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# Analysis, Complex Geometry, and Mathematical Physics: A Conference in Honor of Duong H. Phong

### Conference Program

8:00AM-12:00PM Conference Registration in 307 Mathematics (Tuesday through Saturday)

We ask that all conference attendees please stop by our registration desk in 307 Mathematics to confirm their registration and collect their conference packets, including banquet tickets, campus map, conference schedule, local dining information, name badges, and reimbursement information.

All lectures take place in 312 Mathematics.

All coffee breaks take place in 508 Mathematics.

#### Tuesday

8:15AM-8:45AM Coffee and Bagels in 508 Mathematics
8:50AM-9:00AM Welcome Remarks
9:00AM-10:00AM Edward Witten (Institute for Advanced Study)
On The Work of Phong and D'Hoker On Superstring Perturbation Theory
view abstract
10:00AM-11:00AM Igor Krichever (Columbia University)
The universal Whitham hierarchy and geometry of moduli spaces of curves with punctures
view abstract
11:00AM-11:30AM Coffee Break in 508 Mathematics
11:30AM-12:30PM Eric D'Hoker (University of California, Los Angeles)
Superstring Perturbation Theory at Two Loops
view abstract
12:30PM-2:10PM Lunch
2:15PM-3:15PM Andrei Okounkov (Columbia University)
Quantum groups and quantum cohomology
view abstract
3:15PM-4:00PM Group Photo near Mathematics Building and Coffee Break in 508 Mathematics
4:00PM-5:00PM Clément Hongler (Columbia University)
Planar Ising model: discrete and continuous structures
view abstract
5:00PM-6:00PM Zhiqin Lu (University of California, Irvine)
The essential spectrum of the Laplacian
view abstract
6:00PM-7:30PM Wine and Cheese Reception
508 Mathematics

#### Wednesday

8:15AM-8:55AM Coffee and Bagels in 508 Mathematics
9:00AM-10:00AM Elias M. Stein (Princeton University)
The development of some ideas of Phong in the theory of singular integral and pseudo-differential operators
10:00AM-11:00AM Allan Greenleaf (University of Rochester)
Is there a general theory of Fourier integral operators?
view abstract
11:00AM-11:30AM Coffee Break in 508 Mathematics
11:30AM-12:30PM Gunther Uhlmann (University of Washington)
Travel Time Tomography and Boundary Rigidity
view abstract
12:30PM-2:20PM Lunch
2:30PM-3:30PM Michael E. Taylor (University of North Carolina)
Toeplitz operators on uniformly rectifiable domains.
view abstract
3:30PM-4:00PM Coffee Break in 508 Mathematics
4:00PM-5:00PM Tien-Cuong Dinh (Université Pierre et Marie Curie)
Positive closed currents and dynamics of Henon maps in higher dimension
view abstract
5:00PM-6:00PM Tristan C. Collins (Columbia University)
The boundary of the Kahler cone
view abstract

#### Thursday

8:15AM-8:55AM Coffee and Bagels in 508 Mathematics
9:00AM-10:00AM Charles Fefferman (Princeton University)
Fitting a smooth function to data.
view abstract
10:00AM-11:00AM Zbigniew Błocki (Uniwersytet Jagielloński)
Hormander's d-bar-estimate, some generalizations and new applications
view abstract
11:00AM-11:30AM Coffee Break in 508 Mathematics
11:30AM-12:30PM Jacob Sturm (Rutgers University)
Some applications of the Bergman kernel expansion
view abstract
12:30PM-2:20PM Lunch
2:30PM-3:30PM Xu-Jia Wang (Australian National University)
Potential theory for nonlinear elliptic equations
view abstract
3:30PM-4:00PM Coffee Break in 508 Mathematics
4:00PM-5:00PM Vincent Guedj (Institut de Mathématiques de Toulouse)
Regularizing properties of the twisted Kaehler-Ricci flow
view abstract
5:00PM-6:00PM Yu Wang (Columbia University)
Small Perturbation Solutions of the Complex Monge-Ampère Equation
view abstract

#### Friday

8:15AM-8:55AM Coffee and Bagels in 508 Mathematics
9:00AM-10:00AM Neil S. Trudinger (Australian National University)
Weak continuity of nonlinear operators
view abstract
10:00AM-11:00AM Sławomir Kołodziej (Uniwersytet Jagielloński)
The complex Hessian equations
view abstract
11:00AM-11:30AM Coffee Break in 508 Mathematics
11:30AM-12:30PM Lei Ni (University of California, San Diego)
Entropy and Gauss curvature flow
view abstract
12:30PM-2:20PM Lunch
2:30PM-3:30PM Pengfei Guan (McGill University)
New curvature estimates for Weingarten equations
view abstract
3:30PM-4:00PM Coffee Break in 508 Mathematics
4:00PM-5:00PM Ahmed Zeriahi (Institut de Mathématiques de Toulouse)
Convergence of the normalized Kaehler-Ricci flow on Fano varieties
view abstract
On the Bubbling set of the Yang Mills flow
view abstract

#### Saturday

8:15AM-8:55AM Coffee and Bagels in 508 Mathematics
9:00AM-10:00AM Shing-Tung Yau (Harvard University)
On the pseudonorm project towards birational classification of algebraic varieties
view abstract
10:00AM-11:00AM Valentino Tosatti (Northwestern University)
The Chern-Ricci Flow
view abstract
11:00AM-11:30AM Coffee Break in 508 Mathematics
11:30AM-12:30PM Steve Zelditch (Northwestern University)
Complex geometry of Laplace eigenfunctions
view abstract
12:30PM-1:50PM Lunch
2:00PM-3:00PM Joseph J. Kohn (Princeton University)
Weakly Pseudoconvex CR Manifolds
view abstract
3:00PM-3:45PM Group Photo near Mathematics Building and Coffee Break in 508 Mathematics
3:45PM-4:45PM Mei-Chi Shaw (University of Notre Dame)
Non-closed range property for the Cauchy-Riemann operator in L2 on a Stein domain
view abstract
4:45PM-5:45PM Bernard Shiffman (Johns Hopkins University)
Critical points of random sections of holomorphic line bundles
view abstract
5:45PM-7:15PM Cocktail reception
Italian Academy, first floor, 1161 Amsterdam Avenue (between 116th and 118th streets).
7:15PM-10:00PM Banquet

#### Abstract:

We will present some new applications of the classical Hörmander's $L^2$-estimate for the $\bar\partial$-equation. Among them the Ohsawa-Takegoshi extension theorem with optimal constant, one-dimansional Suita conjecture, as well as Nazarov's approach to the Bourgain-Milman inequality in convex geometry.

#### Abstract:

Complex Henon maps in higher dimension are very interesting from the dynamical point of view. One may expect that several properties in dimension 2 still hold in any dimension. However most of known techniques break down in this general setting. Calculus with positive closed currents of bi-degree (p,p) is more tricky than calculus with bi-degree (1,1) currents which in dimension 2, are also of bi-dimension (1,1). I will explain recent progress in pluripotential theory which allow to obtain several dynamical properties of Henon maps, e.g. the equidistribution properties for generic orbits of varieties and for periodic saddle points. The talk is based on joint works with Nessim Sibony.

#### Abstract:

Let X=C^m(R^n) or W^m,p(R^n). Let f be a real-valued function defined on an arbitrary given subset E of R^n. How can we tell whether f extends to a function F in X? If such F exists, what can we say about it? Can we compute an F with close-to-least-possible norm in X assuming E finite? For X=C^m, these questions were answered in the last decade. This talk explains what we know for X=W^m,p. Joint work with Arie Israel and Kevin Luli.

#### Abstract:

The most commonly occurring Fourier integral operators (FIO) are associated with Lagrangian manifolds which are graphs of canonical transformations, for which there is a full functional calculus. These have many applications throughout analysis, such as spectral theory on Riemannian manifolds, normal forms for linear PDE and inversion formulas for Radon transforms. However, degenerate FIO are unavoidable, arising naturally in harmonic analysis, tomography and inverse problems. I will describe such operators, their relation to some of Phong's work, and address the question of whether a complete functional calculus of FIO is possible.

#### Abstract:

Let X be a compact Kaehler manifold. We show that the Kaehler-Ricci flow (as well as its twisted versions) can be run from an arbitrary positive closed current with zero Lelong numbers and immediately smoothes it. This is joint work with Ahmed Zeriahi.

#### Abstract:

Given a holomorphic vector bundle over a compact Kahler manifold, I will discuss how to identify the set of points where the Yang Mills flow becomes singular. In particular one sees that this set is uniquely determined by the isomorphism class of the bundle itself, and does not depend on any initial data. This work is joint with T.C. Collins.

#### Abstract:

This is joint work with S. Dinew. The L1 and C1 a priori estimates will be discussed, as well as the existence and stability theorems for the solutions of the complex Hessian equations in domains of Cn and on compact K ahler manifolds. The main result is the Calabi-Yau type theorem for Hessian equations.

#### Abstract:

The range of the Cauchy-Riemann operator for a domain in a complex manifold X is well understood if the complex manifold X is C^n or a Stein manifold. Recently, some progress has been made for the L^2 theory and the boundary regularity of the Cauchy-Riemann equations on product domains in complex manifolds. In this talk we will present an example of a pseudoconvex domain whose range of d-bar is not closed in L^2. The domain is Stein but the manifold X is not Stein. This is joint work with Debraj Chakrabarti.

#### Abstract:

We will discuss a series of joint works with Phong concerning applications of the Bergman kernel expansion to problems in Kahler geometry.

#### Abstract:

If $M$ is a compact Riemannian manifold, $D$ a first order elliptic differential operator (between sections of vectors ${\mathcal E}_j$) on $M$, and $\Omega\subset M$ an open subset with uniformly rectifiable boundary, then (under a few technical hypotheses), there is associated a projection ${\mathcal P}$ of $L^p(\partial\Omega, {\mathcal E}_0\otimes{\mathbb C}^\ell)$ onto the space of boundary values of Hardy spaces" $${\mathcal H}^p(\Omega, D)=\{ u\in C(\Omega, {\mathcal E}_0\otimes{\mathbb C}^\ell) : Du=0, {\mathcal N}(u)\in L^p(\partial\Omega)\},$$ where ${\mathcal N}(u)$ is the nontangential maximal function (and $p\in (1,\infty)$). Given also $\Phi\in C(\partial\Omega, M(\ell,{\mathbb C}))$, then we define the Toeplitz operator $T_\Phi f={\mathcal P}\Phi f$. We show such operators are Fredholm if $\Phi$ takes values in $GL(\ell,{\mathbb C})$, and discuss properties of the index. We also consider the more general case $$\Phi\in L^\infty\cap \text{vmo}(\partial \Omega, GL(\ell,{\mathbb C})),$$ obtaining simultaneously extensions to higher dimensions and to domains with rough boundary of index results of Brezis-Nirenberg. This is a descriptioin of joint work with S. Hofmann, I. Mitrea, and M. Mitrea.

#### Abstract:

We are concerned with the weak continuity of nonlinear operators acting on associated classes of subharmonic functions. Such results enable us to extend the operators as measures on non-smooth functions and can be the basis for an ensuing potential theory. Particular classical examples are the real and complex Monge –Ampere operators on convex and plurisubharmonic functions. The programme was initiated in collaboration with Xu-jia Wang in the late 1990s in the context of Hessian measures in Euclidean space, extending the Monge-Ampere measure of Aleksandrov. In particular we will report on recent developments related to mean curvature measure, with Qui-yi Dai and Xu-jia Wang, and the discovery of a new measure on Heisenberg groups, with Wei Zhang.

#### Abstract:

We will survey some recent results on the inverse problem of determining the index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometric inverse problem: Can we determine the Riemannian metric of a Riemannian manifold with boundary from the distance function between boundary points. This is the boundary rigidity problem. We will also consider the linearized problem which consists on determining a symmetric two tensor from its integral along geodesics.

#### Abstract:

Let X be a Fano manifold whose Mabuchi functional is proper. A deep result of Perelman-Tian-Zhu asserts that the normalized Kaehler-Ricci flow, starting from an arbitrary Kaehler form in c_1 (X), converges towards the unique Kaehler-Einstein metric on X. We will give an alternative proof of a weaker convergence result which applies to the broader context of (log)-Fano varieties. This is a joint work with R. Berman, S.Boucksom, P.Eyssidieux and V.Guedj.

#### Abstract:

We study the distribution of critical points of random sections on compact complex manifolds. We show how the topology and geometry of the manifold influences the asymptotic expansion of the expected number of critical points of random sections of high tensor powers of a positive line bundle L (joint work with Douglas and Zelditch). We also discuss the statistical properties of the excursion sets of the norms of random holomorphic sections. Our methods involve the multidimensional Kac-Rice formula for random vector-valued fields and the asymptotics of the Bergman kernel for holomorphic sections of powers of L.

#### Abstract:

Let $\Omega\subset {\mathbb C}^n$ be a domain with a smooth pseudoconvex boundary $b\Omega$ which is weakly pseudoconvex at $P\in b\Omega$ (i.e.\ the Levi form is not positive definite at $P$). Then the regularity of solutions of $\bar\partial\varphi=\alpha$ near $P$, where $\varphi$ is a $(0,q)$-form and $\alpha$ a $(0,q+1)$-form, depends on the behavior of the germs of complex analytic varieties of dimension $q$ through $P$. If $U$ is a neighborhood of $P$ regularity on $U\cap \overline \Omega$ is studied by means of subelliptic estimates of the `energy'' form $$Q(\varphi,\varphi)=\Vert\bar\partial \varphi\Vert^2 + \Vert\bar\partial^\ast \varphi\Vert^2 ,$$ defined on $(q,0)$-forms in $C^\infty(\overline \Omega)$ which are supported in $U\cap \overline \Omega$ and belong to the domain of $\bar\partial^\ast$. the subelliptic estimate holds if there exist constants $\varepsilon$ and $C$ such that $\Vert\varphi\Vert_\varepsilon^2\leq CQ(\varphi,\varphi)$, for all such forms, where the left hand side denotes the $\varepsilon$ Sobolev norm. The D'Angelo $q$-type of $P$ is the maximum order of contact that a complex $q$-dimensional variety through $P$ can have with $b\Omega$. The main theorem, due to Catlin, is that a subelliptic estimate holds at $P$ if and only if the D'Angelo type of $P$ is finite. In the case when $b\Omega$ is real analytic in a neighborhood of $P$ there is an equivalent necessary and sufficient condition for subellipticity, namely that the ideal type of $P$ is finite. This condition is expressed in terms of ideals of subelliptic multipliers, these consist of germs of functions at $P$. When the ideal $q$-type is infinite the above subelliptic estimate does not hold. This follows from the fact that infinite ideal $q$-type of $P$ is equivalent to the existence of a $q$-dimensional complex analytic variety through $P$ that is contained in $b\Omega$. Here we present an explicit construction of such a variety which gives insight into the relation between the D'Anglelo type and the ideal type. The study of these ideals has a direct application to the study of singularities of complex analytic varieties. Consider the variety $V\subset{\mathbb C}^{n-1}$ given by $h_1(z_1,\ldots, z_{n-1})=\cdots=h_m(z_1,\ldots, z_{n-1})=0$, where the $h_j$ are holomorphic functions that vanish at the origin. Let $\Omega\subset{\mathbb C}^n$ be a pseudoconvex domain defined by $z\in \Omega | r(z)<0$ where in a neighborhood of the origin we have $$r(z_1,\ldots, z_n)=Re(z_n)+\sum |h_j(z_1,\ldots, z_{n-1})|^2 .$$ The multiplier ideals at the origin are generated by ideals of germs of holomorphic functions which are invariants of $V$.

#### Abstract:

If M is a compact Riemannian manifold, D a ﬁrst order elliptic dif- ferential operator (between sections of vector bundles Ej ) on M , and Ω ⊂ M an open subset with uniformly rectiﬁable boundary, then (under a few technical hy- potheses), there is associated a projection P of Lp (∂Ω, E0 ⊗ Cl ) onto the space of boundary values of “Hardy spaces” Hp (Ω, D) = {u ∈ C(Ω, E0 ⊗ Cl ) : Du = 0, N (u) ∈ Lp (∂Ω)}, where N (u) is the nontangential maximal function (and p ∈ (1, ∞)). Given also Φ ∈ C(∂Ω, M (l, C)), then we deﬁne the Toeplitz operator TΦ f = PΦf . We show such operators are Fredholm if Φ takes values in Gl(l, C), and discuss properties of the index. We also consider the more general case Φ ∈ L∞ ∩ vmo(∂Ω, Gl(l, C)), obtaining simultaneously extensions to higher dimensions and to domains with rough boundary of index results of Brezis-Nirenberg. This is a description of joint work with S. Hofmann, I. Mitrea, and M. Mitrea.

#### Abstract:

We establish $C^2$ a priori estimate for convex hypersurfaces whose principal curvatures $\kappa=(\kappa_1,\cdots, \kappa_n)$ satisfying Weingarten curvature equation $\sigma_k(\kappa(X))=f(X,\nu(X))$. We also obtain such estimate for the admissible $2$-convex hypersurfaces in the case $k=2$. Our estimates resolve a longstanding problem in geometric fully nonlinear elliptic equations.

#### Abstract:

A real analytic Riemannian manifold $(M, g)$ admits a complexification $M_{\mathbb C}$ and a distinguished Kahler potential $\rho$ whose square root (known as the Grauert tube function) satisfies a homogeneous complex Monge-Ampere equation away from the real locus $M$. We analytically continue eigenfunctions $\phi_{\lambda}^{\mathbb C}$ of $\Delta_g$ to Grauert tubes and study their growth as $\lambda \to \infty$ and the complex geometry of their zero sets. When the geodesic flow is ergodic, the growth of $\phi_{\lambda}^{\mathbb C}$ is governed by the Grauert tube function. When the geodesic flow is completely integrable, there is another type of growth behavior in regions separated by anti-Stokes surfaces.

#### Abstract:

I will discuss the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci curvature. This is an evolution equation which coincides with the Ricci flow if the initial metric is Kahler, and was first studied by M.Gill. I will describe the maximal existence time for the flow in terms of the initial data, and then discuss the behavior of the flow on (mostly non-Kahler) complex surfaces as one approaches the maximal existence time. This is joint work with Ben Weinkove and Xiaokui Yang.

#### Abstract:

We will discuss a geometric characterization of classes of positive volume on the boundary of the Kahler cone of a compact Kahler manifold. The main technical ingredient in this work is a new theorem for extending certain classes of Kahler currents defined on submanifolds. This extension theorem relies on some ideas developed by Phong-Stein and Phong-Stein-Sturm in the study of singular integral operators. As an application, we will show that finite time singularities of the Kahler-Ricci flow always form along analytic subvarieties. This work is joint with Valentino Tosatti.

#### Abstract:

I will review the work of D. Phong and E. D'Hoker on superstring perturbation theory, adding a few comments at the end on contemporary developments.

#### Abstract:

In this talk I shall discuss how an entropy functional can be used to prove the convergence of the Gauss curvature flow. This is a joint work with Pengfei Guan at McGill.

#### Abstract:

The planar Ising model is exactly solvable,as was shown by Onsager. It exhibits many interesting algebraic, probabilistic and analytic discrete structures that have been investigated for many decades. It has been conjectured that its phase transition point, the Ising model converges to a continuous limit, and acquires conformal symmetry. This leads to continuous theories, such as Conformal Field Theory and Schramm-Loewner Evolutions, that possess beautiful and different structures, and yield spectacular results, in particular exact formulae. In this talk, I will explain some recent progress in the investigation of the Ising model and the corresponding continuous theories. In particular I will focus on their connections: formulating the lattice integrability in terms of discrete complex analysis,(one can develop tools to establish rigorously the existence and conformal symmetry of the continuous limit of the model. This approach also sheds a different light on the discrete and continuous theories, which yields alternative formulations of certain problems and new structures. Based on joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, K. Izyurov, F. Johansson Viklund, A. Kemppainen, K. Kytölä, D.H. Phong and S.Smirnov

#### Abstract:

The Newton potential theory is an important ingredient in the study of the Laplace equation. The potential theory for nonlinear elliptic equations has also been a subject of extensive study in many decades. For example, the pluripotential theory associated with the complex Monge-Ampère equation has been developed by Bedford, Taylor and others. Federer extended curvature measures from convex bodies to more general sets of positive reach. In this talk I will discuss recent developments on the potential theory associated with nonlinear elliptic equations including the k-Hessian equations and k-curvature equations.

#### Abstract:

In this talk, we discuss the regularity of the solutions $u$ to the complex Monge-Ampère equation

($i$$\partial$$\bar{\partial}$$u)^n = f dx that are uniformly close to quadratic polynomials. By study these solutions, we produce two regularity results for the complex Monge-Ampère equation. The first result is a Liouville-type theorem. It considers uniqueness of the global solutions on C$$^n$ with certain growth at infinity. The second result is a $C^{2,\alpha}$-estimate of $W^{2,p}$-solutions. It is a perturbation result of Blocki-Dinew's estimate of $W^{2,p}$-solutions.

#### Abstract:

The moduli spaces of algebraic curves with a pair of meromorphic differentials is central for the Hamiltonian theory of integrable systems, Whitham perturbation theory of soliton equations, quantum topological field theories, Sieberg-Witten solution of N=2 SUSY gauge theories and Laplacian growth problems. In the talk their applications to a study of geometry of moduli spaces of curves with punctures will be presented.

#### Abstract:

Counting rational curves in algebraic varieties is a source of many very deep mathematical structures. In particular, Nekrasov and Shatashvili proposed a general vision for how this field connects with quantum integrable systems. Some of these expectations were confirmed in our joint work with Davesh Maulik. The talk will be an introduction to this circle of ideas.

#### Abstract:

Progress in the understanding of superstring perturbation theory at two loop order will be briefly reviewed, and ongoing work towards the calculation of the vacuum energy in Calabi-Yau orbifold models with tree-level supersymmetry will be discussed.

#### Abstract:

We derive an exact implied volatility expansion for any model whose European call price can be expanded analytically around a Black-Scholes call price. Two examples of our framework are provided (i) exponential Lévy models and (ii) CEV-like models with local stochastic volatility and local stochastic jump-intensity.

#### Abstract:

"We consider a problem of optimal investment with intermediate consumption and random endowment in an incomplete semimartingale model of a financial market. We establish the key assertions of the utility maximization theory assuming that both primal and dual value functions are finite in the interiors of their domains as well as that random endowment at maturity can be dominated by the terminal value of a self-financing wealth process. In order to facilitate verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold."

#### Abstract:

We present a new method of pricing plain vanilla call and put options when the underlying asset returns follow a stochastic volatility process. The method is based on stochastic dominance insofar as it does not need any assumption on the utility function of a representative investor apart from risk aversion. This approachdevelops discrete time multiperiod reservation write and reservation purchase bounds on option prices. The bounds are evaluated recursively and the limiting forms of the bounds are found as time becomes continuous. We discuss the implications of this result on the pricing of volatility risk. Joint work with S. Perrakis.

#### Abstract:

It is well-known that short-rate interest rate models with log-normally distributed rates in continuous time are afflicted with divergences which result from infinite accumulation factors in a finite time (the Hogan-Weintraub singularity). Examples of such models are the Dothan model and the Black-Karasinski model. We show explicitly the appearance of this singularity in the Black, Derman, Toy model, which is the discrete time version of the Dothan model, in the limit of a very small time step. A novel singular behavior is shown to appear in the BDT model at large volatility, which is similar to a phase transition in condensed matter physics.

#### Abstract:

We develop an arbitrage pricing theory for liquidity risk and price impacts on fixed income markets. We define a liquidity term-structure of interest rates by hypothesizing that liquidity costs arise from the quantity impact of trading of bonds with different maturities on the interest rates and the associated risk-return premia. We derive no arbitrage conditions which gives a number of theoretical relation satisfied between the impact on risk premia and the volatility structure of the term structure and prices. We calculate the quantity impact of trading a zero-coupon on prices of zero-coupons of other maturities and represent this quantity as a supermartingale. We give conditions under which the market is complete, and show that the replication cost of an interest rate derivative is the solution of a quadratic backward stochastic differential equation. Joint work with Robert Jarrow.

#### Abstract:

Cheridito (Finance Stoch. 7: 533-553, 2003) studies a financial market that consists of a money market account and a risky asset driven by a fractional Brownian motion (fBm). It is shown that arbitrage pos-sibilities in such markets can be excluded by suitably restricting the class of allowable trading strategies. In this note, we show an analogous result in a multi-asset market where the discounted risky asset prices follow more general non-semimartingale models. In our framework, investors are allowed to trade between a risk-free asset and multiple risky assets by following simple trading strategies that require a minimal deterministic waiting time between any two trading dates. We present a condition on the discounted risky asset prices that guarantee absence of arbitrage in this setting. We give examples that satisfy our condition and study its invariance under certain transformations.

#### Abstract:

We present a comprehensive methodology and approach to tackle ordinary differential equations (ODE), partial differential equations (PDE), partial integro-differential equations (PIDE) and related topics analytically. These approaches are used in both physics and quantitative finance with mutual benefit, both theoretically and practically. In our presentation, we will show that these analytical methodologies are making both research in physics and research and its implementation in quantitative finance much more efficient and are critical to substantial advances in physics and quantitative finance, as well as assure a trading and risk optimization success across asset classes.

#### Abstract:

In the talk, I will introduce the Feynman-Kac Formula for SPDE driven by fractional Brownian motion, and show its long term behavior.

#### Abstract:

We study the problem of an investor who hires a fund manager to manage his wealth. The latter is paid by an incentive scheme based on the performance of the fund. Manager and investor have different risk aversions; the manager may invest in a financial market to form a portfolio optimal for his expected utility whereas the investor is free to choose the incentives -- taking only into account that the manager is paid enough to accept the managing contract. We discuss the problem of existence of optimal incentives in general semimartingale models and give an assertive answer for some classes of incentive schemes. This is joint work with Maxim Bichuch (Princeton University).

#### Abstract:

A hedge fund manager, who receives performance fees proportional to the fund’s profits, invests optimally for both the fund and his own account, as to maximize the expected power utility of personal wealth. If separate and constant investment opportunities are available for each account, it is optimal for the manager to hold a constant fraction of the fund in risky assets, which corresponds to an effective risk aversion between one and the manager’s own risk aversion. For the personal account, the optimal policy is to accumulate performance fees in safe assets, and invest remaining wealth in a constant portfolio corresponding to the manager’s risk aversion. Under the optimal policy, the manager’s welfare is the maximum between the welfare he would obtain from either keeping fees in safe assets only, or investing his personal wealth alone. This result is robust to correlation between investment opportunities in both accounts, suggesting that the manager does not tend to hedge exposure to the funds’ performance with personal investments.

#### Abstract:

This is the first comprehensive study of the SABR (Stochastic Alpha-Beta-Rho) model (Hagan et. al (2002)) on the pricing and hedging of interest rate caps. We implement several versions of the SABR interest rate model and analyze their respective pricing and hedging performance using two years of daily data with seven different strikes and ten different tenors on each trading day. In-sample and out-of-sample tests show that the fully stochastic version of the SABR model exhibits excellent pricing accuracy and more importantly, captures the dynamics of the volatility smile over time very well. This is further demonstrated through examining delta hedging performance based on the SABR model. Our hedging result indicates that the SABR model produces accurate hedge ratios that outperform those implied by the Black model.

#### Abstract:

I shall talk about some birational geometric invariants.

TBA