Abstract: In this talk, we consider the finite field theta correspondence between principle series. Joint with Jiajun Ma and Congling Qiu, we explicitly describe this correspondence by analyzing the relevant Hecke algebra bimodules and applying a deformation argument. Joint with Jiajun Ma and Congling Qiu, Zhiwei Yun, we geometrized the whole picture. Consequently, we obtained a relation between the Springer correspondence and theta correspondence.

]]>Since the 1920's, it has been known that the spatially homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes generically develop curvature singularities in the contracting time direction along spacelike hypersurfaces, known as big bang singularities, both in vacuum and for a wide range of matter models. For many years, it remained unclear if the big bang singularities were physically relevant. It was thought by some that big bang singularities were due to the unphysical assumption of spatial homogeneity and that they would disappear in non-homogenous spacetimes, or in other words, big bang singularities were unstable under nonlinear perturbations as solutions to the Einstein field equations. A partial resolution to this situation came in 1967 when Hawking established his singularity theorem that guarantees a cosmological spacetime will be geodesically incomplete for a large class of matter models and initial data sets, including highly anisotropic ones. While Hawking's singularity theorem guarantees that cosmological spacetimes are geodesically incomplete (i.e. at least one observer will experience something pathological at a finite time in the past) for a large class of initial data sets, it is silent on the cause of the geodesic incompleteness. It has been widely anticipated that the geodesics incompleteness is due to the formation of curvature singularities, and it is an outstanding problem in mathematical cosmology to rigorously establish the conditions under which this expectation is true and to understand the dynamical behaviour of cosmological solutions near singularities.

In this talk, I will begin by introducing the FLRW and Kasner solutions of the Einstein-scalar field equations, which are exact, spatially homogeneous solutions that play a distinguished role in the analysis of big bang singularities. After briefly providing context for the FLRW and Kasner solutions in the historical development of the field of cosmology,

I will define what it means for a FLRW/Kasner big bang singularity to be stable. With this notion in hand, I will then discuss the recent influential FLRW and Kasner big bang stability proofs of Rodnianski-Speck and Fournodavlos-Rodnianski-Speck. One aspect of these stability results that I will pay particular attention to is their global nature. To conclude the talk, I will discuss some recent work done in collaboration with Florian Beyer where we improve the Rodnianski-Speck FLRW big bang stability result by establishing that the FRLW big bang is locally stable, which is a significantly stronger notion of stability with important physical consequences that I will briefly discuss. Time permitting, I will also briefly discuss more recent work with Florian Beyer on the stability of big bang singularities when relativistic fluids are present.

We first present a combinatorial bijective proof of the Vandermonde determinant. Then, we present a combinatorial bijective proof of the Cauchy determinant. Finally, we show that the Cauchy identity is an easy corollary of the Cauchy determinant.

]]>In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number.

In my talk, I will discuss this root number correlation bias when the average is taken over all weight k modular newforms. I will point to a source of this phenomenon in this case and compute the correlation function exactly.

One of the major questions over a field of mixed characteristics is, "Is there a divisor D such that the pair (R, D) is BCM-regular when R is a splinter?" Inspired by this, we introduce compatible systems of divisors on finite covers of R, which have data of divisors on all finite covers of the ring. With this, we will define +-test/adjoint ideals for the system of divisors, and discuss the behavior of the +-test ideals including the inverse of adjunction. This is a joint work with Karl Schwede.

]]>Abstract: In this talk I will discuss a categorification of integral generic Hecke algebras using motivic sheaves on affine flag varieties. This eliminates the dependence on the chosen cohomology theory, such as l-adic or Betti cohomology, in previous works. At the unramified level this amounts to a motivic version of the geometric Satake equivalence. On the other extreme, at the Iwahori level we construct a motivic version of Gaitsgory's central functor. This is joint work with T. van den Hove and J. Scholbach.

]]>Non-abelian Hodge theory relates topological and algebro-geometric objects associated to a compact Riemann surface. More precisely, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This yields a transcendental matching between two very different moduli spaces: the character variety (parametrizing representations of the fundamental group) and the Hitchin moduli space (parametrizing Higgs bundles). In 2010, de Cataldo, Hausel, and Migliorini proposed the P=W conjecture, which links precisely the topology of the Hitchin integrable system and the Hodge theory of the character variety. I will introduce the conjecture, review its recent proofs, and discuss how the geometry hidden behind the P=W phenomenon is connected to other branches of mathematics.

]]>Free energy is an important quantity in the study of spin glasses. Results for many spin models show that free energy, as a random variable, behaves differently in low versus high temperature regimes. In this talk, we discuss the low temperature regime. In particular, we will consider an upper bound argument for the order of fluctuations of free energy in the Ising SK model.

]]>Certain classes of groups, such as the special linear groups over integers, satisfy a (co)homological duality property that allows one to study their rational cohomology groups. I'll discuss a criterion to test when a group satisfies this duality, and see how in the case of SLnZ this comes down to the fact that the Tits building is homotopy equivalent to a wedge of spheres.

]]>The main goal of this talk is to introduce a method to compute the canonical form of a convex polytope by computing the volume of its polar dual polytope. We will first define the notion of a dual polytope and then compute the volume in a concrete example. Then we will give a proof that the dual volume actually gives the canonical form of the original convex polytope. Finally, we will introduce a more efficient way to compute the dual volume when the polytope has many vertices but few sides, via the Filliman Duality, which roughly says the volume of the polytope can be obtained from triangulating its dual.

]]>(Co)homology groups are important algebraic invariants associated to a group G. In this talk we will look at how to define group (co)homology algebraically and topologically, prove the equivalence of the two definitions, and briefly see examples where having both perspectives can help study a group various finiteness properties and duality properties of a group.

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]]>Given a proper algebraic map f : X--> Y, the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber provides powerful tools to the study of its topology. This endows the cohomology of X with an extra structure, known as the perverse filtration, which measures the singularities of the map f. In recent years, the decomposition theorem and the induced perverse filtration have been found to share surprising connections to other branches of mathematics; these include non-abelian Hodge theory (the P=W conjecture), enumerative geometry (Donaldson-Thomas and BPS invariants), planar singularities (DAHA, knot invariants), and hyper-Kähler geometries (Hodge modules, motivic techniques). In this lecture series, I will discuss some of these developments. If time permits, open questions will be presented and discussed.

]]>Given a proper algebraic map f : X--> Y, the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber provides powerful tools to the study of its topology. This endows the cohomology of X with an extra structure, known as the perverse filtration, which measures the singularities of the map f. In recent years, the decomposition theorem and the induced perverse filtration have been found to share surprising connections to other branches of mathematics; these include non-abelian Hodge theory (the P=W conjecture), enumerative geometry (Donaldson-Thomas and BPS invariants), planar singularities (DAHA, knot invariants), and hyper-Kähler geometries (Hodge modules, motivic techniques). In this lecture series, I will discuss some of these developments. If time permits, open questions will be presented and discussed.

]]>Heegaard Floer Homology is a useful invariant for three manifolds, and it has many variants that can be used to study four dimensional cobordism and knots in three spheres. In this talk I will first introduce Heegaard decomposition and Heegaard diagram and state some basic properties of them. Then, I will make an analogy to the Lagrangian Floer Homology to define the ingredients of the Heegaard Floer Homology. Finally, I will define the hat version of Heegaard Floer chain complex and homology.

]]>We show that every Schubert polynomial has a decomposition as a sum of polynomials, each of which is given by a tableau formula. This corresponds to a decomposition of its set of bumpless pipe dreams into a disjoint union of classes we call drift classes. (The vexillary case, studied in the last two lectures, is the case when there is one drift class.)

]]>We will be discussing Getz-Hahn's "Introduction to Automorphic Representations."

]]>In this talk, we consider an expansion of a dense monolayer of cells: a collective multicellular phenomenon, where cells divide, grow, and maintain contacts with their neighbors. During migration, cells display complex behavior, adjusting both their division rate and their growth after division to the local mechanical environment. Experimental observations show that cells near the edge of the expanding monolayer are larger and move faster than cells deep inside the colony. To explain these observations and describe cell migration patterns, we formulate a spatio-temporal theoretical model for multicellular dynamics in terms of the cell area distribution; the model includes cell growth after division and effective pressure. Numerical simulations of the model predict both the speed of invasion and the width of the outer proliferative rim; these predictions are in a good agreement with experimental observations. Theoretical analysis yields the equation for density of cells and reveals a novel type of propagating front with compact support. The velocity of front propagation (monolayer expansion) is obtained analytically and its dependence on all the relevant parameters is determined.

]]>It is not hard to appreciate the beauty of singularities when you see their pictures. In this talk, we will describe important invariants of isolated hypersurface singularities, notably the mixed Hodge structure of the Milnor fiber and its corresponding spectrum, with an application to bounding the number of ordinary double points on projective hypersurfaces of fixed dimension and degree.

]]>Given a positive Laplace eigenfunction on a hyperbolic manifold, there exists a horospherically invariant measure corresponding to it, known as the Burger-Roblin measure. A question attributed to Babillot concerns whether every horospherically invariant measure arises from such a correspondence. In this talk, I will survey previous works where affirmative answers to this question were found and present a new result extending it to a broad class of subgroups in rank one Lie groups. This is joint work with Or Landesberg, Elon Lindenstrauss, and Hee Oh.

]]>Given a proper algebraic map f : X--> Y, the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber provides powerful tools to the study of its topology. This endows the cohomology of X with an extra structure, known as the perverse filtration, which measures the singularities of the map f. In recent years, the decomposition theorem and the induced perverse filtration have been found to share surprising connections to other branches of mathematics; these include non-abelian Hodge theory (the P=W conjecture), enumerative geometry (Donaldson-Thomas and BPS invariants), planar singularities (DAHA, knot invariants), and hyper-Kähler geometries (Hodge modules, motivic techniques). In this lecture series, I will discuss some of these developments. If time permits, open questions will be presented and discussed.

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]]>A central question in the field of optimal transport studies optimization problems involving two measures on a common metric space, a source and a target. The goal is to find a mapping from the source to the target, in a way that minimizes distances. A remarkable fact discovered by Caffarelli is that, in some specific cases of interest, the optimal transport maps on a Euclidean metric space are Lipschitz. Lipschitz regularity is a desirable property because it allows for the transfer of analytic properties between measures. This perspective has proven to be widely influential, with applications extending beyond the field of optimal transport.

In this talk, we will further explore transport maps with low distortion. The key point which we shall highlight is that, for low distortion mappings, the optimality conditions mentioned above do not play a major role. Instead of minimizing distances, we will consider a general construction of transport maps based on probabilistic flows, and introduce a set of techniques to analyze their distortion. In particular, we will go beyond the Euclidean setting and consider Riemannian manifolds as well as infinite-dimensional spaces.

We shall also discuss the emerging and intimate connections between our construction and recent advances in algorithms for generative modeling.

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]]>Existing algebraic multigrid (AMG) methods rely on assumptions about the near-kernel components of a given linear system. Namely, that these components are "smooth" in the sense that they can be sufficiently approximated by few degrees of freedom. PDEs with higher order terms violate these assumptions, causing an unbounded number of $V$-cycles for convergence. As an example, we introduce a PDE that arises in kinetic-edge plasma simulation. This PDE contains an isotropic fourth-order term, making existing methods infeasible. In this work, we propose an $O(n)$ highly-parallelizable exact method to solve the system solely containing the isotropic fourth-order term. We then extend this algorithm to solve the original system, including periodic boundary conditions. Our algorithm obtains drastic improvement over existing methods.

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]]>Mean field games (MFGs) study strategic decision-making in large populations where individual players interact via specific mean-field quantities. They have recently gained enormous popularity as powerful research tools with vast applications. For example, the Nash equilibrium of MFGs forms a pair of PDEs, which connects and extends variational optimal transport problems. This talk will present recent progress in this direction, focusing on computational MFG and engineering applications in robotics path planning, pandemics control, and Bayesian/AI sampling algorithms. This is based on joint work with the MURI team led by Stanley Osher (UCLA).

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