Let A be an Abelian variety over a field K. The group A(K) of K-rational points on A, known as the Mordell-Weil group of A, is known to be finitely generated if K  is an algebraic number field of finite degree. It is known to be of infinite rank if K is a certain type of algebraic number field of infinite degree. If K  is "too large", then A(K) contains a non-trivial divisible subgroup. I will discuss some reasonable conditions on K which allow A(K) to contain no non-trivial divisible subgroups, and give some examples of such K.

We prove that for each compact connected one-manifold M and for each real number a >=1, there exists a finitely generated group G inside Diff^a(M) such that G admits no injective homomorphisms into the group cup_{b>a} Diff^b(M). This is a joint work with Thomas Koberda.

Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. We give an O(n log n)-time algorithm for this problem, as well as an Ω(n log n) lower bound in the algebraic computation tree model. We prove that the generalization where two cones of directions per face are allowed is NP-hard.

In these two lectures, I will first introduce the Ricci flow. I will discuss how it can be applied to geometrization of 3-manifolds. Next, I will give a tour of the Analytic Minimal Model Program whose goal is to classify Kaehler manifolds birationally through geometric methods. I will discuss some results and open problems. Finally, I will discuss other and newer curvature flows and their applications.

In these two lectures, I will first introduce the Ricci flow. I will discuss how it can be applied to geometrization of 3-manifolds. Next, I will give a tour of the Analytic Minimal Model Program whose goal is to classify Kaehler manifolds birationally through geometric methods. I will discuss some results and open problems. Finally, I will discuss other and newer curvature flows and their applications.

We will discuss limits of finite groups and their connections with Theoretical Computer Science, Number Theory etc.

For more than one hundred years, the Poincare conjecture was a driving force for topologists and its study led to many progresses on topology. It was finally solved by Perelman using differential geometric methods. In this lecture, I will tell what is the Poincare conjecture and a brief history of pursuing it. I will explain geometric ideas involved in solving the conjecture, particularly, geometrization of 3-spaces. I will end up with some speculations on future developments in geometry. This lecture is aimed at general audience.

In this talk, we introduce some recent results on a 1D stochastic particle model, the totally asymmetric simple exclusion process (TASEP). Contrary to the usual TASEP, in the TASEP with second class particles, first class particles have priority over second class particles when they move. In this talk, we introduce some techniques to find the exact formulas of the transition probabilities and the block probabilities.

We prove a generalization of Pal’s 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360 degrees inside Q. We also prove a lower bound of Ω(m n²) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.

The present talk introduces a localized version of la méthode des multiplicateurs (known as method of Lagrange multipliers) and its recent applications in computational engineering. We will, first, offer a brief review of a variational formulation for the partitioned equations of motion for multi-physics and/or multi-domains utilizing the method of localized Lagrange multipliers, with some of its earlier applications: pore fluid-soil, structure-control, acoustic-structure, structural-thermal and structure-electromagnetic problems. We then focus on recent advances: regularization for stiff coupled systems, reduced-order modeling, nonmatching interfaces, a direct generation of inverse mass matrices for explicit transient analysis, and uncertainty quantification analysis. The presentation concludes with potential areas of further developments in partitioned analysis employing the method of localized Lagrange multipliers.

Michael Artin and Barry Mazur's classical comparison theorem tells us that for a pointed connected finite type $C$-scheme $X$, there is a map from the singular complex associated to the underlying topological spaces of the analytification of $X$ to the 'etale homotopy type of $X$, and it induces an isomorphism on profinite completions. I'll begin with a brief review on Artin-Mazur's 'etale homotopy theory of schemes, and explain how I extended it to algebraic stacks under model category theory. Finally, I'll provide a formal proof of the comparison theorem for algebraic stacks using a new characterization of profinite completions.

We start by introducing general determinantal point processes in one dimension and their relation to random matrices following Borodin. Several examples with increasing level of complexity will be discussed as the classical Gaussian Unitary Ensemble, products of several independent and uncorrelated Gaussian random matrices and the effect of introducing correlations. We will then display the corresponding double contour integral representation of the respective kernels and discuss the issue of universality in the limit of large matrix size. This is based on several joint works with Eugene Strahov as well as a work including also Tomasz Checinski and Dang-Zheng Liu.

For a Hecke character of a totally real field, we consider its twist by a line of characters of p-power order. Following the method of Rohrlich approximate functional equation for family of  the twisted L-values.  We develop a method to count units whose residue have bounded norms in the p-adic expansion w.r.t. a nonsingular cone, namely (C,p)-adic expansion, we obtain the nonvanishing of the L-values when the conductor goes to the infinity.  Finally, we discuss how to apply the result to generation of the coefficient field. This is a joint work with Jungyun Lee and Hae-Sang Sun.

Let $F(t,X)$ be an irreducible polynomial in two variables over a number field $k$. Famously, Hilbert's irreducibility theorem asserts that there exist infinitely many $t_0in k$ such that $f(t_0,X)$ remains irreducible.

In fact, stronger versions of the theorem assert that the ``exceptional" Hilbert set $mathcal{R}_f:=

{t_0in kmid f(t_0,X) text{ is reducible}}$ is small in several well-defined ways.

We will focus on polynomials of the form $F(t,X)=F_1(X)-tF_2(X)$, i.e. $t=f(x):=F_1(x)/F_2(x)$ for a root $x$ of $F$. Using the classification of monodromy groups, we show the following:\

If $f=f_1circ ... circ f_r$ is a decomposition of $f$ into indecomposable rational functions, and all $f_i$ are ``sufficiently generic" and of sufficiently large degree, then up to finitely many values, the set $mathcal{R}_f$ consists only of the $k$-rational values of $f_1$.\

This generalizes in several ways previous finiteness results, such as M"uller's results on reducible {it integral} specializations.

This talk is based on joint work in progress with Danny Neftin.

Reorganization of neuronal circuits through experience-dependent modification of synaptic connections has been thought to be one of the basic mechanisms for learning and memory. This idea is supported by in-vitro experimental works that show long-term changes of synaptic strengths in different slice preparations. However, a single neuron receives inputs from many neurons in cortical circuits, and it is difficult to identify the rule governing synaptic plasticity of an individual synapse from in vivo studies.

In this talk, I would discuss a novel method to infer synaptic plasticity rules and principles of neural dynamics from neural activities obtained in vivo.  The method was applied to the data obtained in monkeys performing visual learning tasks.  This study can connect several experimental works of learning and long-term memory at cellular and system level, and could be applicable to other cortical circuits to further our understanding the interactions between circuit dynamics and synaptic plasticity rules.

We consider the 3D axisymmetric Euler equations on exterior domains $ { (x,y,z) : (1 + epsilon |z|)^2 le x^2 + y^2 } $ for any $epsilon > 0$ so that we can get arbitrarily close to the exterior of a cylinder. We construct a strong local well-posedness class, and show that within this class there exist compactly supported initial data which blows up in finite time. The local well-posedness class consists of velocities which are uniformly Lipschitz in space and have finite energy. Our results were inspired by recent works of Hou-Luo, Kiselev-Sverak, and many others, and the proof builds up on our previous works on 2D Euler and Boussinesq systems. This is joint work with Tarek Elgindi. 

Memory refers to the ability to hold information in time long after the stimulus is off, and is essential for a variety of adaptive behaviors including integration, learning and generalization.  Persistent changes in the activity or connectivity of the systems that lasts longer than the triggering events have been suggested as a substrate for memory.  In this talk, I would discuss requirements of the memory system and theoretical principles that can allow the brain to construct persistent states for memory.  I would review dominant theories based on attractor dynamics suggested for various types of memory as well as reviewing alternative theories.  Also, I would discuss open problems and experimental evidence or tests that can distinguish different mechanisms.

Networked systems, including social, biological, and computer networks, are subjects of study in many disciplines. One of the key properties of the network is the community structure, which refers to the occurrence of natural division of a network into groups of nodes that are more densely connected internally. In this talk, I will briefly introduce methods for finding communities with emphasis on the spectral method using adjacency matrices. In addition, I will also describe how the sparsity of the network affect the community detection problem.

Coordinated and/or cooperative control of multiple unmanned vehicles has been spotlighted as a means to accomplish complex mission objectives in a cost- and resource-effective way, and a rich set of theories and algorithms have been proposed. This talk briefly introduces recent advances in the coordinated decision making for networked autonomous vehicles, in the context of mission & task allocation and informative planning of mobile sensor networks. The particular emphasis is on how advanced mathematical frameworks such as game theory have been adopted in analyzing/synthesizing such coordinated systems. In addition, a potential link of network analysis methodologies with secure and resilient coordination of networked vehicles will be discussed.

We study the asymptotic translation length on curve complexes of the pseudo-Anosov surface homeomorphisms. We first show that the minimal asymptotic translation length of Torelli groups and pure braid groups are asymptotically 1/chi(S) where chi(S) is the Euler characteristic of the surface. If the time permits, we also discuss the asymptotic translation length of pseudo-Anosov monodromies of primitive elements in Thurston’s fibered cone. This talk represents joint work with Hyunshik Shin and Chenxi Wu.

The chromatic number of a graph is the minimum k such that the graph has a proper k-coloring. It is known that if T is a tree, then every graph with large chromatic number contains T as a subgraph. In this talk, we discuss this phenomena for star-coloring (a proper coloring forbidding a bicolored path on four vertices) and acyclic-coloring (a proper coloring forbidding bicolored cycles). Specifically, we will characterize all graphs T such that every graph with sufficiently large star-chromatic number (acyclic-chromatic number) contains T as a subgraph.

In this talk, I will explain an approach of Poincare to prove the uniformization theorem for punctured spheres, and how it is related to the action functional in the Liouville theory.