Abstract: In game theory, an Evolutionarily Stable Set (ES set) is a set of Nash Equilibrium (NE) strategies that give the same payoffs. Similar to an Evolutionarily Stable Strategy (ES strategy), an ES set is also astrict NE. This work investigates the evolutionary stability of classical and quantum strategies in the quantum penny flip games. In particular, we developed an evolutionary game theory model to conduct a series of simulations where a population of mixed classical strategies from the ES set of the game were invaded by quantum strategies. We found that when only one of the two players� mixed classical strategies were invaded, the results were different. In one case, due to the interference phenomenon of superposition, quantum strategies provided more payoff, hence successfully replaced the mixed classical strategies in the ES set. In the other case, the mixed classical strategies were able to sustain the invasion of quantum strategies and remained in the ES set. Moreover, when both players� mixed classical strategies were invaded by quantum strategies, a new quantum ES set was emerged. The strategies in the quantum ES set give both players payoff 0, which is the same as the payoff of the strategies in the mixed classical ES set of this game.

Mirror symmetry, notably in the work of Mark Gross and Berndt Siebert, identifies involutory "mirror pairs" (X,Y) of (degenerating, polarized)
compact Calabi-Yau manifolds and makes predictions relating the symplectic
geometry of X with the algebraic geometry of Y. Those predictions range
from those of the "open string sector", where homological mirror symmetry
(HMS) relates the Lagrangian submanifolds of X to algebraic vector bundles
on Y, to those of the "closed string sector" where, for example, counts of
holomorphic spheres in X are predicted to equal certain period integrals on Y.
I'll report recent work, joint with Nick Sheridan, which says the following: if one can prove a certain fragment of HMS (a fragment which
we expect to fit neatly into the Gross-Siebert program) then, without
knowing anything more about the geometry of X and Y, one can deduce (i) the
full statement of HMS; and (ii) certain algebraic and enumerative claims
from closed-string mirror symmetry.

The product replacement graph (PRG) of a group G is the set of generating k-tuples of G, with edges corresponding to Nielsen moves. It is conjectured that PRGs of infinite groups are nonamenable. We verify that PRGs have exponential growth when G has polynomial growth or exponential growth, and show that this also holds for a group of intermediate growth: the Grigorchuk group. We also provide some sufficient conditions for nonamenability of the PRG, which cover elementary amenable groups, linear groups, and hyperbolic groups.

***Integral equation methods for singular problems and application to the evaluation of Laplace eigenvalues***

Boundary integral equation methods represent a powerful tool for the numerical solution of a variety of problems in acoustics, electromagnetics, fluid mechanics etc. Together with Oscar Bruno at Caltech, we developed a numerical method for the solution of scattering problems with mixed Dirichlet/Neumann boundary conditions. We investigated and obtained the exact form of the solution singularities -- which arise at transition points where Dirichlet and Neumann boundary conditions meet. These singularities are incorporated in the numerical approach and are resolved via Fourier Continuation technique. The resulting method exhibits spectral convergence. Additionally, jointly with Nilima Nigam at SFU, we applied the mixed boundary value solver to Laplace eigenvalue problems. The challenge is in eigenvalue search as a result of the properties of the objective function, which requires scanning through a range of frequencies. We introduce an improved search algorithm, that allows to locate the eigenvalues using standard root-finding methods. We also apply another integral equation method for domains with corners for a mode matching problem in electromagnetics. Jointly with Ahmed Akgiray at Caltech, we calculated TE and TM modes for domains of specific geometry. Those are further used for antenna design.

A Sidon set is a set of integers such that no difference of two
distinct numbers in the set equals the difference of any other two
numbers in the set. Constructions of large Sidon sets have
interesting connections to geometry and number theory. I will show
how to construct Sidon sets and discuss these connections.

Genus one curves, defined over the rationals, need not have rational points.
The set of all such curves, whose Jacobian is a fixed elliptic curve E, form
a group, called the Weil-Chatelet group. It has an important subgroup, the
Tate-Shafarevich group, formed by those curves which have points over all
completions of the rationals.

This talk will address two aspects of the arithmetic of genus one curves:
(1) (with J. Stix) the divisibility of the Tate-Shafarevich group inside
the Weil-Chatelet group; (2) (with A. Wiles) work in progress on the existence
of points defined over number fields with solvable Galois group over the
rationals. Earlier work, also with Wiles, proved existence when the curve
represents an element of the Tate-Shafarevich group; we now aim to extend
this to the whole Weil-Chatelet group.

More than 25 years ago, Kenig, Ruiz, and Sogge established uniform $L^p$ resolvent estimates for the Laplacian in the Euclidean space. Taking their remarkable estimate as a starting point, we shall describe more recent developments concerned with the problem of controlling the resolvent of elliptic self-adjoint operators in $L^p$ spaces in the context of a compact Riemannian manifold. Here some new interesting difficulties arise, related to the distribution of eigenvalues of such operators. Applications to inverse boundary problems and to the absolute continuity of spectra for periodic Schr\"odinger operators will be presented as well."

In a report published in 1937, Doeblin - who is regarded the father of the "coupling method" - introduced an ergodicity coefficient that provided the first necessary and sufficient condition for the weak ergodicity of non-homogeneous Markov chains over finite state spaces.

In today's jargon, Doeblin's coefficient corresponds to the maximal coupling of the probability transition kernels associated with a Markov chain, and the Monte Carlo literature has (often implicitly) used it to draw perfectly from the stationary distribution of a homogeneous Markov chain over a Polish state space.

In this talk, I will show how Doeblin's coefficient can be used to approximate the distribution of additive functionals of homogeneous Markov chains, particularly sojourn-times, instead of characterizing asymptotic objects such as stationary distributions. The methodology leads to easy to compute and explicit error bounds in total variation distance, and gives access to approximations in Markov chains that are too long for exact calculation but also too short to rely on Normal approximations or stationary assumptions underlying Poisson approximations.

This is a report on joint work with Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, and Vytautas Paskunas. We will explain some ideas around the global construction of new representation-theoretic objects called "patched modules" by a variant of the Taylor-Wiles-Kisin method. They are in many ways better suited than p-adically completed cohomology for a global attempt to understand the p-adic local Langlands correspondence. As an application, we obtain new cases of the Breuil-Schneider conjecture.

``Schubert calculus'' is the (somewhat misleading) name given to the classical branch of enumerative geometry which counts intersections of certain simple varieties like points, lines, and planes. Rather than actually doing any geometry, we'll find these intersection numbers with a combinatorial rule known as the Littlewood-Richardson rule. In particular, we'll show why the problem of counting the number of lines in $\mathbb{C}^3$ that intersect four generic lines is now considered to be ``just combinatorics.''

We extend a model for the morphology and dynamics of a crawling eukaryotic cell to describe cells on micropatterned substrates. This model couples cell morphology, adhesion, and cytoskeletal flow in response to active stresses induced by actin and myosin. We propose that protrusive stresses are only generated where the cell adheres, leading to the cell's effective confinement to the pattern. Consistent with experimental results, simulated cells exhibit a broad range of behaviors, including steady motion, turning, bipedal motion, and periodic migration, in which the cell crawls persistently in one direction before reversing periodically. We show that periodic motion emerges naturally from the coupling of cell polarization to cell shape by reducing the model to a simplified one-dimensional form that can be understood analytically. Additionally, we will discuss a turning instability arising from our model applying onto a free moving cell without interaction with the micropatterned substrates. Some attempts have made to test how the instability depends on the parameters in the model numerically. For a much simplified model, we do find that surface tension is a key factor to stabilize the cell turning.

Algorithms have two costs: arithmetic and communication, i.e. moving data between levels of a memory hierarchy or processors over a network. Communication costs (measured in time or energy per operation) already greatly exceed arithmetic costs, and the gap is growing over time following technological trends. Thus our goal is to design algorithms that minimize communication. We present algorithms that attain provable lower bounds on communication, and show large speedups compared to their conventional counterparts. These algorithms are for direct and iterative linear algebra, for dense and sparse matrices, as well as direct n-body simulations. Several of these algorithms exhibit perfect strong scaling, in both time and energy: run time (resp. energy) for a fixed problem size drops proportionally to the number of processors p (resp. is independent of p). Finally, we describe extensions to algorithms involving arbitrary loop nests and array accesses, assuming only that array subscripts are affine functions of the loop indices.

Suppose that G is a connected reductive p-adic group. We will
describe the classification of irreducible admissible smooth mod p
representations of G in terms of supercuspidal representations. This is
joint work with N. Abe, G. Henniart, and M.-F. Vigneras.

The study of mean curvature flow not only is fundamental in geometry, topology and analysis, but also has important applications in applied mathematics, for instance, image processing. One of the most important problems in mean curvature flow is to understand the possible singularities of the flow and self-shrinkers, i.e., self-shrinking solutions of the flow, provide the singularity models.

In this talk, I will describe the rigidity of asymptotic structures of self-shrinkers. First, I show the uniqueness of properly embedded self-shrinkers asymptotic to any given regular cone. Next, I give a partial affirmative answer to a conjecture of Ilmanen under an infinite order asymptotic assumption, which asserts that the only two-dimensional properly embedded self-shrinker asymptotic to a cylinder along some end is itself the cylinder. The feature of our results is that no completeness of self-shrinkers is required.

The key ingredients in the proof are a novel reduction of unique continuation for elliptic operators to backwards uniqueness for parabolic operators and the Carleman type techniques. If time permits, I will discuss some applications of our approach to shrinking solitons of Ricci flow.