# 세미나 및 콜로퀴엄

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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.

The theory of error-correcting codes, also known as Coding Theory, was invented by R. Hamming and C. Shannon around 1948. Since then, we can communicate information or data reliably. It has been applied to satellite communication, mobile phone, compact disc, high definition TV, and Artificial Intelligence.

In this talk, we will give two other applications of error-correcting codes. One is a game based on the binary Hamming [7,4,3] code. The other is code-based cryptography based on product codes. We only assume graduate level Algebra.

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E6-1 Room 4415
정수론
김연수 (전남대학교)
Classification of discrete series representations of odd GSpin groups

The classification of discrete series is one important subject with numerous application in the harmonic analysis and in the theory of automorphic forms. Recently, with Ivan Mati ́c (University of Osijek, Croatia), we obtain a classification of discrete series of odd general spin groups, generalizing the Mœglin-Tadi ́c classification for classical groups. Our approach presents a simplified and uniform proof of a bijective correspondence between isomorphism classes of the non-cuspidal discrete series and the admissible triples, which mostly relies on purely algebraic methods, available in both classical and the odd general spin case. If time permits, I am going to explain what Mati ́c and I recently did for other connected reductive groups.

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E6-1, ROOM 3434
Discrete Math
Michael Dobbins (Binghamton University, Binghamton, NY, USA)
Grassmanians and Pseudosphere Arrangements

In this talk I will present a metric space of pseudosphere arrangements, as in the topological representation theorem of oriented matroids, where each pseudosphere is assigned a weight. This gives an extension of the space of full rank vector configurations of fixed size in a fixed dimension that has nicer combinatorial and topological properties. In rank 3 these spaces, modulo SO(3), are homotopy equivalent to Grassmanians, and the subspaces representing a fixed oriented matroid are contractible. Work on these spaces was partly motivated by combinatorial tools for working with vector bundles.

The Siegel series is defined to be a local factor of the Fourier coefficient of the Siegel-Eisenstein series. It is usually described as a polynomial, whose coefficients are known to be difficult to compute.

In this talk, I will explain a precise reformulation of the Siegel series using stratification of p-adic scheme, geometric description of each stratum, Grassmannian, and lattice counting argument. This would give a conceptual interpretation of each coefficient of the Siegel series.

If time permits, then I will discuss about ongoing work related to this topic.

The Dirichlet prime number theorem states that if two integers a and b are relatively prime, then there are infinitely many primes of the form an+b. It is a natural question after the Dirichlet theorem to find an upper bound for the smallest prime having the form an+b. Linnik answered the question, which is called 'Linnik problem' now. This problem can be stated in terms of a certain character, and so the Linnik problem is extended to a question on automorphic representations. In this talk, I will talk about application of Langalnds program to the Linnik problem for automorphic representations.

In this talk, we discuss about global $L^p$-gradient estimates and local piece-wise gradient H"{o}lder continuity results for elliptic equations from composite materials.

First we prove global $L^{p}$-gradient estimates for the weak solutions to linear elliptic equations under the assumption that the domain is composed of a finite number of disjoint Reifenberg flat boundaries while the coefficients have small BMO-semi norms in each subdomain.

Also we show local piece-wise gradient H"{o}lder continuity results for the weak solutions to nonlinear elliptic equations if the domain is composed of a finite number of disjoint $C^{1,alpha}$-boundaries and the nonlinearities are H"{o}lder continuous in each subdomain.

We decompose any object in the wrapped Fukaya category of a 2n-dimensional Weinstein manifold as a twisted complex built from the cocores of the n-dimensional handles in a Weinstein handle decomposition. If time permits, we will also discuss how to generalize this result to Weinstein sectors. This is joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Paolo Ghiggini.

We examine the role of international capital mobility in shaping the relation between economic growth and structural transformation. We build a small open economy Ramsey model with two goods, tradables and nontradables. We show that if the long-run autarky interest rate of a small open economy is higher than the world interest rate, the employment and value-added shares of the tradables sector will rise over time. In the opposite case, the shares will fall. Because the autarky interest rate increases with the rate of technological progress, our result suggests that cross-country differences in the rate of technological progress may be a significant factor in accounting for diverse patterns of structural changes among countries.

In the 1960’s, V.I. Arnold announced several fruitful conjectures in symplectic topology concerning the number of fixed point of a Hamiltonian diffeomorphism in both the absolute case (concerning periodic Hamiltonian orbits) and the relative case (concerning Hamiltonian chords on a Lagrangian submanifold). The strongest form of Arnold conjecture for a closed symplectic manifold (sometimes called the strong Arnold conjecture) says that the number of fixed points of a generic Hamiltonian diffeomorphism of a closed symplectic manifold X is greater or equal than the number of critical points of a Morse function on X. We will discuss the stable version of Arnold conjecture in both the absolute case and the relative case, which is closely related to the strong Arnold conjecture. This is joint work with Georgios Dimitroglou Rizell.

Mazur and Rubin introduced the so-called Selmer-companion curves in 2015. Let $E$ be an elliptic curve over a number field $K$. Suppose there is a function that sends a quadratic character of $K$ to the $p$-Selmer rank of $E$ twisted by that character. How much information of $E$ can be read off from the function? In this talk, we give a sketch of proof of a conjecture on $p$-Selmer near-companion posed by Mazur and Rubin when $p=2$. If time allows, we will discuss an application of the technique in the proof to the Shafarevich-Tate groups.

The Black-Scholes formula for option pricing is given by a solution of the Black-Scholes partial differential equation, which is of a parabolic type, and the Black-Scholes equation is converted into a diffusion equation via change of variables. Then the diffusion equation is solved by the Laplace transformation or the Fourier transformation, and in this talk we explain the similarity and difference of two approaches.

This is a gentle introduction to the Langlands program based on selected historical developments. In particular attention will be drawn to the birth of the Langlands program in a letter of Langlands to Weil in 1967. Time permitting a snapshot of some current developments will be given.

In this talk, we present certain rigidity results for group actions via geometric group theory. We will prove a topological version of Popa's measurable cocycle superrigidity theorem for full shifts. In the first part, we provide a new characterization of one end groups via continuous cocycle superrigidity of their full shifts. As a consequence, we have an application in continuous orbit equivalence rigidity. In the second part, we show that every Holder continuous cocycle for the full shifts of every finitely generated group G that has one end, undistorted elements and sub-exponential divergence function is rigidity. This is joint work with Yongle Jiang.

Every properly convex domain in RP^2 carries a pair consisting of a complete Riemannian metric and a holomorphic cubic differential satisfying certain PDE, given by the hyperbolic affine sphere in R^3 projecting to that domain. An intriguing object of study is the interaction between the flat geometry of the cubic differential and the projective geometry of the convex domain. We will explain a local version of a theorem of Dumas and Wolf, showing that if an open subset U of the convex domain is conformal to certain sector region in C with the cubic differential dz^3, then U gives rise to a line segment on the boundary of the convex domain.

Thurston's hyperbolic Dehn filling theorem states that if the

interior of a compact 3-manifold M with toral boundary admits a complete

finite volume hyperbolic structure, then all but finitely many Dehn

fillings on each boundary component of M yield 3-manifolds which admit

hyperbolic structures. In this talk, I will explain that although Dehn

filling is not possible in d-dimensional hyperbolic geometry for d > 3,

it is possible in the category of convex real projective d-orbifolds for

d = 4, 5, 6. Joint work with Suhyoung Choi and Ludovic Marquis.

This talk is about improving the efficiency of the optimization methods that is mainly characterized by their worst-case convergence rates. In particular, this talk will present how the proposed accelerated gradient methods, named optimized gradient method (OGM) and OGM-G, with the best-known worst-case convergence rates for smooth convex optimization are developed by optimizing the efficiency of the first-order methods in terms of the cost function and the norm of the gradient respectively. This is based on the performance estimation problem approach that casts the worst-case convergence rate analysis into a finite-dimensional semidefinite optimization problem. This talk will further discuss about extending the approach to more general classes of problems such as nonsmooth composite convex optimization problems and monotone inclusion problems.

In this talk, we will see that there is a unique global strong solution to the 2D Navier-Stokes system coupled with diffusive Fokker-Planck equation of a Hookean type potential. This system regards a polymeric fluid as a dilute suspension of polymers in an incompressible solvent, which is governed by the Navier-Stokes equation, and distribution of polymer configuration is governed by the Fokker-Planck equation, where spatial diffusion effects of polymers are also considered. Well-known Oldroyd-B models can be rewritten in the form of this system. We will discuss an appropriate notion for the solution for this multiscale system, and we will discuss how to find such solution.

In financial trading of assets such as stocks and bonds, many participants are interconnected by a complicated network of computer terminals, and it is necessary to give an ordering of events. In this talk we consider the influence of the special relativity theory on the task of the synchronization of events in a distributed system based on L. Lamport’s paper “Time, Clocks, and the Ordering of Events in a Distributed System”.

Many modern applications such as machine learning, inverse problems, and control require solving large-dimensional optimization problems. First-order methods such as a gradient method are widely used to solve such large-scale problems, since their computational cost per iteration mildly depends on the problem dimension. However, they suffer from slow convergence rates, compared to second-order methods such as Newton's method. Therefore, accelerating a gradient method has received a great interest in the optimization community, and this led to the development and extension of a conjugate gradient method, a heavy-ball method, and Nesterov's fast gradient method, which we review in this talk. This talk will then present new proposed accelerated gradient methods, named optimized gradient method (OGM) and OGM-G, that have the best known worst-case convergence rates for smooth convex optimization among any accelerated gradient methods.

I shall discuss the role of geometry in creating the space time that is fundamental to the physics of general relativity. I shall also discuss fundamental concepts such as mass, linear momentum and angular momentum in general relativity. The lack of continuous symmetries in general spacetime makes it difficult to define such quantities and I shall explain how the difficulty can be overcome by the works of Brown-York, Liu-Yau and Wang -Yau.