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Landau-Ginzburg orbifold is just another name for a holomorphic function W with its abelian symmetry G. Its Fukaya category can be viewed as a categorification of a homology group of its Milnor fiber. In this introductory talk, we will start with some classical results on the topology of isolated singularities and its Fukaya-Seidel category. Then I will explain a new construction for such category to deal with a non-trivial symmetry group G. The main ingredients are classical variation map and the Reeb dynamics at the contact boundary. If time permits, I will show its application to mirror symmetry of LG orbifolds and its Milnor fiber. This is a joint work with C.-H. Cho and W. Jeong.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

I will report on a recent joint work with Sorin Popa in which we undertake a systematic study of W*-rigidity paradigms for the embedding relation between II₁ factors and their amplifications. We say that a II₁factor M stably embeds into a II₁ factor N if M may be realized as a subfactor of an amplification of N, not necessarily of finite index. This is a preorder relation and we prove that it is as complicated as it can be: under the appropriate separability assumptions, we concretely realize any partially ordered set inside the preordered class of II₁ factors with embeddability.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Cliques are important structures in network science that have been used in numerous applications including spam detection, graph analysis, graph modeling, community detection among others. Obtaining the counts of k-cliques in real-world graphs with millions of nodes and edges is a challenging problem due to combinatorial explosion. Essentially, as k increases, the number of k-cliques goes up exponentially. Existing techniques are (typically) able to count k-cliques for only up to k=5.

In this talk, I will present two algorithms for obtaining k-clique counts that improve the state of the art, both in theory and in practice. The first method is a randomized algorithm that gives a (1+ε)-approximation for the number of k-cliques in a graph. Its running time is proportional to the size of an object called the Turán Shadow, which, for real-world graphs is found to be small. In practice, this algorithm works well for k<=10 and gives orders of magnitude improvement over existing methods. This paper won the Best Paper Award at WWW, 2017.

The second method, a somewhat surprising result, is a simple but powerful algorithm called Pivoter that gives the exact k-clique counts for all k and runs in O(n3^{n/3}) time in the worst case. It uses a classic technique called pivoting that drastically cuts down the search space for cliques and builds a structure called the Succinct Clique Tree from which global and local (per-vertex and per-edge) k-clique counts can be easily read off. In practice, the algorithm is orders of magnitude faster than even other parallel algorithms and makes clique counting feasible for a number of graphs for which clique counting was infeasible before. This paper won the Best Paper Award at WSDM, 2020.

Zoom: https://harvard.zoom.us/j/98231541450

High-temperature superconductivity in the cuprates remains an unsolved problem because the cuprates start off their lives as Mott insulators in which no organizing principle such a Fermi surface can be invoked to treat  the electron interactions.  Consequently, it would be advantageous to solve even a toy model that exhibits both Mottness and superconductivity.  Part of the problem is that the basic model for a Mott insulator, namely the Hubbard model is unsolvable in any dimension we really care about.  To address this problem, I will start by focusing on the overlooked Z_2 emergent symmetry of a Fermi surface first noted by Anderson and Haldane.  Mott insulators  break this emergent symmetry.  The simplest model of this type is due to Hatsugai/Kohmoto.  I will argue that this model can be thought of a fixed point for Mottness.  I will then show exactly[1] that this model when appended with a weak pairing interaction exhibits not only the analogue of Cooper’s instability but also a superconducting ground state, thereby demonstrating that a model for a doped Mott insulator can exhibit superconductivity.  The properties of the superconducting state differ drastically from that of the standard BCS theory.  The elementary excitations of this superconductor are not linear combinations of particle and hole states but rather are superpositions of doublons and holons, composite excitations signaling that the superconducting ground state of the doped Mott insulator inherits the non-Fermi liquid character of the normal state. Additional unexpected features of this model are that it exhibits a superconductivity-induced transfer of spectral weight from high to low energies and a suppression of the superfluid density as seen in the cuprates.

[1] PWP, L. Yeo, E. Huang, Nature Physics, 16, 1175-1180 (2020).

Zoom: https://harvard.zoom.us/j/977347126

We will outline a definition of analytic spaces that relates
to complex- or rigid-analytic varieties in the same way that schemes
relate to algebraic varieties over a field. Joint with Dustin Clausen.

Zoom: https://harvard.zoom.us/j/99334398740

Password: The order of the permutation group on 9 elements.

We will explore the idea of universality in probability theory, as seen in the central limit theorem, in which wide and varied input distributions can all give identical limiting outputs. We will then explore examples of universality in random algebraic structures, such as random vector spaces over finite fields. We will explain how this is connected to recent research on random integral matrices and distributions of class groups of number fields.

Please go to the College Calendar to register.

4d N=1 super Yang-Mills has multiple gapped vacua arising from the spontaneously broken discrete R-symmetry. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We propose an explicit answer to this question: the domain walls support specific topological quantum field theories. We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions (twisted by all global symmetries).

Zoom: https://harvard.zoom.us/j/977347126

I will discuss applications of Vojta’s conjecture to some problems in arithmetic dynamics, concerning the growth of sizes of coordinates of orbits, greatest common divisors among coordinates, and prime factors of coordinates.  These problems can be restated and generalized in terms of (local/global) height functions, and I proved estimates on asymptotic behavior of height functions along orbits assuming Vojta’s conjecture.  As corollaries, I showed that Vojta’s conjecture implies the Dynamical Lang-Siegel conjecture for projective spaces (the sizes of coordinates grow in the same speed) and existence of primitive prime divisors in higher dimensional settings.

Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for Zoom information.