# 세미나 및 콜로퀴엄

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Imaginary geometry [Miller-Sheffield '16] provides a coupling between Schramm-Loewner evolutions (SLE) and Gaussian free field (GFF), and even can be combined with Liouville quantum gravity (LQG) through mating of trees [Duplantier-Miller-Sheffield '18]. This talk gives a brief survey on this program. Then we suggest a construction of north-going flow line in imaginary geometry from alternating west and east-going flow lines, using an excursion theory for planar Brownian motions. This leads a convergence of multiple trees in peanosphere which has been employed in specific settings [Gwynne-Holden-Sun '16, Li-Sun-Watson '17]. Joint work with E. Gwynne, N. Holden, X. Sun, and S. Watson.

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E6-1, ROOM 3434
Discrete Math
Tillmann Miltzow (Université Libre de Bruxelles, Brussels)
The Art Gallery Problem is ∃R-complete

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p∈P is seen by at least one guard g∈G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃R. The class ∃R consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP⊆∃R. We prove that the art gallery problem is ∃R-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃R. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many geometric approaches to the problem. This is joint work with Mikkel Abrahamsen and Anna Adamaszek.

The standard action of a torus $T := (\mathbb{C}^{\ast})^n$ on the complex vector space $\mathbb{C}^n$ induces an action of $T$ on the flag variety $\mathcal{F}\ell(\mathbb{C}^n)$. The Schubert variety $X_w$ associated to a permutation $w \in \mathfrak{S}_n$ admits the action of $T$. In this talk, we study the closure $Y_w$ of a generic $T$-orbit in the Schubert variety $X_w$. We associate a graph $\Gamma_w(u)$ to each $u \leq w$, and show that $Y_w$ is smooth at the fixed point $uB$ if and only if the graph $\Gamma_w(u)$ is acyclic. This is joint work with Mikiya Masuda.

There have been many exciting developments on the arithmetic of Shimura varieties in the recent decade, which have contributed to better understandings of the (still largely conjectural) Langlands programme. In this talk, I’ll try to give an overview of recent developments in mod p reduction of Shimura varieties (in both good reduction and bad reduction cases), and conclude the talk by introducing my joint work with Paul Hamacher on Mantovan’s formula (a cohomological formula conjecturally encoding the local-global compatibility of Langlands correspondences).

We introduce a new method to derive a lower bound of the mean curvature for the solutions of inverse mean curvature flow. This estimate for IMCF, an expanding curvature flow, corresponds to interior curvature estimates of Ecker-Huisken and Caffarelli-Nirenberg-Spruck for shrinking curvature flow and elliptic problem. It yields the smoothness of solutions and the existence of the flow for complete non-compact convex initial hypersurfaces.

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that also adaptively finds the number of the underlying communities. Central to our method is fused l1 penalty applied on an induced graph from the given data.

The classical Grunwald–Wang theorem is an example of a local–global principle stating that except in some special cases which are precisely determined, an element m in a number field K is an a-th power in K if and only if it is an a-th power in the completion K℘ of K for all but finitely many primes ℘ of K. In this talk, based on the analogue between the power map and the Carlitz module, I introduce an analogue of the Grundwald-Wang theorem in the Cartlitz module setting.

The original version of Waring's problem asks whether, for every positive integer n, there exists M(n) such that every non-negative integer is of the form a_1^n + ......+ a_{M(n)}^n, where the $a_i$ are non-negative integer. Since 1909, Hilbert proved that such a bound exists. In this talk, I introduce an analogue of Waring's problem for an algebraic group G, which is a generalization of the original Waring's Problem in the algebraic-group setting. (Joint work with Michael Larsen)

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

We show that for any connected sum of lens spaces L there exists a connected sum of lens spaces X such that X is rational homology cobordant to L and if Y is rational homology cobordant to X, then there is an injection from H_1(L; Z) to H_1(Y; Z). Moreover, as a connected sum of lens spaces, X is uniquely determined up to orientation preserving diffeomorphism. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.

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산업경영학동(E2) Room 3221
확률론
남동훈 (Princeton University)
Cutoff phenomenon for the Swendsen-Wang dynamics

The Swendsen-Wang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once, as opposed to the classical single-site Glauber dynamics. Although widely used in practice due to efficiency, the mixing time of the Swendsen-Wang dynamics is far from being well-understood, mainly because of its non-local behavior. In this talk, we prove cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from “unmixed” to “well-mixed.” The proof combines two earlier methods of proving cutoff, the update support [Lubetzky-Sly ’13] and information percolation [Lubetzky-Sly ’16], to establish cutoff in a non-local dynamics. Joint work with Allan Sly.

In this talk we shall try to give a survey of Voevodsky's smash nilpotence conjecture. Considerable time will be spent on explaining the statement of the conjecture, and a result of Kimura which is very fundamental. After this we shall see some examples where the conjecture holds, the work of Kahn-Sebastian, Sebastian, Laterveer, Vial and others.

We discuss the number of proper colorings of hypercubes

given q colors. When q=2, it is easy to see that there are only 2

possible colorings. However, it is already highly nontrivial to figure

out the number of colorings when q=3. Since Galvin (2002) proved the

asymptotics of the number of 3-colorings, the rest cases remained open

so far. In this talk, I will introduce a recent work on the number of

4-colorings, mainly focusing on how entropy can be used in counting.

This is joint work with Jeff Kahn.

From Huygens' observation on two synchronous pendulum clocks in the middle of 17th century, its rigorous studies have been started only in several decades ago by two pioneers Winfree and Kuramoto. The Kuramoto model is extensively studied because of its good properties, such as the gradient structure. In this talk, we look over the Kuramoto's conjecture and sychronization results of this model in the particle (ODE) and kinetic (PDE) descriptions. The main concern is to present a sufficient condition which leads to the synchronization phenomena. We will see how Lyapunov functional approach works on the particle-path analysis. Three related topics will be suggested, the model in the higher dimensions, on the random environment, and for the phase frustrated interactions.

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E2 Room 3434
Discrete Math
Eric Vigoda (Georgia Institute of Technology)
Learning a graph via random colorings

For an unknown graph G on n vertices, given random k-colorings of G, can one learn the edges of G? We present results on identifiability/non-identifiability of the graph G and efficient algorithms for learning G. The results have interesting connections to statistical physics phase transitions.

This is joint work with Antonio Blanca, Zongchen Chen, and Daniel Stefankovic.

The conjecture of Prasanna-Venkatesh predicts that the rational cohomology ring of an arithmetic manifold has special endomorphisms of a motivic origin. Their motivic nature allows one to test the conjecture indirectly through various regulator maps, but the source of these endomorphisms remains mysterious. We propose that the discrete Hodge star operator supplies the predicted endomorphisms in the case when the arithmetic manifold in question is uniformized by the upper half-space. In order to justify the proposal, we analyze how two different rational structures in cohomology interact with the discrete and geometric Hodge star operators, and show that the difference between them is measured by a special value of an $L$-function. Numerical examples will be given to illustrate the result.

One can simulate harmonic analysis on a Riemannian manifold by replacing the de Rham complex with the cochain complex of a triangulated manifold, following the ideas that go back at least to A. Whitney and D. Sullivan, and more recently to S. Wilson. As a result, one obtains the discrete Hodge star operator acting on simplicial cochains, which is an analogue of the usual Hodge star operator acting on differential forms. We will show that the discrete Hodge star operator is a topological invariant a 3-manifold; its action on the cohomology is determined by the underlying manifold together with its orientation, independently of the choice of a triangulation. Like the usual geometric Hodge star operator, it commutes with correspondences, and it is compatible with the Poincare duality. Furthermore, it induces a canonical positive definite bilinear pairing on the singular cohomology.

Toric orbifolds are topological generalization of projective toric varieties. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure an invariant CW-structure of the toric orbifold. In this talk I will discuss 3 different equivariant cohomology theories of toric orbifolds. This is a joint work with V. Uma.

In this presentation, we shall analyze random processes exhibiting metastable /tunneling behaviors among several metastable valleys. Such behaviors can be described by a Markov chain after a suitable rescaling. We will focus on three models: random walks in a potential field, condensing zero-range processes, and metastable diffusion processes.

Fock and Goncharov (2006) introduced the notion of positive framed PGL(n,R) representations. In this talk we exhibit framed PGL(3,R) representations of the 3-holed sphere group that are "negative" in a certain sense. If we require the boundary holonomies be all quasi-unipotent, then the boundary-embedded and transversal representations in the corresponding relative character variety form an open subset. These examples may be called "relatively Anosov" and properly include the Pappus representations studied by R. Schwartz (1993). If we further restrict to a certain real 1-dimensional subvariety consisting of representations with 2-fold symmetry, then we obtain a PGL(3,R) analogue of the Goldman-Parker conjecture (solved by R. Schwartz in 2001) on the ideal triangle reflection groups in PU(2,1). Joint work in progress with Sungwoon Kim.

The main interest of this talk is the behavior of Selmer groups of families of twists of elliptic curves. Mazur and Rubin show that there are infinitely many quadratic twists of arbitrary 2-Selmer ranks, under the some conditions on the given elliptic curve.

In this talk, I will introduce a cubic analogue of this result. A naive cubic analogue of the result of Mazur--Rubin does not hold, since all elliptic curves in our settings have ``constant 3-Selmer parity''. I will explain why this problem happens, and how we can manage it.

Formal orbifolds are normal varieties $X$ over perfect fields with a branch data $P$ which encodes compatible system of finite Galois extensions of function fields of formal neighbourhoods of points of $X$. I will introduce these objects and demonstrate how these objects can be used to study (wild) ramification theory in an organised way. In particular I will define etale site, fundamental group, etc. of formal orbifolds. I will discuss a reasonable formulation of Lefschetz theorem for fundamental group of quasi-projective varieties over fields of positive characteristic in the language of formal orbifolds. Time permitting some partial results in this direction will also be stated.

The non-symplectic index of an algebraic K3 surface is the order of the image of the representation of the automorphism group of the K3 surface on the global two forms. If the base field is the complex field, the non-symplectic index is finite and its Euler phi value is at most 20. In this talk, we will see if the base field is of odd characteristic and the K3 surface is of finite height, we have a similar result through a lifting argument. Also we calculate the non-symplectic index of all supersingular K3 surfaces over a field of characteristic at least 5 using the crystalline Torelli theorem.