## Seminar information archive

Seminar information archive ～11/14｜Today's seminar 11/15 | Future seminars 11/16～

### 2015/04/22

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Construction of minimal skew products of amenable minimal dynamical systems

**Yuhei Suzuki**(Unvi. Tokyo)Construction of minimal skew products of amenable minimal dynamical systems

### 2015/04/21

#### Tuesday Seminar of Analysis

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Residue current techniques with application to a general theory of

linear delay-differential equations with constant coefficients (Japanese)

**Saiei Matsubara**(Graduate School of Mathematical Sciences, the University of Tokyo)Residue current techniques with application to a general theory of

linear delay-differential equations with constant coefficients (Japanese)

[ Abstract ]

We introduce the ring of differential operators with constant coefficients and commensurate time lags (we use the terminology D$\Delta$ operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary $D\Delta$ operators and observe that various function modules enjoy good cohomological properties over this ring. %After revising the notion of the residue current in the spirit of M. Andersson and E. Wulcan, we introduce the multidimensional version of the ring D$\Delta$ operators.

Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.

We introduce the ring of differential operators with constant coefficients and commensurate time lags (we use the terminology D$\Delta$ operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary $D\Delta$ operators and observe that various function modules enjoy good cohomological properties over this ring. %After revising the notion of the residue current in the spirit of M. Andersson and E. Wulcan, we introduce the multidimensional version of the ring D$\Delta$ operators.

Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

**Yoshikata Kida**(The University of Tokyo)Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

[ Abstract ]

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

#### Lie Groups and Representation Theory

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Norm computation and analytic continuation of vector valued holomorphic discrete series representations

(English)

**Ryosuke Nakahama**(the University of Tokyo, Department of Mathematical Sciences)Norm computation and analytic continuation of vector valued holomorphic discrete series representations

(English)

[ Abstract ]

The holomorphic discrete series representations is realized on the space of vector-valued holomorphic functions on the complex bounded symmetric domains. When the parameter is sufficiently large, then its norm is given by the converging integral, but when the parameter becomes small, then the integral does not converge. However, if once we compute the norm explicitly, then we can consider its analytic continuation, and can discuss its properties, such as unitarizability. In this talk we treat the results on explicit norm computation.

The holomorphic discrete series representations is realized on the space of vector-valued holomorphic functions on the complex bounded symmetric domains. When the parameter is sufficiently large, then its norm is given by the converging integral, but when the parameter becomes small, then the integral does not converge. However, if once we compute the norm explicitly, then we can consider its analytic continuation, and can discuss its properties, such as unitarizability. In this talk we treat the results on explicit norm computation.

### 2015/04/20

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Weighted Laplacians on real and complex complete metric measure spaces (Japanese)

**Akito Futaki**(The Univ. of Tokyo)Weighted Laplacians on real and complex complete metric measure spaces (Japanese)

[ Abstract ]

We compare the weighted Laplacians on real and complex (K¥"ahler) metric measure spaces. In the compact case K¥"ahler metric measure spaces are considered on Fano manifolds for the study of K¥"ahler Ricci solitons while real metric measure spaces are considered with Bakry-¥'Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.

We compare the weighted Laplacians on real and complex (K¥"ahler) metric measure spaces. In the compact case K¥"ahler metric measure spaces are considered on Fano manifolds for the study of K¥"ahler Ricci solitons while real metric measure spaces are considered with Bakry-¥'Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

TBA (JAPANESE)

**Tetsuya Hattori**(Faculty of Economics, Keio University)TBA (JAPANESE)

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Fano 5-folds with nef tangent bundles (日本語)

**Akihiro Kanemitsu**(University of Tokyo)Fano 5-folds with nef tangent bundles (日本語)

### 2015/04/17

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Mean dimension of the dynamical system of Brody curves (日本語)

**Masaki TSUKAMOTO**(Kyoto University)Mean dimension of the dynamical system of Brody curves (日本語)

[ Abstract ]

Mean dimension is a topological invariant of dynamical systems with infinite dimension and infinite entropy. Brody curves are Lipschitz entire holomorphic curves, and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. A key novel ingredient is an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss.

Mean dimension is a topological invariant of dynamical systems with infinite dimension and infinite entropy. Brody curves are Lipschitz entire holomorphic curves, and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. A key novel ingredient is an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss.

### 2015/04/15

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

On the tricategory of coordinate free conformal nets

(English)

**Juan Orendain**(UNAM/Univ. Tokyo)On the tricategory of coordinate free conformal nets

(English)

#### Mathematical Biology Seminar

14:55-16:40 Room #128演習室 (Graduate School of Math. Sci. Bldg.)

Mathematical modeling of life history and population dynamics: Effects of individual difference on carrying capacity in semelparous species (JAPANESE)

**Ryo Oizumi**(Graduate School of Mathematical Sciences, University of Tokyo)Mathematical modeling of life history and population dynamics: Effects of individual difference on carrying capacity in semelparous species (JAPANESE)

### 2015/04/14

#### Seminar on Mathematics for various disciplines

13:30-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

**Yosuke Hasegawa**#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Pin(2)-monopole invariants for 4-manifolds (JAPANESE)

**Nobuhiro Nakamura**(Gakushuin University)Pin(2)-monopole invariants for 4-manifolds (JAPANESE)

[ Abstract ]

The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations

which can be considered as a real version of the SW equations. A Pin(2)-mono

pole version of the Seiberg-Witten invariants is defined, and a special feature of

this is that the Pin(2)-monopole invariant can be nontrivial even when all of

the Donaldson and Seiberg-Witten invariants vanish. As an application, we

construct a new series of exotic 4-manifolds.

The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations

which can be considered as a real version of the SW equations. A Pin(2)-mono

pole version of the Seiberg-Witten invariants is defined, and a special feature of

this is that the Pin(2)-monopole invariant can be nontrivial even when all of

the Donaldson and Seiberg-Witten invariants vanish. As an application, we

construct a new series of exotic 4-manifolds.

#### Lie Groups and Representation Theory

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Visible actions of compact Lie groups on complex spherical varieties (English)

**Yuichiro Tanaka**(Institute of Mathematics for Industry, Kyushu University)Visible actions of compact Lie groups on complex spherical varieties (English)

[ Abstract ]

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds.

In this talk we consider visible actions of a compact real form U of a connected complex reductive algebraic group G on G-spherical varieties. Here a complex G-variety X is said to be spherical if a Borel subgroup of G has an open orbit on X. The sphericity implies the multiplicity-freeness property of the space of polynomials on X. Our main result gives an abstract proof for the visibility of U-actions. As a corollary, we obtain an alternative proof for the visibility of U-actions on linear multiplicity-free spaces, which was earlier proved by A. Sasaki (2009, 2011), and the visibility of U-actions on generalized flag varieties, earlier proved by Kobayashi (2007) and T- (2013, 2014).

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the theory of visible actions on complex manifolds.

In this talk we consider visible actions of a compact real form U of a connected complex reductive algebraic group G on G-spherical varieties. Here a complex G-variety X is said to be spherical if a Borel subgroup of G has an open orbit on X. The sphericity implies the multiplicity-freeness property of the space of polynomials on X. Our main result gives an abstract proof for the visibility of U-actions. As a corollary, we obtain an alternative proof for the visibility of U-actions on linear multiplicity-free spaces, which was earlier proved by A. Sasaki (2009, 2011), and the visibility of U-actions on generalized flag varieties, earlier proved by Kobayashi (2007) and T- (2013, 2014).

### 2015/04/13

#### Tokyo Probability Seminar

16:50-17:50 Room #128 (Graduate School of Math. Sci. Bldg.)

Modern Monte Carlo methods -- Some examples and open questions (ENGLISH)

**Hans Rudolf Kuensch**(ETH Zurich)Modern Monte Carlo methods -- Some examples and open questions (ENGLISH)

[ Abstract ]

Probability and statistics once had strong relations, but in recent years the two fields have moved into opposite directions. Despite this, I believe that both fields would profit if they continued to interact. Monte Carlo methods are one topic that is of interest to both probability and statistics: Statisticians use advanced Monte Carlo methods, and analyzing these methods is a challenge for probabilists. I will illustrate this, using as examples rare event estimation by sample splitting, approximate Bayesian computation and Monte Carlo filters.

Probability and statistics once had strong relations, but in recent years the two fields have moved into opposite directions. Despite this, I believe that both fields would profit if they continued to interact. Monte Carlo methods are one topic that is of interest to both probability and statistics: Statisticians use advanced Monte Carlo methods, and analyzing these methods is a challenge for probabilists. I will illustrate this, using as examples rare event estimation by sample splitting, approximate Bayesian computation and Monte Carlo filters.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Campana's Multiplicity and Integral Points on P^2 (English)

**Yu Yasufuku**(Nihon Univ.)Campana's Multiplicity and Integral Points on P^2 (English)

[ Abstract ]

We analyze when the complements of (possibly reducible) curves in P^2 have Zariski-dense integral points. The analysis utilizes the structure theories for affine surfaces based on logarithmic Kodaira dimension. When the log Kodaira dimension is one, an important role is played by Campana's multiplicity divisors for fibrations, but there are some subtleties. This is a joint work with Aaron Levin (Michigan State).

We analyze when the complements of (possibly reducible) curves in P^2 have Zariski-dense integral points. The analysis utilizes the structure theories for affine surfaces based on logarithmic Kodaira dimension. When the log Kodaira dimension is one, an important role is played by Campana's multiplicity divisors for fibrations, but there are some subtleties. This is a joint work with Aaron Levin (Michigan State).

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

An orbifold version of Miyaoka's semi-positivity theorem and applications (English)

**Frédéric Campana**(Université de Lorraine)An orbifold version of Miyaoka's semi-positivity theorem and applications (English)

[ Abstract ]

This `orbifold' version of Miyaoka's theorem says that if (X,D)

is a projective log-canonical pair with K_X+D pseudo-effective,

then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.

The definitions will be given. The original proof of Miyaoka, which

mixes

char 0 and char p>0 arguments could not be adapted. Our proof is in char

0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,

in which case the cotangent sheaf is the classical Log-cotangent sheaf:

if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line

bundle, then K_X+D is 'big' too. This implies, together with work of

Viehweg-Zuo,

the `hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a

non-uniruled smooth divisor in aprojective hyperkaehler manifold with

symplectic form s,

then its characteristic foliation is algebraic only if X is a K3 surface.

This was shown previously bt Hwang-Viehweg assuming D to be of general

type. This result has some further consequences.

This `orbifold' version of Miyaoka's theorem says that if (X,D)

is a projective log-canonical pair with K_X+D pseudo-effective,

then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.

The definitions will be given. The original proof of Miyaoka, which

mixes

char 0 and char p>0 arguments could not be adapted. Our proof is in char

0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,

in which case the cotangent sheaf is the classical Log-cotangent sheaf:

if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line

bundle, then K_X+D is 'big' too. This implies, together with work of

Viehweg-Zuo,

the `hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a

non-uniruled smooth divisor in aprojective hyperkaehler manifold with

symplectic form s,

then its characteristic foliation is algebraic only if X is a K3 surface.

This was shown previously bt Hwang-Viehweg assuming D to be of general

type. This result has some further consequences.

### 2015/04/10

#### Seminar on Probability and Statistics

14:50-16:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Principal Component Analysis of High Frequency Data (joint with Dacheng Xiu)

**Yacine Ait-Sahalia**(Princeton University)Principal Component Analysis of High Frequency Data (joint with Dacheng Xiu)

[ Abstract ]

We develop a methodology to conduct principal component analysis of high frequency financial data. The procedure involves estimation of realized eigenvalues, realized eigenvectors, and realized principal components and we provide the asymptotic distribution of these estimators. Empirically, we study the components of the constituents of Dow Jones Industrial Average Index, in a high frequency version, with jumps, of the Fama-French analysis. Our findings show that, excluding jump variation, three Brownian factors explain between 50 and 60% of continuous variation of the stock returns. Their explanatory power varies over time. During crises, the first principal component becomes increasingly dominant, explaining up to 70% of the variation on its own, a clear sign of systemic risk.

We develop a methodology to conduct principal component analysis of high frequency financial data. The procedure involves estimation of realized eigenvalues, realized eigenvectors, and realized principal components and we provide the asymptotic distribution of these estimators. Empirically, we study the components of the constituents of Dow Jones Industrial Average Index, in a high frequency version, with jumps, of the Fama-French analysis. Our findings show that, excluding jump variation, three Brownian factors explain between 50 and 60% of continuous variation of the stock returns. Their explanatory power varies over time. During crises, the first principal component becomes increasingly dominant, explaining up to 70% of the variation on its own, a clear sign of systemic risk.

### 2015/04/08

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

On treeable equivalence relations arising from the Baumslag-Solitar groups

(English)

**Yoshikata Kida**(Univ. Tokyo)On treeable equivalence relations arising from the Baumslag-Solitar groups

(English)

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Integrality of $p$-adic multiple zeta values and application to finite multiple zeta values.

(English)

**Seidai Yasuda**(Osaka University)Integrality of $p$-adic multiple zeta values and application to finite multiple zeta values.

(English)

[ Abstract ]

I will give a proof of an integrality of p-adic multiple zeta values. I would also like to explain how it can be applied to give an upper bound of the dimension of finite multiple zeta values.

I will give a proof of an integrality of p-adic multiple zeta values. I would also like to explain how it can be applied to give an upper bound of the dimension of finite multiple zeta values.

### 2015/04/07

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Potential functions for Grassmannians (JAPANESE)

**Kazushi Ueda**(The University of Tokyo)Potential functions for Grassmannians (JAPANESE)

[ Abstract ]

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

Potential functions are Floer-theoretic invariants

obtained by counting Maslov index 2 disks

with Lagrangian boundary conditions.

In the talk, we will discuss our joint work

with Yanki Lekili and Yuichi Nohara

on Lagrangian torus fibrations on the Grassmannian

of 2-planes in an n-space,

the potential functions of their Lagrangian torus fibers,

and their relation with mirror symmetry for Grassmannians.

#### Lie Groups and Representation Theory

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Branching laws and elliptic boundary value problems

(English)

**Bent Orsted**(Aarhus University)Branching laws and elliptic boundary value problems

(English)

[ Abstract ]

Classically the Poisson transform relates harmonic functions in the complex upper half plane to their boundary values on the real axis. In

some recent work by Caffarelli et al. some new generalizations of this appears in connection with the fractional Laplacian. In this lecture we

shall explain how the symmetry-breaking operators introduced by T. Kobayashi for studying branching laws may shed new light on the situation for elliptic boundary value problems. This is based on joint work with J. M\"o{}llers and G. Zhang.

Classically the Poisson transform relates harmonic functions in the complex upper half plane to their boundary values on the real axis. In

some recent work by Caffarelli et al. some new generalizations of this appears in connection with the fractional Laplacian. In this lecture we

shall explain how the symmetry-breaking operators introduced by T. Kobayashi for studying branching laws may shed new light on the situation for elliptic boundary value problems. This is based on joint work with J. M\"o{}llers and G. Zhang.

### 2015/04/06

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Analytic torsion for K3 surfaces with involution (Japanese)

**Ken-ichi Yoshikawa**(Kyoto Univ.)Analytic torsion for K3 surfaces with involution (Japanese)

[ Abstract ]

In 2004, I introduced a holomorphic torsion invariant for 2-elementary K3 surfaces, i.e., K3 surfaces with involution. In the talk, I will report a recent progress in this invariant. Namely, for all possible deformation types, the holomorphic torsion invariant viewed as a function on the moduli space, is expressed as the product of an explicit Borcherds lift and an explicit Siegel modular form. If time permits, I will interpret the result in terms of the BCOV invariant, i.e., the genus-one string amplitude in B-model, for Calabi-Yau threefolds of Borcea-Voisin. This is a joint work with Shouhei Ma.

In 2004, I introduced a holomorphic torsion invariant for 2-elementary K3 surfaces, i.e., K3 surfaces with involution. In the talk, I will report a recent progress in this invariant. Namely, for all possible deformation types, the holomorphic torsion invariant viewed as a function on the moduli space, is expressed as the product of an explicit Borcherds lift and an explicit Siegel modular form. If time permits, I will interpret the result in terms of the BCOV invariant, i.e., the genus-one string amplitude in B-model, for Calabi-Yau threefolds of Borcea-Voisin. This is a joint work with Shouhei Ma.

### 2015/03/24

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Knots and Mirror Symmetry (ENGLISH)

**Mina Aganagic**(University of California, Berkeley)Knots and Mirror Symmetry (ENGLISH)

[ Abstract ]

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.

#### Lie Groups and Representation Theory

18:00-19:30 Room #126 (Graduate School of Math. Sci. Bldg.)

A Gysin formula for Hall-Littlewood polynomials

**Piotr Pragacz**(Institute of Mathematics, Polish Academy of Sciences)A Gysin formula for Hall-Littlewood polynomials

[ Abstract ]

Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.

We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.

With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").

We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.

Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.

We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.

With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").

We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.

### 2015/03/20

#### Numerical Analysis Seminar

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Asymmetric Auctions (English)

**Gadi Fibich**(Tel Aviv University)Asymmetric Auctions (English)

[ Abstract ]

Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.

In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.

Joint work with A. Gavious and N. Gavish.

Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.

In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.

Joint work with A. Gavious and N. Gavish.

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