17 OCT 2024
14:00 - 16:00



https://www.arxiv.org/abs/2408.15014
Zoom room information:
https://us06web.zoom.us/j/81916335336?pwd=5zQT4lutMiBoY5Xp1pkFGtbqiaGozg.1
Meeting ID: 819 1633 5336
Passcode: 559618
Venue: Niavaran, Lecture Hall 1

16 OCT 2024
15:30 - 17:00

The second order elliptic partial differential equations have been studied for more than a century. The most important and basic example of such equations is the Laplace equation. It is a linear equation related to many topics in mathematics and Physics. The main property of such an equation is that solutions of Laplace equations are regular. Another important example of elliptic equations is the Monge-Ampere equation which is fully nonlinear. In contrast to the Laplace equation, the solutions of the Monge-Ampere equation may fail to be smooth. In this talk, we go over basic properties of Monge-Ampere equations. At last we will go over the celebrated results of Caffarelli on interior regularity of solutions of the Monge-Ampere equation.
Venue: Niavaran, Lecture Hall 1

16 OCT 2024
17:30 - 19:00

Tropical geometry is a combinatorial counterpart of algebraic geometry, transforming polynomials into piecewise linear functions and their solutions (varieties) into polyhedral fans. This transformation is intricately linked to the concept of Grobner bases, which provide a powerful tool in computational algebra. Specifically, all possible Grobner bases of an ideal are encoded within a polyhedral fan, with the tropical variety appearing as a subfan. Despite its significance, the computational complexity of tropical varieties often limits computations to small-scale instances. In this talk, we introduce a geometric approach that enables the effective computation of various points within tropical varieties. One application of this method is the computation of toric degenerations, which are important objects in algebraic geometry. These degenerations can be modeled on polytopes, and there exists a dictionary between their geometric properties and the combinatorial invariants of the corresponding polytopes. This dictionary can be extended from toric varieties to arbitrary varieties through toric degenerations.

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Meeting ID: 9086116889
Passcode: 362880
Venue: , (Zoom Lecture)

16 OCT 2024
14:00 - 15:00

As a landmark in probability theory, the Central Limit Theorem (CLT) asserts that the normalized sum of independent and identically distributed random variables with zero mean converges to a Gaussian random variable. Depending on the structure of the underlying distribution, this convergence can be formulated with respect to different topologies or metrics. To name two famous such CLTs, the Lindberg--Levy CLT states convergence in distribution assuming the finiteness of the second moment, and the Berry--Esseen CLT asserts the uniform convergence of cumulative distribution functions at the rate of $O(1/sqrt n)$ assuming the finiteness of the third absolute moment. In this talk, we focus on the CLT in terms of stronger metrics, namely the total variation distance and the relative entropy. In fact, we are interested in the quantum version of CLT in terms of such strong metrics. We prove that the convergence rate of the quantum CLT in terms of the trace distance is $O(1/sqrt{n})$, and in terms of the relative entropy is $O(1/n)$, assuming some conditions on the moments. In the talk we mostly explain the results and techniques in the classical case, to make it accessible for the wide range of audiences of the seminars, and at the end explain the techniques in the quantum case.

Zoom room information:
https://us06web.zoom.us/j/85237260136?pwd=MFSZoKdmRXAjfaSaBzbf19lTaaKglf.1
Meeting ID: 852 3726 0136
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

9 OCT 2024
14:00 - 15:00

Optimal transport is the problem of moving a mass of objects from an initial mass distribution to a final mass distribution with minimum cost. The input to the problem is the initial and final distributions, as well as the distance metric or cost of transportation between each initial position and each final position. We should match points in the initial mass with points in the final mass so as to minimize the total cost. This problem was first proposed in the context of economics and recently in the context of machine learning to compare and transform probability distributions. I shall mainly talk about these applications.
If I have time, in the second part of the lecture, I will talk about a new method for computing transportations between distributions in high dimensions. This approach is different from previous works in that it does not assume that the distributions are given explicitly. Rather, it assumes that we can just query or sample the distribution (through what is often called an oracle in the computer science literature.) This allows the method to work for inputs that may be exponentially larger than explicitly given inputs. The simple main technique behind our method is to change the components in the initial vector point one by one and as little as possible, while making sure we attain the final distribution. Our method can be turned into a specific algorithm for some parameterized classes of distributions and distance metrics. For each class, our method only guarantees that the transportation cost is not worse than the optimal transport of a worst-case instance among that class up to a constant multiplicative factor. This work is related to previous work on computational concentration of measure, which appeared in the context of adversarial machine learning. The second part of this talk is joint work with Salman Beigi, Mohammad Mahmoody, and Amir Najafi and is work in progress.
Zoom room information:
https://us06web.zoom.us/j/85237260136?pwd=MFSZoKdmRXAjfaSaBzbf19lTaaKglf.1
Meeting ID: 852 3726 0136
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

2 OCT 2024
14:00 - 15:00

‎A signed graph is a pair $G_{\sigma}=(G‎, \sigma)$‎, ‎where $G$ is a simple graph and $\sigma$‎, ‎called the signature‎, ‎is a function which assigns 1 or $-1$ to each edge of $G$‎. ‎We call $\sigma$ as an it orthogonal signature of $G$ if‎ $A(G_{\sigma})^2=D$‎, ‎where $A(G_{\sigma})$ is the signed adjacency matrix of $G_{\sigma}$ and $D$ is the diagonal matrix with vertex degrees on the diagonal‎.
‎We have shown that if $G$‎ has an orthogonal signature‎, ‎then $G$ should be regular‎.


‎Graphs with orthogonal signature were crucial in the proof of sensitivity conjecture by Huang in 2019‎. ‎Hunag used an orthogonal signature of hypercube in his proof‎. Alon and Zheng extended it from hypercubes to Cayley graphs of $\mathbb{Z}^{n}_2$‎. ‎Other researchers had considered signed graphs with just two distinct eigenvalues‎. ‎Note that the signed adjacency matrix of an orthogonal signature‎
‎of a $k$-regular graph has only two distinct eigenvalues which are $\pm \sqrt{k}$‎. ‎We also note the signed adjacency matrix of any signature‎ of a $k$-regular graph has spectral radius in the interval $[ \sqrt{k},k]$‎. ‎So an orthogonal signature has the minimum spectral radius among all signatures of a regular graph‎. Orthogonal signatures have been classified for $3$-regular and 4-regular graphs‎. ‎A partial answer for 5-regular graphs is given by Stanic‎.


‎We will discuss general constructions for graphs with orthogonal‎
‎signature and also the classification of orthogonal signatures of $k$-regular‎
‎graphs for a fixed $k$‎, ‎specially for $k = 5‎, ‎6$‎.
‎This is a joint work with Bojan Mohar and Maxwell Levit‎.





Zoom room information:


https://us06web.zoom.us/j/85237260136?pwd=MFSZoKdmRXAjfaSaBzbf19lTaaKglf.1


Meeting ID: 852 3726 0136


Passcode: 362880
Venue: Niavaran, Lecture Hall 1

25 SEP 2024
16:00 - 17:00

حدس الیوت در سال 1990 مطرح شد و پیش بینی می کرد که کلاس سی استار جبرهای جدایی پذیر یکدار ساده میانگین پذیر با استفاده از پایاهای K-نظریه قابل طبقه بندی هستند. این حدس بر پایه نتایجی بود که برای کلاسهای کوچکتر اثبات شده بود. این حدس برای کلاسهای بزرگی ثابت شده است و مثالهای نقضی در سالهای 2002 و 2008 برای آن پیدا شد. برنامه طبقه بندی الیوت که از زمینه های پژوهشی پویا و گسترده در جبر عملگرها می باشد، به بررسی این حدس و کاربردهای آن در جبر عملگرها، سیستم های دینامیکی و سایر شاخه های ریاضی می پردازد. در این سخنرانی، ضمن بیان این برنامه و نتایج بدست آمده، ارتباط و کاربردهای آن در برخی از شاخه های ریاضی را شرح می دهیم.
Venue: Niavaran, Lecture Hall 1