Suppose $L$ is a link with n components and the rank of $Kh(L;Z/2)$ is $2^n$, we show that $L$ can be obtained by disjoint unions and connected sums of Hopf links and unknots. This result gives a positive answer to a question asked by Batson-Seed, and generalizes the unlink detection theorem of Khovanov homology by Hedden-Ni and Batson-Seed. The proof relies on a new excision formula for the singular instanton Floer homology introduced by Kronheimer and Mrowka. This is joint work with Yi Xie.

In this talk I will survey recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, which was originally due to Hatcher. Second, we show that the space of metrics with positive scalar curvature on every 3-manifold is either contractible or empty. This completes work initiated by Marques. Our proof is based on a new uniqueness theorem for singular Ricci flows, which I have previously obtained with Kleiner. Singular Ricci flows were inspired by Perelman's proof of the Poincar\'e and Geometrization Conjectures, which relied on a flow in which singularities were removed by a certain surgery construction. Since this surgery construction depended on various auxiliary parameters, the resulting flow was not uniquely determined by its initial data. Perelman therefore conjectured that there must be a canonical, weak Ricci flow that automatically ``flows through its singularities'' at an infinitesimal scale. Our work on the uniqueness of singular Ricci flows gives an affirmative answer to Perelman's conjecture and allows the study of continuous families of singular Ricci flows leading to the topological applications mentioned above.

The study of stochastic processes and their expected suprema arises in many natural contexts. Some examples include understanding the modulus of continuity for Brownian motion, bounding the maximum singular value of a random matrix, or quantifying discrepancies between a distribution and its corresponding empirical distribution. Early attempts at understanding Gaussian processes dates back as far as Kolmogorov and more recently to Dudley, Fernique, Pisier, and Marcus, among many others. Michel Talagrand has provided a powerful and elegant framework known as the Generic Chaining which unifies and extends the work of these mathematicians to give optimal bounds on the expected suprema of Gaussian processes. The aspirational goal of this talk is to outline chaining \'a la Talagrand while focusing on understanding and not getting lost in the details of the set-up. Material is drawn mostly from Chapter 2 of Talagrand's text ``Upper and Lower Bounds for Stochastic Processes.''

Graph distances have proven quite useful in machine learning/statistics, particularly in the estimation of Euclidean or geodesic distances. The talk will include a partial review of the literature, and then present more recent developments on the estimation of curvature-constrained distances on a surface, and well as on the estimation of Euclidean distances based on an unweighted and noisy neighborhood graph.

Using the Torelli theorem for K3 surfaces of Pyatetskii-Shapiro and
Shafarevich one can describe the automorphism group of a K3 surface over
${\mathbb C}$ up to finite error as the quotient of the orthogonal group
of its Picard lattice by the subgroup generated by reflections in
classes of square -2. We will give a similar description valid over an
arbitrary field in which the reflection group is replaced by a certain
subgroup. We will then illustrate this description by giving several
examples of interesting behaviour of the automorphism group, and by
showing that the automorphism groups of two families of K3 surfaces that
arise from Diophantine problems are finite. This is joint work with
Martin Bright and Ronald van Luijk (University of Leiden).

Thromboembolism, one of the leading causes of morbidity and mortality worldwide, is characterized by formation of obstructive intravascular clots (thrombi) and their mechanical breakage (embolization). A novel two-dimensional multi-phase computational model will be described that simulates active interactions between the main components of the clot, including platelets and fibrin. It can be used for studying the impact of various physiologically relevant blood shear flow conditions on deformation and embolization of a partially obstructive clot with variable permeability. Simulations provide new insights into mechanisms underlying clot stability and embolization that cannot be studied experimentally at this time. In particular, multi-phase model simulations, calibrated using experimental intravital imaging of an established arteriolar clot, show that flow-induced changes in size, shape and internal structure of the clot are largely determined by two shear-dependent mechanisms: reversible attachment of platelets to the exterior of the clot and removal of large clot pieces [1]. Model simulations also predict that blood clots with higher permeability are more prone to embolization with enhanced disintegration under increasing shear rate. In contrast, less permeable clots are more resistant to rupture due to shear rate dependent clot stiffening originating from enhanced platelet adhesion and aggregation. Role of platelets-fibrin network mechanical interactions in determining shape of a clot will be also discussed and quantified using analysis of experimental data [2,3]. These results can be used in future to predict risk of thromboembolism based on the data about composition, permeability and deformability of a clot under specific local hemodynamic conditions.

1. Xu S, Xu Z, Kim OV, Litvinov RI, Weisel JW, Alber M. Model predictions of deformation, embolization and permeability of partially obstructive blood clots under variable shear flow. J. R. Soc. Interface 14 (2017) 20170441.

2. Oleg V. Kim, Rustem I. Litvinov, Mark S. Alber & John W. Weisel, Quantitative structural mechanobiology of platelet driven blood clot contraction, Nature Communications 8 (2017) 1274.

3. Samuel Britton, Oleg Kim, Francesco Pancaldi, Zhiliang Xu, Rustem I. Litvinov, John W.Weisel, Mark Alber [2019], Contribution of nascent cohesive fiber-fiber interactions to the non-linear elasticity of fibrin networks under tensile load, Acta Biomaterialia 94 (2019) 514--523.

Combining a multi-armed bandit task and Bayesian computational modeling, we find that humans systematically under-estimate reward availability in the environment. This apparent pessimism turns out to be an optimism bias in disguise, and one that compensates for other idiosyncrasies in human learning and decision-making under uncertainty, such as a default tendency to assume non-stationarity in environmental statistics as well as the adoption of a simplistic decision policy. In particular, reward rate underestimation discourages the decision-maker from switching away from a ``good'' option, thus achieving near-optimal behavior (which never switches away after a win). Furthermore, we demonstrate that the Bayesian model that best predicts human behavior is equivalent to a particular class of reinforcement learning models, thus giving statistical, normative grounding to phenomenological models of human behavior.

By a ``modular form'' for a reductive group $G$ we mean an
automorphic form that has some sort of very nice Fourier expansion. The
classic example are the holomorphic Siegel modular forms, which are
special automorphic functions for the group $\mathrm{Sp}_{2g}$.
Following work of Gan, Gross, Savin, and Wallach, it turns out that
there is a notion of modular forms on certain real forms of the
exceptional groups. I will define these objects and explain what is
known about them.

We introduce the normalized Laplacian matrix and study how its eigenvalues relate to random walks on graphs and isoperimetric inequalities.

In the early '60s, Abraham Robinson scrapped his standard transmission
calc-mobile and built a non-standard (aka automatic) model. His new model
allowed him and others to shift many proofs into a more intuitive gear.
We'll discuss the first time he took his model through the
car- \L o\'s; and d-rive the transfer principle. We'll also get
infinitely close to talking about the hyperreal numbers.

I will discuss compactifications of the moduli space of smooth plane curves of degree d at least 4. We will regard a plane curve as a log Fano pair $(\mathbb{P}^2,aC)$, where a is a rational number, and study the compactifications coming from K stability for general log Fano pairs. We establish a wall crossing framework to study these spaces as a varies and show that, when a is small, the moduli space coming from K stability is isomorphic to the GIT moduli space. We describe all wall crossings for degree 4, 5, and 6 plane curves and discuss the picture for general Q-Gorenstein smoothable log Fano pairs. This is joint work with Kenneth Ascher and Yuchen Liu.