# Documents

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## Summarized Talks

### Constructive analysis

Delivered by Fengyang Wang on Wednesday November 1, 2017

Constructive mathematics, as the name would suggest, is centered on the philosophy that mathematical proofs should be able to be turned into algorithms. We will contextualize constructive approaches to analysis, roughly following Bridges and Vîţă. This talk has no formal prerequisites beyond an elementary understanding of the real numbers and the usual concept of completeness. In particular, no logical background is assumed; intuitionistic logic will be overviewed in the talk. We will finish with a discussion of the ramifications of completeness of the real numbers.

A summary of this talk is available here.

### Cantor Set and Dynamical Systems

Delivered by James Bai on Friday March 31, 2017

The talk will be begin on the cosnstruction of the most commonly used tenary Cantor set. The talk will then talk about the common properties of Cantor set and methods of evaluating the size of the set. Then, depending on time, a brief introduction will be given on the dynamical system and chaos.

A summary of this talk is available here.

### Universal Property of Quotients

Delivered by Lirong Yang on Friday March 17, 2017

In this talk, we generalize universal property of quotients (UPQ) into arbitrary categories. UPQs in algebra and topology and an introduction to categories will be given before the abstraction. As in the discovery of any universal properties, the existence of quotients in the category of sets and that of groups will be presented.

If you have not yet been exposed to group theory, please read Monoids and Groups for an introduction.

A summary of this talk is available here.

### General Secure Multi-Party Computation from any Linear Secret-Sharing Scheme

Delivered by Zihao Zhu on Friday February 17, 2017

As more and more sensitive data gets digitized, there is a need to ensure privacy and reliability of the data, especially in the face of adversarial parties who attempt to corrupt or unwanted access to sensitive secrets.

In many instances such as online gambling, bidding, and even Google's targeted advertisements, a client wants to be able to take inputs from multiple sources (for example, auction bids) and produce an output (for example, the highest bidder) without revealing any information about the other inputs. We will use such scenarios as well as more cryptography related ones in order to motivate Multi-Party Computation as a method to compute on encrypted data. With MPC, we will quickly see it's limitations with unsecure channels and first develop secret sharing schemes (specifically linear secret sharing schemes) such as Shamir's scheme, and soon after, verifiable secret sharing schemes.

We will introduce the different types of adversarial structures and explore both the robustness and limitations of secret sharing schemes against them.

Finally, we will show that all Linear Secret Sharing Schemes can be constructed to be verifiable. We will explore the consequences of this and discuss techniques in their construction.

Prereqs: Math136 used in proofs

A summary of this talk is available here.

### Ordinals

Delivered by Eddie Onochie on Friday February 10, 2017

In this talk we will talk about ordinals and the philosophy of infinity. We will define what ordinals are and how to construct them. We will also define transfinite recursion and use the axiom of choice to give meaning to the "cardinality" of a set.

A summary of this talk is available here.

### $q$-analogs

Delivered by Fengyang Wang on Friday February 3, 2017

Some of the most interesting results in combinatorics are generalizations of simple problems or theorems to problems or theorems parameterized by a parameter $q$ (often complex-valued). By taking the limit as $q\to1$, the simple problems can be recovered. Surprisingly, many of these $q$-analogs take similar forms, and are seen across a wide variety of problems that may initially seem unrelated.

Concepts from MATH 239 will be used. It is heavily recommended that people who have not taken MATH 239 or an equivalent course read some material on ordinary generating functions.

A summary of this talk is available here.

### Hypercomplex Numbers

Delivered by Fengyang Wang on Friday November 18, 2016

There are exactly three distinct two-dimensional unital algebras over the reals, up to isomorphism. Each of these algebras corresponds to a unique geometry, with applications. This talk will develop the concepts needed to understand two-dimensional algebras over the reals, starting from the definitions of key concepts. We will rediscover the familiar complex numbers and generalize its construction to find the other hypercomplex number systems. We will then prove the result that these are the unique hypercomplex number systems, up to isomorphism. Finally, we will discuss possible generalizations to $n$ dimensions. Please ensure that you have a good understanding of fundamental concepts of two-dimensional linear algebra.

We will use Catoni, F., Cannata, R., Catoni, V., & Zampetti, P. (2004) as a reference.

A summary of this talk is available here.

### Simplicial Homology

Delivered by Kai Rüsch on Friday November 11, 2016

How do we count how many holes a shape has? We can answer this question using homology groups, whose order's count the number of $n$-dimensional holes.

A summary of this talk is available here.

### Game Theory (Part 1)

Delivered by Koosha Totonchi on Friday November 4, 2016

A game is a “mathematical model between interacting decision makers” where each player must make choices based on a set of rules. Every individual in a game must also have a preferred reaction to any combination of actions taken by other agents. Game theory is about the study and application of these models. It involves various solution concepts and methods that can be employed to predict the outcomes of strategic engagements. This talk will introduce the major ideas in the field. There will be a focus on basic definitions, types of games, and how we can “solve games” using the Nash equilibrium. Hopefully we’ll get to “play” some ourselves. Towards the end, we can also review some neat unsolved problems in game theory that are very easy to understand, but prove really difficult to solve.

A summary of this talk is available here.

### Groups

Delivered by Fengyang Wang on Wednesday October 12, 2016

This talk will cover the basics of group theory. There are no official prerequisites for this talk, but MATH 145 and MATH 146 are an asset. The group theory part of the talk will mostly be based on Alekseev, 2004, specifically sections 1.1 (motivation), 1.2 (transformation groups), 1.5, 1.9, and 1.13 (various morphisms), and 1.11 (quotient groups).

Due to time constraints, it is inevitable that some content must be rushed. I will not go over proofs and derivations in full rigour, and I highly advise studying the reference material after the talk to catch up on what you may have missed. Please message me in advance if there are any other subjects in particular that you would like me to discuss.

A summary of this talk is available here.

## Documents

### Galois Theory

This is a reference document on Galois Theory by Akshay Tiwary and Fengyang Wang. It covers a subset of the material of PMATH 348.

### Lie Theory

This is a reference document on Lie Theory by Aidan Patterson. It covers a subset of the material of PMATH 763.

### Monoids and Groups

This is a reference document on Monoids and Groups by Fengyang Wang. It covers a subset of the material of PMATH 347.

### Ordinary Generating Functions

This is a reference document on Ordinary Generating Functions by Fengyang Wang. It covers a subset of the material of MATH 239, MATH 249, or CO 330.

### Quadratic Reciprocity

This is a reference document on Quadratic Reciprocity by Fengyang Wang. It covers a subset of the material of PMATH 340 or PMATH 440.

### Two-Dimensional Linear Algebra

This is a reference document on Two-Dimensional Linear Algebra by Fengyang Wang. It covers a subset of the material of MATH 136 or MATH 146.