., Room 211)
Combinatorics: Stephanie van Willigenburg (Memorial Bldg., Room 312)
Differential Geometry: Lu Wang (Memorial Bldg., Room 313)
Mathematical Biology: Xi Huo (Memorial Bldg., Room 315)
2:45pm - 3:00pm Coffee Break (Cox Science Veranda)
3:00pm - 3:45pm Special Session Talks, Part II
Algebraic Geometry: Andrew Harder (Memorial Bldg., Room 211)
Combinatorics: Carolina Benedetti (Memorial Bldg., Room 312)
Differential Geometry: Martín Reiris (Memorial Bldg., Room 313)
Mathematical Biology: Jorge Velasco-Hernandez (Memorial Bldg., Room 315)
3:45pm - 4:00pm Coffee Break (Cox Science Veranda)
4:00pm - 4:45pm Special Session Talks, Part III
Algebraic Geometry: Bernardo Uribe (Memorial Bldg., Room 211)
Combinatorics: Alejandro Morales (Memorial Bldg., Room 312)
Differential Geometry: Chao Li (Memorial Bldg., Room 313)
Mathematical Biology: Zhisheng Shuai (Memorial Bldg., Room 315)

Friday, July 27, 2018
8:30am - 9:30am Breakfast and Registration (Cox Science Lobby)
9:30am - 10:45am Mark Lewis, Population Dynamics in Moving Environments, Part I
(Cox Science Bldg., Room 145)
10:45am - 11:15am Coffee Break (Cox Science Lobby)
11:15am - 12:30pm Mark Lewis, Population Dynamics in Moving Environments, Part II
(Cox Science Bldg., Room 145)
12:30pm - 2:00pm Lunch (Cox Science Lobby)
2:00pm - 2:45pm Special Session Talks, Part IV
Algebraic Geometry: Gabriel Kerr (Memorial Bldg., Room 211)
Combinatorics: John Shareshian (Memorial Bldg., Room 312)
Differential Geometry: Robert Haslhofer (Memorial Bldg., Room 313)
Mathematical Biology: King-Yeung Lam (Memorial Bldg., Room 315)
2:45pm - 3:00pm Coffee Break (Cox Science Lobby)
3:00pm - 3:45pm Special Session Talks, Part V
Algebraic Geometry: Richard (Paul) Horja (Memorial Bldg., Room 211)
Combinatorics: Rafael González D'León (Memorial Bldg., Room 312)
Differential Geometry: Christos Mantoulidis (Memorial Bldg., Room 313)
Mathematical Biology: Salomé Martínez (Memorial Bldg., Room 315)
3:45pm - 4:00pm Coffee Break (Cox Science Lobby)
4:00pm - 4:45pm Special Session Talks, Part VI
Algebraic Geometry: Andrei Teleman (Memorial Bldg., Room 211)
Combinatorics: Marcelo Aguiar (Memorial Bldg., Room 312)
Differential Geometry: Otis Chodosh (Memorial Bldg., Room 313)
Mathematical Biology: Nancy Rodriguez (Memorial Bldg., Room 315)
7:30pm BANQUET – Shalala Student Center – Ballroom 3rd Floor

Saturday, July 28, 2018
8:00am - 9:00am Breakfast (Cox Science Lobby)
9:00am - 10:15am Michael Eichmair, On Scalar Curvature, Minimal Surfaces, and the Isoperimetric Problem in the Large, Part I
(Cox Science Bldg., Room 145)
10:15am - 10:45am Coffee Break
10:45am - 12:00pm Michael Eichmair, On Scalar Curvature, Minimal Surfaces, and the Isoperimetric Problem in the Large, Part II
(Cox Science Bldg., Room 145)
12:00pm - 1:30pm Lunch (Cox Science Lobby)
1:30pm - 2:30pm Panel Discussion on Graduate Education (Cox Science Bldg., Room 145)
2:30pm - 2:45pm Coffee Break (Cox Science Lobby)
2:45pm - 3:30pm Special Session Talks, Part VII
Algebraic Geometry: Manuel Rivera (Memorial Bldg., Room 211)
Combinatorics: Sara Billey (Memorial Bldg., Room 312)
Differential Geometry: Lucas Ambrozio (Memorial Bldg., Room 313)
Mathematical Biology: Suzanne Lenhart (Memorial Bldg., Room 315)
3:30pm - 3:45pm Coffee Break (Cox Science Lobby)
3:45pm - 4:30pm Special Session Talks, Part VIII
Algebraic Geometry: Jacob Mostovoy (Memorial Bldg., Room 211)
Combinatorics: José Alejandro Samper Casas (Memorial Bldg., Room 312)
Differential Geometry: Siyuan Lu (Memorial Bldg., Room 313)
Mathematical Biology: Carlos Castillo-Chavez (Memorial Bldg., Room 315)
4:30pm - 4:45pm Coffee Break (Cox Science Lobby)
4:45pm - 5:30pm Special Session Talks, Part IX
Algebraic Geometry: Lino Grama (Memorial Bldg., Room 211)
Combinatorics: Patricia Hersh (Memorial Bldg., Room 312)
Differential Geometry: Abraão Mendes (Memorial Bldg., Room 313)
Mathematical Biology: Juan Gutierrez (Memorial Bldg., Room 315)

Sunday, July 29, 2018
8:00am - 9:00am Breakfast (Cox Science Lobby)
9:00am - 10:15am Denis Auroux, An Invitation to Homological Symmetry, Part I
(Cox Science Bldg., Room 145)
10:15am - 10:45am Coffee Break (Cox Science Lobby)
10:45am - 12:00pm Denis Auroux, An Invitation to Homological Symmetry, Part II
(Cox Science Bldg., Room 145)
12:00pm - 1:00pm Lunch (Cox Science Lobby)
1:00pm - 2:00pm Yuri Tschinkel, Building Institutes, (Cox Science Bldg., Room 145)
2:00pm Conference Farewell



Plenary and Special Public Lectures
Thursday, July 26, 2018
9:30am - 10:45am Victor Reiner, Finite Reflection Groups and General Linear Groups, Part I
(Cox Science Bldg., Room 145)
11:15am - 12:30pm Victor Reiner, Finite Reflection Groups and General Linear Groups, Part II
(Cox Science Bldg., Room 145)
Friday, July 27, 2018
9:30am - 10:45am Mark Lewis, Population Dynamics in Moving Environments, Part I
(Cox Science Bldg., Room 145)
11:15am - 12:30pm Mark Lewis, Population Dynamics in Moving Environments, Part II
(Cox Science Bldg., Room 145)
Saturday, July 28, 2018
9:00am - 10:15am Michael Eichmair, On Scalar Curvature, Minimal Surfaces, and the Isoperimetric Problem in the Large, Part I
(Cox Science Bldg., Room 145)
10:45am - 12:00pm Michael Eichmair, On Scalar Curvature, Minimal Surfaces, and the Isoperimetric Problem in the Large, Part II
(Cox Science Bldg., Room 145)
Sunday, July 29, 2018
9:00am - 10:15am Denis Auroux, An Invitation to Homological Symmetry, Part I
(Cox Science Bldg., Room 145)
10:45am - 12:00pm Denis Auroux, An Invitation to Homological Symmetry, Part II
(Cox Science Bldg., Room 145)
1:00pm - 2:00pm Yuri Tschinkel, Building Institutes, (Cox Science Bldg., Room 145)
Algebraic Geometry and Topology
Thursday, July 26, 2018
2:00pm - 2:45pm Ernesto Lupercio (Memorial Bldg., Room 211)
3:00pm - 3:45pm Andrew Harder (Memorial Bldg., Room 211)
4:00pm - 4:45pm Bernardo Uribe (Memorial Bldg., Room 211)
Friday, July 27, 2018
2:00pm - 2:45pm Gabriel Kerr (Memorial Bldg., Room 211)
3:00pm - 3:45pm Richard (Paul) Horja (Memorial Bldg., Room 211)
4:00pm - 4:45pm Andrei Teleman (Memorial Bldg., Room 211)
Saturday, July 28, 2018
2:45pm - 3:30pm Manuel Rivera (Memorial Bldg., Room 211)
3:45pm - 4:30pm Jacob Mostovoy (Memorial Bldg., Room 211)
4:45pm - 5:30pm Lino Grama (Memorial Bldg., Room 211)
Combinatorics
Thursday, July 26, 2018
2:00pm - 2:45pm Stephanie van Willigenburg (Memorial Bldg., Room 312)
3:00pm - 3:45pm Carolina Benedetti (Memorial Bldg., Room 312)
4:00pm - 4:45pm Alejandro Morales (Memorial Bldg., Room 312)
Friday, July 27, 2018
2:00pm - 2:45pm John Shareshian (Memorial Bldg., Room 312)
3:00pm - 3:45pm Rafael González D'León (Memorial Bldg., Room 312)
4:00pm - 4:45pm Marcelo Aguiar (Memorial Bldg., Room 312)
Saturday, July 28, 2018
2:45pm - 3:30pm Sara Billey (Memorial Bldg., Room 312)
3:45pm - 4:30pm José Alejandro Samper Casas (Memorial Bldg., Room 312)
4:45pm - 5:30pm Patricia Hersh (Memorial Bldg., Room 312)
Differential Geometry and Geometric Analysis
Thursday, July 26, 2018
2:00pm - 2:45pm Lu Wang (Memorial Bldg., Room 313)
3:00pm - 3:45pm Martín Reiris (Memorial Bldg., Room 313)
4:00pm - 4:45pm Chao Li (Memorial Bldg., Room 313)
Friday, July 27, 2018
2:00pm - 2:45pm Robert Haslhofer (Memorial Bldg., Room 313)
3:00pm - 3:45pm Christos Mantoulidis (Memorial Bldg., Room 313)
4:00pm - 4:45pm Otis Chodosh (Memorial Bldg., Room 313)
Saturday, July 28, 2018
2:45pm - 3:30pm Lucas Ambrozio (Memorial Bldg., Room 313)
3:45pm - 4:30pm Siyuan Lu (Memorial Bldg., Room 313)
4:45pm - 5:30pm Abraão Mendes (Memorial Bldg., Room 313)
Mathematical Biology
Thursday, July 26, 2018
2:00pm - 2:45pm Xi Huo (Memorial Bldg., Room 315)
3:00pm - 3:45pm Jorge Velasco-Hernandez (Memorial Bldg., Room 315)
4:00pm - 4:45pm Zhisheng Shuai (Memorial Bldg., Room 315)
Friday, July 27, 2018
2:00pm - 2:45pm King-Yeung Lam (Memorial Bldg., Room 315)
3:00pm - 3:45pm Salomé Martínez (Memorial Bldg., Room 315)
4:00pm - 4:45pm Nancy Rodriguez (Memorial Bldg., Room 315)
Saturday, July 28, 2018
2:45pm - 3:30pm Suzanne Lenhart (Memorial Bldg., Room 315)
3:45pm - 4:30pm Carlos Castillo-Chavez (Memorial Bldg., Room 315)
4:45pm - 5:30pm Juan Gutierrez (Memorial Bldg., Room 315)



Plenary and Special Public Lectures
Denis Auroux, University of California at Berkeley, USA
An Invitation to Homological Mirror Symmetry
We will give a gentle introduction to some recent developments in the area of mirror symmetry, focusing on two key conjectures in the field: Kontsevich's homological mirror symmetry (1994), which relates the Fukaya category of a symplectic manifold to the derived category of coherent sheaves of a mirror space, and the Strominger-Yau-Zaslow (SYZ) conjecture (1996), which gives a geometric underpinning for the construction of mirror spaces. We will use simple examples to illustrate these conjectures and their extension beyond the Calabi-Yau setting in which they were first formulated. Specifically, we will focus on two one-dimensional examples, the cylinder and the pair of pants, to give a flavor of the geometric concepts involved in a general formulation of homological mirror symmetry.
Michael Eichmair, University of Vienna, Austria
On Scalar Curvature, Minimal Surfaces, and the Isoperimetric Problem in the Large
View PDF
Mark Lewis, University of Alberta, Canada
Population Dynamics in Moving Environments
Classical population dynamics problems assume constant unchanging environments. However, realistic environments fluctuate in both space and time. My lectures will focus on the analysis of population dynamics in environments that shift spatially, due either to advective flow (eg., river population dynamics) or to changing environmental conditions (eg., climate change). The emphasis will be on the analysis of nonlinear advection-diffusion-reaction equations in the case where there is strong advection and environments are heterogeneous. I will use methods of spreading speed analysis, net reproductive rate and inside dynamics to understand qualitative outcomes. Applications will be made to river populations in one- and two-dimensions and to the genetic structure of populations subject to climate change.
Victor Reiner, University of Minnesota, USA
Finite Reflection Groups and General Linear Groups
We discuss some remarkable counting formulas over finite fields that have arisen in recent years, coming from thinking of finite general linear groups as reflection groups, and pursing their analogy to Weyl groups.
Yuri Tschinkel, New York University and Simons Institute, USA
Building Institutes
I will discuss the role of foundations in the creation of mathematical centers.
Algebraic Geometry and Topology
Lino Grama, Universidade Estadual de Campinas, Brazil
On the Construction of LG Models on Coadjoint Orbits
In this talk we describe the LG models associated to coadjoint orbits of complex simple Lie groups. We also discuss its Fukaya-Seidel category in low dimensional examples as well as geometric information about the mirror manifold.
Andrew Harder, University of Miami, USA
Pseudolattices, del Pezzo Surfaces, and Elliptic Fibrations
I will explore the relationship between factorizations of certain elements in SL2(Z), elliptic fibrations over the disc, and del Pezzo surfaces. It turns out that all three of these things can be modelled linear algebraically by objects called pseudolattices, and this fact implies us that their classifications are closely related. Finally, I will explain how this can be seen as a manifestation of homological mirror symmetry.
Paul Horja, University of Miami, USA
Toric Schobers and D-modules
Many classical mirror symmetry results can be recast using the recent language of perverse sheaves of categories and schobers. In this context, I will explain a Riemann-Hilbert type conjectural connection with the D-modules naturally appearing in mirror symmetry. This is joint work with Ludmil Katzarkov.
Gabriel Kerr, Kansas State University, USA
Spheres in Complex Hypersurfaces
Given a hypersurface X of the complex torus, mirror symmetry predicts a quasi-equivalence between the Fukaya category F(X) of X and a certain category of graded matrix factorizations MF (W) on a toric Calabi-Yau variety. In this talk, I will describe this correspondence, as well as how it fits in the larger picture of homological mirror symmetry. Exploring the algebraic side, one finds there are many spherical objects in MF (W) which have combinatorial descriptions. Using phase tropical varieties, I will provide a prediction of the topological mirrors to these objects in F(X) and discuss some generalizations. This is based on joint work with Ilia Zharkov.
Ernesto Lupercio, Cinvestav, México
Self-Organized Critical Complex Systems and Algebraic Geometry in the Tropical Limit
In this talk I will survey our recent work relating the mathematics of complex systems and power laws and the tropical geometry of curves in toric manifolds. Collaborators on this project include N. Kalinin, A. Guzman, Y. Prieto, M. Shkonivkov, V. Kalinina, L. Katzarkov, L. Meersseman, and A. Verjovsky.
Jacob Mostovoy, Cinvestav, México
The Pure Braid Group and the Pure Cactus Group
The topology of the real part of the moduli space of stable curves of genus zero with n marked points is known to be determined completely by its fundamental group, known as the pure cactus group. In this talk I will describe the analogy between the pure cactus group and the pure braid group and show how it leads to an elementary proof of the residual nilpotency of the pure cactus group, conjectured by Etingof et al.
Manuel Rivera, University of Miami, USA, and Cinvestav, México
Higher Categories, Loop Spaces, and Local Systems
I will describe how basic results of the "brave new homotopy theory" a la Lurie may be applied to improve classical results. More precisely, I will explain how unraveling the combinatorics behind the "rigidification functor" of Lurie (the left adjoint of the homotopy coherent nerve functor) leads to the improvement of a classical result of Adams which relates the based loop space on a space and the algebraic cobar construction. Then I will explain the following applications: 1) we obtain algebraic models for different types of path spaces of connected (possibly non-simply connected) spaces, 2) our results lead to a transparent and concrete approach to the homotopy theory of (infinity) local systems, 3) we may deduce that the singular chains on a space with its natural algebraic structure, under a notion of weak equivalence stronger than quasi-isomorphism, encodes the fundamental group.
Andrei Teleman, Aix-Marseilles Université, France
New Methods in the Classification of Class VII Surfaces
View PDF
Bernardo Uribe, Universidad del Norte, Colombia
The Evenness Conjecture on Equivariant Unitary Bordism
The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring in even dimensional generators. In this talk I will review the cases where the conjecture is known to hold and I will highlight the properties that permit one to prove the conjecture in these cases.
Combinatorics
Marcelo Aguiar, Cornell University, USA
Topics in Hyperplane Arrangements
We will discuss a number of geometric and algebraic constructions associated to real hyperplane arrangements, focusing on the monoid of faces and the category of lunes of the arrangement. Basics on hyperplane arrangements will be reviewed. We will then discuss the beginnings of a theory of noncommutative Mobius functions and its connections to the structure of the algebra of faces. We will also discuss an extension of a theorem of Joyal, Klyachko and Stanley relating the homology of the partition lattice to free Lie algebras. These topics are from recent and ongoing joint work with Swapneel Mahajan.
Carolina Benedetti, Universidad de Los Andes, Colombia
A Murnaghan-Nakayama Rule for Quantum Cohomology of the Flag Manifold
In this talk, we provide a combinatorial rule for the product of a quantum power-sum by a quantum Schubert polynomial. This expansion is known, in the classical setting as Murnaghan-Nakayama rule. Our expansion involves chains and intervals in the quantum Bruhat order, and cyclic shifts of those. In geometry, a Murnaghan-Nakayama formula computes the intersection of Schubert cycles with tautological classes coming from the Chern character.
Sara Billey, University of Washington, USA
Boolean Product Polynomials and Schur-Positivity
We study a family of symmetric polynomials that we refer to as the Boolean product polynomials. The motivation for studying these polynomials stems from the computation of the characteristic polynomial of the real matroid spanned by the nonzero vectors in Rn all of whose coordinates are either 0 or 1. To this end, one approach is to compute the zeros of the Boolean product polynomials over finite fields. The zero loci of these polynomials cut out hyperplane arrangements known as resonance arrangements, which show up in the context of double Hurwitz polynomials. By relating the Boolean product polynomials to certain total Chern classes of vector bundles, we establish their Schur-positivity by appealing to a result of Pragacz relying on earlier work on numerical positivity by Fulton-Lazarsfeld. Subsequently, we study a two-alphabet version of these polynomials from the viewpoint of Schur-positivity. As a special case of these polynomials, we recover symmetric functions first studied by Desarmenien and Wachs in the context of descents in derangements.
This is based on joint work with Lou Billera and Vasu Tewari.
Rafael González D'León, Universidad Sergio Arboleda, Colombia
The Whitney Duals of a Graded Poset
Two posets are Whitney duals to each other if the (absolute value of their) Whitney numbers of the first and second kind are switched between the two posets. We introduce new types of edge and chain-edge labelings of a graded poset which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual and we show how to explicitly construct a Whitney dual using a technique that involves quotient posets. As an application of our main theorem, we show that geometric lattices, the lattice of noncrossing partitions, the poset of weighted partitions studied by González D'León-Wachs and the R*S-labelable posets studied by Simion-Stanley all have Whitney duals. We also show that a graded poset P with a Whitney labeling admits a local action of the 0-Hecke algebra on the set of maximal chains of P. The characteristic of the associated representation is Ehrenborg's flag quasisymmetric function of P. This is joint work with Josh Hallam (Wake Forest University).
Patricia Hersh, North Carolina State University, USA
Topology and Combinatorics of Totally Nonnegative Spaces
We will discuss results, both old and new, regarding the topological and combinatorial structure of totally nonnegative varieties. Interest in these varieties comes from geometric representation theory and from the theory of cluster algebras. In many cases, these varieties arise as images of quite interesting maps, with the fibers of these maps describing relations for instance amongst exponentiated Chevalley generators. We will discuss not only the structure of some of these spaces themselves but also of these fibers. Parts of this are joint work with James Davis and Ezra Miller and with Drew Armstrong.
Alejandro Morales, University of Massachusetts Amherst, USA
Hook Formulas for Skew Shapes
The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of straight shape. No such product formula exists for general skew shapes, though there are determinantal formulas or positive formulas involving Littlewood-Richardson coefficients. In 2014, Naruse announced a new positive formula coming from geometry that is very close to the formula for straight shapes and has spurred new interest in tableaux enumeration. I will talk about this formula and how it has led to new results about semistandard tableaux and reverse plane partitions, asymptotics of the number of standard tableaux, and new product formulas for the number of tableaux of certain skew shapes. This is joint work with Igor Pak and Greta Panova.
José Alejandro Samper Casas, University of Miami, USA
Matroid Independence Complexes with Prescribed Homotopy Type
It is well known that the independence complex of any matroid without coloops is homotopy equivalent to a wedge of k > 0 equidimensional spheres. We prove that if the dimension and the number of spheres is fixed, then only finitely many such independence complexes exist. This counterintuitive property leads to new structural questions such as upper and lower bound theorems/conjectures for matroids based on the two parameters mentioned. New theorems about the face numbers of the independence complex also show up. If time permits we will discuss similar results for geometric lattices. This is joint work with F. Castillo.
John Shareshian, Washington University, USA
Regular Hessenberg Varieties and Characters of Hecke Algebras
Combining results of Brosnan-Chow and Clearman-Hyatt-Shelton-Skandera, one sees that Poincaré polynomials of type A regular Hessenberg varieties give values of parabolic characters of the type A Hecke algebras evaluated at certain Kazhdan-Lusztig basis elements. I will describe joint work with Ryan Schneider, along with work of Alex Woo, in which it is shown that, although the corresponding result does not always hold in all Lie types, it does hold in many cases.
Stephanie van Willigenburg, University of British Columbia, Canada
The e-Positivity of Chromatic Symmetric Functions
The chromatic polynomial was generalized to the chromatic symmetric function by Stanley in his seminal 1995 paper. This function is currently experiencing a flourishing renaissance, in particular the study of the positivity of chromatic symmetric functions when expanded into the basis of elementary symmetric functions, that is, e-positivity.
In this talk we approach the question of e-positivity from various angles. Most pertinently we resolve the 1995 statement of Stanley that no known graph exists that is not contractible to the claw, and whose chromatic symmetric function is not e-positive.
This is joint work with Soojin Cho, Samantha Dahlberg and Angele Foley, and no prior knowledge is assumed.
Differential Geometry and Geometric Analysis
Lucas Ambrozio, University of Warwick, UK
Free Boundary Minimal Surfaces
In recent years, the theory of free boundary minimal surfaces, that is, of critical points of the area functional on the space of surfaces whose boundaries are contained inside the boundary of the ambient manifold, has been developing fast into several directions: construction of new examples by several methods (e.g. maximisation of Steklov eigenvalues), classifications theorems, index estimates, compactness results. In this talk we will present an overview of some of these recent developments.
Otis Chodosh, Princeton University, USA
Minimal Surfaces and the Allen-Cahn Equation on 3 Manifolds, Part II
We will describe recent work on the Allen-Cahn semilinear PDE on 3 manifolds including curvature, multiplicity, and index estimates.
Robert Haslhofer, University of Toronto, Canada
Minimal Two-Spheres in Three-Spheres
We prove that any manifold diffeomorphic to S3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S.T. Yau in 1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp. This is joint work with Dan Ketover.
Chao Li, Stanford University and Princeton Unversity, USA
A Polyhedron Comparison Theorem in Positive Scalar Curvature
We establish a comparison theorem for polyhedra in manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collection of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron. We will start the discussion from 3-manifolds, and illustrate how our result is parallel to the positive mass theorem, and thus generalizable to higher dimensions.
Siyuan Lu, Rutgers University, USA
Isometric Embedding: Old and New
In this talk, we will first review the classic Weyl's isometric embedding theorem, solved by Nirenberg and Pogorelov. A key part of the proof is to derive the mean curvature estimate. We will discuss two different approaches for the curvature estimate: global estimate and interior estimate. We will then discuss how to extend these two approaches to general ambient space. If time permits, we will further discuss Weyl's embedding theorem in general Riemannian manifolds and its applications in general relativity.
Christos Mantoulidis, Massachusetts Institute of Technology
Minimal Surfaces and the Allen-Cahn Equation on 3 Manifolds, Part I
We will describe recent work on the Allen-Cahn semilinear PDE on 3 manifolds including curvature, multiplicity, and index estimates.
Abraão Mendes, Universidade Federal de Alagoas, Brazil
Rigidity of Free Boundary Surfaces
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Martín Reiris, Universidad de la República-Montevideo, Uruguay
New Techniques a la Bakry-Emery on Vacuum Static and Stationary Solutions
The classification of static and stationary solutions of the Einstein equations is one of the oldest and most natural problems in General Relativity. The celebrated uniqueness theorem of the Schwarzschild solutions (Israel, Robinson et al, Bunting/Masood-ul-Alam) classifies all asymptotically flat (3+1) static vacuum black holes. Recently, comparison techniques a la Bakry-Emery have been applied to shown that, without any asymptotic assumption, vacuum static black holes are either (i) Schwarzschild, (ii) Boosts, or (iii) of Myers/Korotkin-Nicolai type, namely their topology is that of a solid three-torus minus a finite number of open three-balls and the asymptotic is Kasner, (see arXiv:1806.00818 arXiv:1806.00819). I will discuss how Bakry-Emery techniques could be used to shed light in the still open problem of the existence of vacuum stationary (rotating) black holes of Myers/Korotkin-Nicolai type. Achieving this goal would be a step towards the classification of vacuum stationary black holes, still an unresolved problem.
Lu Wang, University of Wisconsin, USA
Self-Similar Solutions of Mean Curvature Flow
Mean curvature flow is the negative gradient flow of the volume functional, which decreases the volume of hypersurfaces in the steepest way. The flow starting from any compact hypersurface will develop singularities in finite time. Self-similar solutions of mean curvature flow play an important role in understanding the asymptotic behavior of the flow near singularities. In this talk, I will survey some known results as well as some open problems about self-similar solutions of mean curvature flow – with a particular emphasis on properties of self-shrinking solutions.
Mathematical Biology
Carlos Castillo-Chavez, Arizona State University, USA and Yachay University of Experimental Technical Research, Ecuador
Scaling Up the Impact of Dynamic Individual Decisions in Response to Ongoing Epidemic Outbreaks
The long standing challenge posed by the threat of emergent or re-emergent diseases is intimately linked to the use that individuals make of disease risk information. A modeling framework that accounts for the impact that an ongoing disease outbreak has on the decisions that individuals make based on their real or perceived risk of infection is revisited.
Recent work with collaborators at various institutions including Eli Fenichel, Charles Perrings and Ben Morin is highlighted. The research is based on a behavioral framework where individual decisions are based on the tradeoffs made in response to costs associated with the present or future risk of infection and the potential loss of benefits that may result as a consequence of risk aversion decisions – risks due in part to changes in prevalence. This research project will be highlighted in the context of influenza.
Juan Gutierrez, University of Georgia, USA
The Math of Multi-Scaling: From Molecular Dynamics to Epidemiological Processes of Malaria
The advent of high-throughput molecular technologies in particular, and the broad availability of data, in general, have forced the quantitative biology community to rethink how to conceive, build, and validate mathematical models. In this talk I will demonstrate how molecular and cellular processes are related to the epidemiology of malaria. We will explore (i) asymptomaticity at the epidemiological level, (ii) the cellular models that explain this phenomenon as an interaction between the immune system and infected red blood cells, (iii) mathematical models that link cellular and transcriptional time series, (iv) transcriptomic analysis, and finally (vi) high-throughput in-silico drug discovery to solve an epidemiological problem. All these linked analyses provide a comprehensive picture that no single scale can produce alone. The usefulness of models under this light takes on new meanings, and this broad scope requires the cooperation of scientists coming from very different intellectual traditions. In this talk we will also explore how an information system that delivers Adaptive Learning for Interdisciplinary Collaborative Environments (ALICE) is used to train scientists in this new normal.
Xi Huo, University of Miami, USA
Modelling the Antibiotic Use in Intensive Care Units – Comparing De-escalation and Continuation
Antimicrobial de-escalation refers to the treatment mechanism of switching from empiric antibiotics with good coverage to alternatives based on laboratory susceptibility test results, with the aims of reducing costs and avoiding unnecessary use of broad-spectrum antibiotics. Though widely practiced and recommended, the benefits and tradeoffs of this strategy have not been well understood. In this talk, we will first show our preliminary simulation results of a set of multi-strain-multi-drug models in an intensive care unit setting, to numerically compare de-escalation with the conventional strategy called antimicrobial continuation. Then we simplify the previous models to compare the long-term dynamical behaviors between de-escalation and continuation systems under a double-strain-double-drug scenario. Finally we extend our models to seek for optimal antibiotic use strategies under a triple-strain-triple-drug scenario. The major conclusion of this study shows that, if we suppose there are two identical intensive care units that separately adopt de-escalation and continuation as the major drug use strategy, then the one following de-escalation: (1) could maintain either higher or lower percentage of colonized patients in the two-strain transmission scenario; (2) is superior in preventing outbreaks of strains resistant to the empiric antibiotic.
King-Yeung Lam, Ohio State University, USA
Invasion of Open Space by Two Competing Species
I will discuss a question raised by Shigesada and Kawasaki in Chapter 7 of their monograph, concerning the spreading properties of two competing species on the real line when the initial values are null or exponentially decaying in a right half-line. In the case of compactly supported initial values, we prove that the first species spreads with the KPP speed of the single species, whereas the speed of the second species can be given by an exact formula depending on the speed of the first species. This is joint work with Leo Girardin (Paris VI). If time allows, I will also talk about some recent progress obtained with Qian Liu (OSU and Renmin Univ. of China).
Suzanne Lenhart, University of Tennessee, USA
Optimal control techniques for management strategies in biological models
Two examples with different optimal control techniques to choose management actions will be presented. One model is a PDE system representing Zika spreading across a state in Brazil; the control varying in space and time is a vaccination rate. Data from Brazil were used to estimate parameters. The second model represents a large scale forest fire. We incorporate the stochasticity of the time of a fire into our model and explore the trade-offs between prevention management spending and suppression spending. A large fire event in the past was used to form an illustrative example.
Salomé Martínez, Universidad de Chile, Chile
Multiple Steady States for a Competition System Supporting an Ideal-Free Distribution
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Nancy Rodriguez, University of North Carolina, USA
Birth-Jump Processes in Plant Dynamics
In this talk I will introduce a model for the dynamics of the growth and dispersal of plants in various environments. The discrete model is based on a birth-jump process which exhibits wave-like solutions. After discussing the continuum limit, which is an non-local reaction-diffusion equation, I will present the proof of existence of traveling waves for speeds above a critical threshold (both sharp and continuously differentiable) in the diffusion-limit assuming a logarithmic-type growth term. I will conclude by verifying the theoretical results presented via the use of numerical simulations.
Zhisheng Shuai, University of Central Florida, USA
Biased population movement and infectious disease dynamics
Many recent outbreaks and spatial spread of infectious diseases have been influenced by human movement over air, sea and land transport networks, and/or anthropogenic-induced pathogen/vector movement. These spatial movements in heterogeneous environments and networks are often asymmetric (biased). The effects of asymmetric movement versus symmetric movement will be investigated using several epidemiological models from the literature. These investigations provide a better understanding of disease transmission and control in the real life application.
Jorge Velasco-Hernandez, Universidad Nacional Autónoma de México, México
Dengue Dynamics in Southern Mexico: An Approximation to its Population Dynamics and the Role of Population Movement
We present data and analysis of dengue incidence in several states of Mexico. A metapopulation model is presented along with further commentaries on the construction of a dengue network for mobility in these regions.